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A bivariate process capability vector
... 1. The probability of nonconforming products in process with multivariate distribution (see for example Chen, 6 Pal, 7 Polansky 8 ). 2. The ratio of a tolerance region to a process region (PR) such as the ones proposed by Hubele et al., 9 Taam et al., 10 Shahriari et al., 11 Wang et al., 12 and Shahriari and Abdollahzadeh. 13 Also, Pana and Lee 14 introduced an index similar to the Shahriari and Abdollahzadeh 13 index. ...
... Hubele et al. 9 proposed a composite measure for process capability based on two quality characteristics. In the method of Taam et al. 10 (MC pm ), modified tolerance is depicted based on USL and LSL of the quality characteristic. ...
... In the method of Taam et al. 10 (MC pm ), modified tolerance is depicted based on USL and LSL of the quality characteristic. C pm was proposed by Shahriari et al., 11 based on the original work of Hubele et al. 9 This index is a three-component vector, called multivariate process capability vector. ...
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.
... Since the cut is led through the center, the tangent parallelepiped may not touch the ellipse. In the case of three quality characteristics, the projection can be made onto the planes , we observe and evaluate: a) the process centralization, using the so-called Hotelling's statistic (Hubele, Shahriari and Cheng, 1991), b) the multivariate capability index (Pearn and Lin, 2002): ...
... For 1 X and 2 X , we have from Fig. 11 Figure Working with a multivariate index, Fig. 11 can be used to assess three attributes: p MC , centralization and outliers (Hubele, Shahriari and Cheng, 1991). Fig. 12 represents different views of the plane 2 1 X X × : the minus sign means the observed criterion is not compliant, the plus signs means it is compliant (the criteria are: ...
Purpose: The main objective of the paper is an analysis of the behaviour of capability indices under different conditions. It is assumed that the metrological properties of a check standard are correct, however, the uncertainty of the check standard affects the evaluation of the measurement process capability. The paper analyses individual cases of the influence of the check standard bias and its influence on the measurement process capability.
Methodology/Approach: Statistical analysis of both the measurement process and the check standard is provided at the beginning. Development and analysis of possible cases, when the bias of a check standard affects the calculated capability index of a measurement process follows.
Findings: The paper confirmed the theoretical assumption that a bias of a check standard can affect the calculated capability index of a measurement process, thus shifting the judgment on the measurement process capability.
Research Limitation/Implication: The paper is based on the theoretical assumptions of the measurement process capability as well as on the analysis of the possible behaviour of a respective check standard.
Originality/Value of paper: The paper clarifies that several particular and specifically selected cases of bias of a check standard may affect the resulting capability index negatively/positively, which may lead to inaccurate decisions on measurement process capability. This is confirmed by simulations of a biased check standard, clearly visualizing the shifts in capability indices.
... Since the cut is led through the center, the tangent parallelepiped may not touch the ellipse. In the case of three quality characteristics, the projection can be made onto the planes , we observe and evaluate: a) the process centralization, using the so-called Hotelling's statistic (Hubele, Shahriari and Cheng, 1991), b) the multivariate capability index (Pearn and Lin, 2002): ...
... For 1 X and 2 X , we have from Fig. 11 Figure Working with a multivariate index, Fig. 11 can be used to assess three attributes: p MC , centralization and outliers (Hubele, Shahriari and Cheng, 1991). Fig. 12 represents different views of the plane 2 1 X X × : the minus sign means the observed criterion is not compliant, the plus signs means it is compliant (the criteria are: ...
p> Purpose: The paper focuses on how the problem of process capability assessment can be handled when taught, using convenient numerical and graphical means. The contents of the paper results from the authors’ own academic and practical experience, which suggested that many important steps are overlooked in the process of selecting and using capability indices.
Methodology/Approach: Selected problems in capability assessment are illustrated with suitable examples and graphs.
Findings: The authors’ experience is reflected in the paper, aiming to emphasize what matters and how, and what does not. Also, a new capability index is introduced.
Research Limitation/implication: The style in which the problems are analysed may serve as a guide for further studies in the field and capability index applications.
Originality/Value of paper: The paper also contains, aside from specific examples, some more advanced techniques, and is therefore accompanied by software readouts, since computer support is required in such cases.
... Here, as an example, the Taam's at al. index M C pm[3], which is one of the most used multivariate capability indices[11], has the value of 2.87 for all those cases. Values of the other indices: N M C pm proposed by Pan and Lee[7], Shahriari's multivariate process capability vector[12], and multivariate process capability vector proposed by Ciupke[10] Figure 1: Examples of processes with the same specification limits and target, and with modified specification limits according to[7]processes are shown in Table 1. Traditionally, indices have dimensionless value. ...
... The V R component, as in[12], measures the capability of the process and is defined as the percentage ratio of the volumes of a current process region R P and a modified tolerance region R T : ...
The purpose of this paper is to provide a multivariate process capability index, which could be used regardless on data distribution and also on data correlation. Such an index could be defined because of application of non-parametric methodology that utilizes a data depth concept. Based on this concept, a two-phase methodology was developed. In the first phase the modified tolerance region is estimated, while in the second one, a current process is assessed using the proposed three-component index. Estimation of a modified tolerance region on the basis on historical data allows applying the methodology not only for bilateral quality characteristics but also for unilateral ones, where often in practice, the modified tolerance region could be defined as a closed region. The performance of the proposed index was evaluated using bilateral and unilateral examples. The obtained results showed that the proposed index performs satisfactorily for all the considered cases. Copyright
... Since then, many indices have been proposed; among which the most recognized are by Hubele et al. (1991), Taam et al. (1993), Shahriari et al. (1995), and Chen (1994). Wang et al. (2000) performed a comparative study from these last methods and discussed their usefulness. ...
... The Multivariate Process Capability Vector was introduced by Shahriari et al. (1995) based on the pioneer work of Hubele et al. (1991). It consists of a threecomponent vector which is defined as: ½CpM; PV; LI; ...
In this chapter the most recognized multivariate process capability indices are presented. The first section approaches the computation of these indices in R, and the next ones are dedicated to the indices based on ratios of the volume tolerance region to a process region such as Taam et al. (J Appl Stat 20:339–351, 1993), Shahriari et al. (Proceedings of the 4th Industrial Engineering Research Conference 1:304–309, 1995), and Pan and Lee (Qual Reliab Eng Int 26(1):3–15, 2010). While the last part of the chapter focuses on the indices derived of principal component analysis.
... Ebadi and Shahriari 7 calculated three-component PCI based on predicted values using the classical method to estimate the parameters. It should be noticed that the classical three-component capability vector was proposed by Shahriari et al. 22 based on Hubele et al. 23 Wang et al. 24 compared this multivariate capability index to some other proposed multivariate PCIs. ...
In some statistical process control applications, the quality of a process is described by a linear relationship between the response variable(s) and the independent variable(s), which is called a linear profile. Process capability is a significant issue in statistical process control. The ability of a process to meet customer specifications or standards is measured by the process capability indices (PCIs). There are several attempts for studying the process capability in linear profiles. In this research, two robust PCIs for multiple linear profiles are proposed. In the suggested robust PCIs, the process capability is estimated using the M-estimator and the Fast-τ-estimator. Performances of the proposed robust PCIs in comparison with the classical PCIs in the absence and presence of contamination are evaluated. The results show that the robust PCIs proposed in this research perform as well as the classical PCIs in the absence of contamination and much better in the presence of contamination. The proposed PCIs, using Fast-τ-estimator, perform better in small shifts, and the proposed PCIs, using M-estimator, perform better in large shifts. Introduction of robust indices for multivariate multiple linear profiles is an area for further research.
... Application examples include the manufacturing of semiconductor products (Hoskins, Stuart, and Taylor (1988)), jet-turbine engine components (Hubele, Shahriari, and Cheng (1991)), wood products (Lyth and Rabiej (1995)), audio speaker drivers (Chen and Pearn (1997)) and many others. ...
Control charts are statistical process control (SPC) tools that are widely used in
the monitoring of processes, specifically taking into account stability and dispersion. Control charts signal when a significant change in the process being studied
is observed. This signal can then be investigated to identify issues and to find solutions. It is generally accepted that SPC are implemented in two phases, Phase
I and Phase II. In Phase I the primary interest is assessing process stability, often
trying to bring the process in control by locating and eliminating any assignable
causes, estimating any unknown parameters and setting up the control charts. After that the process move on to Phase II where the control limits obtained in Phase
I are used for online process monitoring based on new samples of data. This thesis
concentrate mainly on implementing a Bayesian approach to monitoring processes
using SPC. This is done by providing an overview of some non-informative priors
and then to specifically derive the reference and probability-matching priors for the
common coefficient of variation, standardized mean and tolerance limits for a normal
population. Using the Bayesian approach described in this thesis SPC is performed,
including derivations of control limits in Phase I and monitoring by the use of run-lengths and average run-lengths in Phase II for the common coefficient of variation,
standardized mean, variance and generalized variance, tolerance limits for normal
populations, two-parameter exponential distribution, piecewise exponential model
and capability indices. Results obtained using the Bayesian approach are compared
to frequentist results.
... However most of them have not been already widely adopted by the industry (de-Felipe & Benedito, 2017a, 2017b. (Hubele, Shahriari, & Cheng, 1991) proposed a capability vector to indicate multivariate process capability. The authors considered rectangular tolerance regions and three separate quantities were needed to represent the process capability. ...
Statistical process control (SPC) techniques have been implemented in a few studies in order to determine action limits on the earned value management (EVM) indices to assist project managers in monitoring project performance. Nevertheless, these indices are correlated, reliance on univariate control charts when more than one variable are involved and they are correlated may lead to unsatisfactory results such as increasing the rate of false alarms. This study presented an approach to overcome this limitation by applying multivariate Hotelling's T ² control chart to take into account possible correlations between EVM indicators and describe the entire capability of the project performance more accurately. Furthermore, in order to quantify how well a project can meet its requirements, some practical multivariate process capability indices (PCIs) are introduced. This approach provides management with more reliable information about the deviations between planned and actual performance both in terms of time and cost simultaneously.
... where USL i and LSL i are the upper and lower speci cation limits for the ith quality characteristic; UPL i and LPL i are the upper and lower specication limits of a modi ed process region for the ith quality characteristic, i = 1; ; v (see [22] for more details). In this case, we may need help from computer software since the mathematical framework is hard to deal with. ...
Comparison of quality for products (supplies and goods) is extremely important for manufacturers and consumers. Based on correct comparisons, manufacturers and consumers can find better suppliers to cooperate and better merchandise to purchase, respectively. Quality is often measured and compared by process capability indices, among which Cp is very efiective, simple to apply, and particularly useful for the first round of comparison. In practice, Cp is unknown and should be estimated from observations. Let dCpi denote the maximum likelihood estimator obtained from normal process, Xi, with index value Cpi; i = 1; 2. If dCp1 > (<)d Cp2 is observed, we will conclude that Cp1 > (<) Cp2 and decide that X1 is better (worse) than X2. Given a small and positive number, , there is no need to make comparison when (1-€) Cp2 < Cp1 < (1+€) Cp2 since Cp1 is close to Cp2. It is desirable to observe dCp1 <d Cp2 with high probability when (1 + €) Cp2 > Cp1 and with low probability when (1-€) Cp2 < Cp1. Given 0 < 1, 2 < 1, based on the table constructed from P(dCp1 >d Cp2), we demonstrate how to find the smallest sample size needed to ensure observing dCp1 d Cp2 with probability greater than 1 €1 when (1 + €) Cp2 < Cp1 and smaller than 2 when (1 €) Cp2 < Cp1.
... Taam et al. [28] introduced a multivariate capability index as a ratio of two volumes. Hubele et al. [11] proposed a process capability vector for bivariate normal processes and afterwards, Shahriari and Lawrence [26] extended it to multivariate case. Thereafter, Shahriari and Abdollahzadeh [25] revised it. ...
... Pearn, Kotz, and Johnson (1992) proposed two MPCIs, which are generalizations of the index introduces by Chan et al. (1991). Hubele, Shahriari, and Cheng (1991) defined a process capability vector for bivariate normal processes and Shahriari and Lawrence (1995) extended it to the multivariate cases. In the literature, numerous studies can be found that worked on multivariate capability indices such as Taam, Subbaiah, and Liddy (1993), Pan and Li (2014), Pan and Hung (2015), Tano andVannman (2013), HorngShiau, Yen, Pearn, andLee (2012), Ciupke (2015), Siman (2014) and Zhang, Alan, Shuguang, and Zhen (2014); however among those various different multivariate capability indices, here the focus is on the one proposed by Taam et al. (1993), mentioned in the subsequent section. ...
Multivariate process capability indices are applied to account the capability of the processes which the quality of the products depends on two or more related characteristics. We call a tolerance region asymmetric, when the target value of at least one characteristic is not the mid-point of the tolerance interval. This paper introduces a superstructure index to measure the capability of multivariate normal process in asymmetric tolerance regions, which could be applied for symmetric cases, too. In addition, the effects of two modification factors in the index which weigh the mean departure from target and process variability are investigated. Furthermore, some examples are presented to demonstrate the applicability and effectiveness of the proposed index.
... Nuehard [19] proposed a method for calculating capability indices for multivariate processes in which the variance is adjusted for correlation by multiplying it by a factor, and then the adjusted variance is used to calculate the indices. Hubele et al [20] discussed the disadvantage of using univariate capability index and the advantage of the bivariate process capability vector. They considered the bivariate normal distribution and analyzed the process for its capability. ...
In many industrial instances product quality depends on a multitude of dependent characteristics and as a
consequence, attention on capability indices shifts from univariate domain to multivariate domain. In this
research fuzzy inference system is used to determine the process capability index. Fuzzy sets can represent
imprecise quantities as well as linguistic terms. Fuzzy inference system (FIS) is a method, based on the
fuzzy theory, which maps the input values to the output values. The mapping mechanism is based on some
set of rules, a list of if-then statements. In this research Mamdani fuzzy inference system is used to derive
the overall output process capability when subjected to six crisp input and one output. This paper deals with
a novel approach to evaluating process capability based on readily available information using fuzzy
inference system.
... Nuehard [19] proposed a method for calculating capability indices for multivariate processes in which the variance is adjusted for correlation by multiplying it by a factor, and then the adjusted variance is used to calculate the indices. Hubele et al [20] discussed the disadvantage of using univariate capability index and the advantage of the bivariate process capability vector. They considered the bivariate normal distribution and analyzed the process for its capability. ...
In many industrial instances product quality depends on a multitude of dependent characteristics and as a
consequence, attention on capability indices shifts from univariate domain to multivariate domain. In this
research fuzzy inference system is used to determine the process capability index. Fuzzy sets can represent
imprecise quantities as well as linguistic terms. Fuzzy inference system (FIS) is a method, based on the
fuzzy theory, which maps the input values to the output values. The mapping mechanism is based on some
set of rules, a list of if-then statements. In this research Mamdani fuzzy inference system is used to derive
the overall output process capability when subjected to six crisp input and one output. This paper deals with
a novel approach to evaluating process capability based on readily available information using fuzzy
inference system
... Nuehard [10] proposed a method for calculating capability indices for multivariate processes, in which the variance is adjusted for correlation by multiplying it by a factor, and then the adjusted variance is used to calculate the indices. Hubele, Shahriari and Cheng [11] discussed the disadvantage of using univariate capability index and the advantage of the bivariate process capability vector. They considered the bivariate normal distribution and analyzed the process for its capability. ...
It is customary to use the 6-sigma spread in the distribution of the product quality characteristics as a measure of process capability. This value represents the collective contribution of many factors (such as machine, tool, setup, labor and many more) involved in the production process. Measuring the process capability, however accurately done, only represents a single combination of these factors. Any changes in one of these factors would affect the resultant process capability. Moreover, the existing evaluation of process capability is a tedious and time consuming process requiring possible interruption and stoppage of production. This paper presents a novel approach for the evaluation of process capability based on readily available information and knowledge of experts. It is a flexible system in that it can consider a number of different factors that may have an effect on process capability. It utilizes the concept of fuzzy logic to transfer the experts' evaluation of the factors into a value representing the process capability. Initial results indicate that the proposed technique is very effective.
... Nuehard [10] proposed a method for calculating capability indices for multivariate processes, in which the variance is adjusted for correlation by multiplying it by a factor, and then the adjusted variance is used to calculate the indices. Hubele, Shahriari and Cheng [11] discussed the disadvantage of using univariate capability index and the advantage of the bivariate process capability vector. They considered the bivariate normal distribution and analyzed the process for its capability. ...
It is customary to use the 6-sigma spread in the distribution of the product quality characteristics as a measure of process capability. This value represents the collective contribution of many factors (such as machine, tool, setup, labor and many more) involved in the production process. Measuring the process capability, however accurately done, only represents a single combination of these factors. Any changes in one of these factors would affect the resultant process capability. Moreover, the existing evaluation of process capability is a tedious and time consuming process requiring possible interruption and stoppage of production. This paper presents a novel approach for the evaluation of process capability based on readily available information and knowledge of experts. It is a flexible system in that it can consider a number of different factors that may have an effect on process capability. It utilizes the concept of fuzzy logic to transfer the experts' evaluation of the factors into a value representing the process capability. Initial results indicate that the proposed technique is very effective.
... A multivariate capability vector, denoted here as Method 1, was proposed by Shahriari, Hubele, and Lawrence (1995), based on the original work of Hubele, Shahriari, and Cheng (1991). The proposed vector consists of three components. ...
Process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. When these variables are related characteristics, the analysis should be based on a multivariate statistical technique. In this expository paper, three recently proposed multivariate methodologies for assessing capability are contrasted and compared. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed.
... A multivariate capability vector, denoted here as Method 1, was proposed by Shahriari, Hubele, and Lawrence (1995), based on the original work of Hubele, Shahriari, and Cheng (1991). The proposed vector consists of three components. ...
Process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. When these variables are related characteristics, the analysis should be based on a multivariate statistical technique. In this expository paper, three recently proposed multivariate methodologies for assessing capability are contrasted and compared. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed.
... When there are two or more process characteristics, a multivariate process capability analysis may be applied to evaluate process capability (Shishebori and Hamadani, 2010). Several different approaches to compute multivariate capability indices to evaluate process capability based on multiple process characteristics may be found in the academic literature (see, e.g., Castagliola and Castellanos, 2005; Shahriari et al., 1995; Chen, 1994; Taam et al., 1993). A process capability measure for the multivariate case is typically obtained by the ratio of the tolerance region's volume and the volume of the specified process region. ...
... A multivariate capability vector was proposed by Shahriari et al. (1995), based on the original work of Hubele et al. (1991). The multivariate capability vector consists of three components. ...
... The ratio of volume of the ellipse and rectangular tolerance area is used as a multivariable process capability index. A correlated data observation ellipse conversion to the smallest rectangular process area is studied in reference [2]. This converted rectangular process area is compared to a rectangular specification area and a three-component vector is reported as a process capability index. ...
Geometrical dimensioning and tolerancing (GD&T) enable possibility to include a statistical quality level parameter into a dimensional drawing. A typical GD&T drawing do not take account a correlation between parameters and do not specify accurately enough a statistical analysis for a two-dimensional design. This paper proposes a two-dimensional process capability index CpkR for correlated data with a circular acceptance area. A case study data is studied to illustrate the difference between a one-dimensional process capability index Cpk and two-dimensional CpkR. The one-dimensional Cpk overestimates the process capability compared to two-dimensional CpkR, since CpkR is based on observed quality level. If a statistical process capability index is included into a GT&D design, then it needs to be specified detailed manner and preferably a specification for a correlation factor should be included.
... Nevertheless we will term them all " multivariate PCIs " (MPCIs). References with a title including the word " multivariate " or " bivariate " are: Beck and Ester (1998); Bernardo and Irony (1996); Boyles (1996b); Chan et al. (1991); Davis et al. (1992); Hellmich and Wolf (1996); Hubele et al. (1991); Karl et al. (1994); Li and Lin (1996); Mukherjee and Singh (1994); Niverthi and Dey (2000); Shariari et al. (1995); Taam et al. (1993); Tang and Barnett (1998); Veevers (1995 Veevers ( , 1998 Veevers ( , 1999); Wang et al. (2000); Wierda (1992 Wierda ( , 1993 Wierda ( , 1994a Wierda ( , 1994b); and Yeh and Chen (1999). Multivariate situations are also discussed in the following references, that do not indicate, explicitly, in their titles that this is so: Chan et al. (1988b); Wang and Chen (1998/9); and Hubele (1999, 2001). ...
Quantifying the "capability" of a manufacturing process is an important
initial step in any quality improvement program. Capability is usually defined
in dictionaries as "the ability to carry out a task, to achieve an objective".
Process capability indices(PCIs) is defined as a combination of materials,
methods, equipments and people engaged in producing a measurable output. PCIs
which establish the relationships between the actual process performance and
the manufacturing specifications, have been a focus of research in quality
assurance and process capability analysis. Capability indices that qualify
process potential and process performance are practical tools for successful
quality improvement activities and quality program implementation. As a matter
of fact, all processes have inherent statistical variability, which can be
identified, evaluated and reduced by statistical methods. Generalized Process
Capability Index, defined as the ratio of proportion of specification
conformance (or, process yield) to proportion of desired (or, natural)
conformance. We review the process capability indices in case of normal,
non-normal, discrete and multivariate process distributions and discuss the
inferential aspects of some of these process capability indices. Relations
among the process capability indices have also been illustrated with examples.
Finally we also consider the process capability indices using conditional
ordering and transforming multivariate data to univariate one using the concept
of structural function.
... However, since most products contain more than one quality characteristics, multivariate PCI are being more investigated in comparison to the univariate indices (2002), respectively. A brief review of multivariate PCI is given as follows: The multivariate PCI can be classified in four main categories (Shahriari and Abdollahzadeh, 2009). 1 the indices which measure the ratio of a tolerance region to a process region, such as the research proposed by Wang et al. (2000), Shahriari et al. (1995, and Taam et al. (1993) 2 the indices based on proportion of conforming items, such as the research proposed by Polansky (2001), Pal (1999), and Chen (1994) 3 the indices based on principal components analysis (PCA), for instance the research proposed by Wang and Chen (1999) 4 the indices based on the extension of univariate PCI such as those introduced by Holmes and Mergen (1999) and Chen et al. (2003). Most of the proposed research works mentioned in the above classification, consider the multivariate normal distribution for the quality characteristics, while in most real processes the quality characteristics might follow multivariate non-normal distributions, or even mixed discrete-continuous distributions. ...
While most of the methods developed for computing process capability indices (PCI) concentrate on cases with normally or continuous non-normally distributed quality characteristics, computing this measure for processes with mixed distributed data has not been investigated so far. In this paper, a new method is proposed for computing (PCI) for mixed binary-normal quality characteristics. In the proposed method, first a mixed binary-normal distribution is considered to be fitted on the available data. Having estimated the unknown parameters of the fitted distribution using maximum likelihood estimation and genetic algorithm, the proportion of the conforming items of thecorresponding distribution is estimated by Monte Carlo simulation runs. Finally, the PCI is computed based on the relationship of PCI and proportion of conforming items. The performance of the proposed method is evaluated using simulation studies as well as a case study in a plastic injection moulding process.
... Wang and Chen provide three examples and a real data application that clarify the assessment of the indices MC p , MC pk , MC pm and MC pmk . As the authors point out, the obtained values of the indices in these examples are in accordance to the indices by Chan et al. (1991), Taam et al. (1993, Shahriari et al. (1995) and Chen (1994). An additional appealing feature of this technique is its simple implementation in comparison to the other approaches suggested for measuring the capability of multivariate processes. ...
... Other MCI which depend upon the correlation structure of the variables have been proposed in the literature for independent processes. Wang et al. (2000) compared three indices: Taam et al (1993), Chen (1994) and Shahriari et al. (1995), considering some particular examples. Pearn et al. (2007) presented a study of distributional and inferential properties of the MC P and MC pm multivariate indices proposed by Taam et al. (1993). ...
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.
... Chan et al. 7 extended their univariate index C pm in Chan et al. 2 to a multivariate version by measuring how far away from the target vector the process mean is in the Mahalanobis distance. Pearn et al. 3 proposed a multivariate version of C p and C pm with an approach they claimed to be more natural than that of Chan et al. 7 Hubele et al. 8 proposed a process capability vector for bivariate normal processes, and later, Shahriari et al. 9 extended it to the multivariate case. Taam et al. 10 proposed a multivariate C p as the ratio of two areas, the area of a modified specification (also called the modified engineering tolerance region by some researchers), defined as the largest ellipsoid centering at the target value and completely within the original specification over the area of the elliptical process region that covers 99.73% of the multivariate normal process. ...
Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate Cpk and Cp (denoted by BCpk and BCp, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MCpk and MCp for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MCpk. An algorithm suitable for such processes is developed to compute the natural estimate of MCpk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MCpk, a measure often used by producers for quality assurance to consumers. As for BCp, we first modify the original definition with a simple preprocessing step to make BCp scale-invariant. A very efficient algorithm is developed for computing a natural estimator of BCp. This new approach of BCp can be easily extended to MCp for multivariate processes. For BCp, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright © 2012 John Wiley & Sons, Ltd.
Providing a single-valued assessment of the performance of a process is often one of the greatest challenges for a quality professional. Process Capability Indices (PCIs) precisely do this job. For processes having a single measurable quality characteristic, there is an ample number of PCIs, defined in literature. The situation worsens for multivariate processes, i.e., where there is more than one correlated quality characteristic. Since in most situations quality professionals face multiple quality characteristics to be controlled through a process, Multivariate Process Capability Indices (MPCIs) become the order of the day. However, there is no book which addresses and explains different MPCIs and their properties. The literature of Multivariate Process Capability Indices (MPCIs) is not well organized, in the sense that a thorough and systematic discussion on the various MPCIs is hardly available in the literature.
Handbook of Multivariate Process Capability Indices provides an extensive study of the MPCIs defined for various types of specification regions. This book is intended to help quality professionals to understand which MPCI should be used and in what situation. For researchers in this field, the book provides a thorough discussion about each of the MPCIs developed to date, along with their statistical and analytical properties. Also, real life examples are provided for almost all the MPCIs discussed in the book. This helps both the researchers and the quality professionals alike to have a better understanding of the MPCIs, which otherwise become difficult to understand, since there is more than one quality characteristic to be controlled at a time.
Multivariate process capability indices (MPCIs) have been proposed to measure multivariate process capability in real‐world application over the past three decades. For the practitioner's point of view, the intention of this paper is to examine the performances and distributional properties of probability‐based MPCIs. Considering issues of construction of capability indices in multivariate setup and computation with performance, we found that probability‐based MPCIs are a proper generalization of univariate basic process capability indices (PCIs). In the beginning of this decade, computation of probability‐based indices was a difficult and time‐consuming task, but in the computer age statistics, computation of probability‐based MPCIs is simple and quick. Recent work on the performance of MPCI NMCpm and distributional properties of its estimator reasonably recommended this index, for use in practical situations. To study distributional properties of natural estimators of probability‐based MPCIs and recommended index estimator, we conducted simulation study. Though natural estimators of probability‐based indices are negatively biased, they are better with respect to mean, relative bias, mean square error. Probability‐based MPCI MCpm is better as compared with NMCpm with respect to performance and as its estimator quality. Hence, in real‐world practice, we recommend probability‐based MPCIs as a multivariate analogue of basic PCIs.
Process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. When these variables are related characteristics, the analysis should be based on a multivariate statistical technique. In this expository paper, three recently proposed multivariate methodologies for assessing capability are contrasted and compared. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed.
En el análisis de capacidad multivariado existen muchos aspectos aún no resueltos en torno a algunos índices, como la no normalidad de los datos y si las variables de calidad están correlacionadas o no están correlacionadas. En este trabajo se pretendió proponer la elaboración de un índice de capacidad multivariado (ICPM) que funcione para ambos casos anteriores por medio de ejemplos de datos simulados. Cabe mencionar que se encontró un amplio desempeño del índice propuesto frente a otros índices de capacidad similares.
Process capability is determined by comparing the actual performance of a process with required specifications. Several indices have been proposed to report the capability of a process in the univariate case. When performance is tracked in several dimensions, an extension of these indices is required. We present several multivariate capability indices, and extend the idea of multivariate tolerance regions for assessing the capability of a process.
Keywords:
process capability;
specification limits;
multivariate statistical process control;
Mahalanobis T2 charts;
multivariate tolerance regions
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.
Process capability is determined by comparing the actual performance of a process with required specifications. Several indices have been proposed to report the capability of a process in the univariate case. When performance is tracked in several dimensions, an extension of these indices is required. We present several multivariate capability indices, and extend the idea of multivariate tolerance regions for assessing the capability of a process.
Higher sigma quality level is generally perceived by customers as improved performance by assigning a correspondingly higher satisfaction score. The process capability indices and the sigma level Z_{st} ave been widely used in six sigma industries to assess process performance. Most evaluations on process capability indices focus on statistical estimation under normal process which may result in unreliable assessments of process performance. In this paper, we consider statistical estimation for bivariate VPCI(Vector-valued Process Capability Index) C_{pkl}=(C_{pklx},\;C_{pklx}) under Marshall and Olkin (1967)'s bivariate exponential process. First, we derive some limiting distribution for statistical inference of bivariate VPCI C_{pkl}. And we propose two asymptotic normal confidence regions for bivariate VPCI C_{pkl}. The proposed method may be very useful under bivariate exponential process. A numerical result based on our proposed method shows to be more reliable.
Process capability assessment is important to statistical quality control. The existing univariate or multivariate process capability indices cannot be directly applied to assess the process for the quality characterized by a simple linear profile. There has been little attempt to study the process capability assessment in simple linear profile. This paper proposes a method to assess process capability for simple linear profile. Since two-dimensional predictions of the slope and the intercept of simple linear profile can represent the prediction profile, the process capability analysis can be treated as a two-dimensional correlated variables problem. Then, a multivariate process capability index based on a vector of three components is applied to assess the process capability in simple linear profile. The proposed method was proved by a simulation study.
A multivariate exponentially weighted moving average (MEWMA) control chart is proposed for detecting process shifts during the phase II monitoring of simple linear profiles (SLPs) in the presence of within-profile autocorrelation. The proposed control chart is called MEWMA-SLP. Furthermore, two process capability indices are proposed for evaluating the capability of in-control SLP processes, and their utilization is demonstrated through examples. Intensive simulations reveal that the MEWMA-SLP chart is more sensitive than existing control charts in detecting profile shifts.
Process capability analysis plays an important role in statistical quality control. We present the process yield index TSpkA to evaluate the process yield for multivariate linear profiles in manufacturing processes. This index provides an exact measure of the process yield. In addition, an approximate confidence interval for TSpkA is constructed. A simulation study is conducted to assess the performance of the proposed method for multivariate linear profiles data under mutually independent normality and multivariate normality. The simulation results confirm that the estimated TSpkA is close to the target value with smaller standard deviation as the sample size increases.
Purpose
– In the past few years, several capability indices have been developed for evaluating the performance of multivariate manufacturing processes under the normality assumption. However, this assumption may not be true in most practical situations. Thus, the purpose of this paper is to develop new capability indices for evaluating the performance of multivariate processes subject to non-normal distributions.
Design/methodology/approach
– In this paper, the authors propose three non-normal multivariate process capability indices (MPCIs) RNMCp, RNMCpm and RNMCpu by relieving the normality assumption. Using the two normal MPCIs proposed by Pan and Lee, a weighted standard deviation method (WSD) is used to modify the NMCp and NMCpm indices for the-nominal-the-best case. Then the WSD method is applied to modify the multivariate ND index established by Niverthi and Dey for the-smaller-the-better case.
Findings
– A simulation study compares the performance of the various multivariate indices. Simulation results show that the actual non-conforming rates can be correctly reflected by the proposed capability indices. The numerical example further demonstrates that the actual quality performance of a non-normal multivariate process can properly reflected by the proposed capability indices.
Practical implications
– Process capability index is an important SPC tool for measuring the process performance. If the non-normal process data are mistreated as a normal one, it will result in an improper decision and thereby lead to an unnecessary quality loss. The new indices can provide practicing managers and engineers with a better decision-making tool for correctly measuring the performance for any multivariate process or environmental system.
Originality/value
– Once the existing multivariate quality/environmental problems and their Key Performance Indicators are identified, one may apply the new capability indices to evaluate the performance of various multivariate processes subject to non-normal distributions.
Process capability indices are useful tools, which provide common quantitative measures on manufacturing capability and production quality. The existing Multivariate Process Capability indices, Cpm, MCpm,[CpM, PV, LI], MCf & (sTpk} are proposed by different authors for different conditions and situations[1,2,3,4,5]. Many of the above multivariate process capability indices are based on the ratios of two areas or volumes for bivariate or in multivariate domain respectively. All above proposed indices assume the process to be normal and there by the indices are expressed in the form of ratios of areas of two ellipses or volume of two ellipsoids. When the process does not follow normal distribution the estimation of the index is difficult as the shape of the data distribution will be not known. The proposed MCsvdd index uses computational geometry and find out the convex hull of the process data to find the volume. Support Vector Data description helps to find the outliers.
We consider gauge repeatability and reproducibility (R&R) analyses for two-dimensional date when the engineering tolerance associated with measurements is a circle. We develop summaries for repeatability, reproducibility, and R&R by employing the diameters of circles that provide 99% capture rates. We derive an inequality between the results of a one-dimensional gauge R&R method and this two-dimensional method. We also show that the additivity-of-variance in the one-dimensional gauge R&R case becomes a sub-additivity-of-variance in the two-dimensional case under certain conditions. We use measurements of unbalance of rotating devices, where the tolerance must be a circle, to motivate the ideas and to illustrate the technique. Our results apply to corresponding coordinate-measuring-machine data when the engineering tolerance is a circle. The method can be extended to other problems, including square and spherical tolerances. We recommend that this method be used in conjunction with graphical, other univariate, and multivariate analyses.
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.
In some measurement processes the measured data by a gauge are two-dimensional and the design tolerance is a circular. When there is some correlation between two measured quality characteristics, the common evaluation indices for univariate measurement system capability cannot be applied directly to analyze and evaluate the two-dimensional measurement system capability with circular tolerance. A method of constructing evaluation indices based on ratios of areas was proposed herein. The measurement system capability evaluation indices were built through the area ratio of ellipse of gauge errors and circular of design tolerance, the ellipse area ratio of gauge error and measured values. Then the common evaluation indices for univariate measurement system capability could be extended to the two-dimensional measurement system. MANOVA was used to estimate the indices to assess the two-dimensional measurement system capability. At last the proposed method was proved by an example. ©, 2015, China Mechanical Engineering Magazine Office. All right reserved.
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.
Process capability indices provide a measure of the output of an in-control process that conforms to a set of specification limits. These measures, which assume that process output is approximately normally distributed, are intended for measuring process capability for manufacturing systems. When the performance of a system results in a product that fails to fall within a given specification range, however, the product is typically scrapped or reworked, and the actual distribution that the customer perceives after inspection is truncated. In this paper, the concept of a truncated measure for three types of quality characteristics is introduced as the key to linking customer perception to process capability. Subsequently, a set of customer-perceived process capability indices is presented as an extension of traditional manufacturer-based counterparts. Finally, data transformation-based process capability indices are also discussed. A comparative study and numerical example reveal considerable differences among the traditional and proposed process capability indices. It is believed that the proposed process capability index for various quality characteristics may more aptly lead to process improvement by facilitating a better understanding of the integrated effects found in engineering design problems. Copyright © 2015 John Wiley & Sons, Ltd.
Most multivariate process capability indices proposed by other researchers provide quality measure in losses due to variability and being off-target. Some suggested loss function models that estimate losses due to rejection and quality costs but failed to consider the correlation between the multivariate quality characteristics. In this paper, we propose a novel new approach for estimating the total expected quality cost that provides quality cost measure not only in losses due to variability and being off-target but also losses due to rejection. The proposed model also estimates the quality costs for correlated quality characteristics with nominal specifications and provides the estimated costs due to individual characteristics; it helps to identify those process quality characteristics which have high variation when compared with their specification spread. The model will also improve the estimate of the rejection cost suggested by earlier researchers as it identifies the reject region as the region outside of the modified tolerance region. We present four different correlated bivariate normal process scenarios where the proposed method can be applicable and demonstrate that our approach provides a robust tool in estimating the risk of the product being rejected and performs as well as existing methods in estimating the total quality cost.
A higher quality level is generally perceived by customers as improved performance by assigning a correspondingly higher satisfaction score. The third generation index C_{pmk} is more powerful than two useful indices C_p and C_{pk} that have been widely used in six sigma industries to assess process performance. In actual manufacturing industries, process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. Since these characteristics are related, it is a risky undertaking to represent the variation of even a univariate characteristic by a single index. Therefore, the desirability of using vector-valued process capability index(PCI) arises quite naturally. In this paper, we consider more powerful vector-valued process capability index C_{pmk} = (C_{pmkx}, C_{pmky})^t that consider the univariate process capability index C_{pmk}. First, we examine the process capability index C_{pmk} and plug-in estimator \hat{C}_{pmk}. In addition, we derive its asymptotic distribution and variance-covariance matrix V_{pmk} for the vector valued process capability index C_{pmk}. Under the assumption of bivariate normal distribution, we study asymptotic confidence regions of our vector-valued process capability index C_{pmk} = (C_{pmkx}, C_{pmky})^t.
In some quality control applications, quality of a product or a process can be characterized by a profile defined as a functional relationship between a response variable and one or more explanatory variables. Many researchers have contributed to the development of linear and nonlinear profiles to monitor a process or product. However, less work has been devoted to the development of process capability indices in profile monitoring to evaluate process performance with respect to specification limits. This paper presents a process capability analysis when the quality characteristic of interest is represented by a linear profile. Simulation analyses along with a real case study in leather industry are used to evaluate the performance of the proposed method. Results indicate satisfactory performance.
The use of principal component analysis in measuring the capability of a multivariate process is an issue initially considered by Wang and Chen (1998). In this article, we extend their initial idea by proposing new indices that can be used in situations where the specification limits of the multivariate process are unilateral. Moreover, some new indices for multivariate processes are suggested. These indices have been developed so as to take into account the proportion of variance explained by each principal component, thus making the measurement of process capability more effective. Copyright © 2011 John Wiley & Sons, Ltd.
In this paper we study two vector-valued process capability indices =(, ) and C/aub pm/=( , ) considering process capability indices and . First, two asymptotic distributions of plug-in estimators =(, ) and =) , ) are derived.. With the asymptotic distributions, we propose asymptotic confidence regions for our indices. Next, obtaining the asymptotic distributions of two bootstrap estimators =(, )and =( , ) with our bootstrap algorithm, we will provide the consistency of our bootstrap for statistical inference. Also, with the consistency of our bootstrap, we propose bootstrap asymptotic confidence regions for our indices. (no abstract, see full-text)see full-text)e full-text)
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