SciELO - Scientific Electronic Library Online

 
vol.18 issue3Knowledge and innovation in local production systems of ceramic tiles and the new challenges of the international competitionSystematic layout planning aided by multicriteria decision analysis author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

Share


Production

Print version ISSN 0103-6513On-line version ISSN 1980-5411

Prod. vol.18 no.3 São Paulo Sept./Dec. 2008

http://dx.doi.org/10.1590/S0103-65132008000300014 

Comparing Mingoti and Glória's and Niverthi and Dey's multivariate capability indexes

 

Comparando os coeficientes de capacidade multivariados de Mingoti e Glória e Niverthi e Dey

 

 

Sueli Aparecida MingotiI; Fernando Augusto Alves GlóriaII

IUFMG
IINestle S/A

 

 


ABSTRACT

In this paper a comparison between Mingoti and Glória's (2003) and Niverthi and Dey's (2000) multivariate capability indexes is presented. Monte Carlo simulation is used for the comparison and some confidence intervals were generated for the true capability index by using bootstrap methodology.

Key words: Quality control, multivariate capability indexes, Monte Carlo, Bootstrap.


RESUMO

Neste artigo é apresentada uma comparação entre os índices de capacidade multivariados de Mingoti e Glória (2003) e Niverthi e Dey (2000). O método de simulação de Monte Carlo é utilizado na comparação, e intervalos de confiança para o verdadeiro valor do índice de capacidade do processo são construídos através da metodologia Bootstrap.

Palavras-chave: Controle de qualidade, índices de capacidade multivariados, Monte Carlo, Bootstrap.


 

 

1. INTRODUCTION

Process capability indexes (PCI) are used to evaluate the process performance according to the required specifications limits. Some well known indexes for the univariate case are Cp, Cpk, Cpm, (MONTGOMERY, 2001; ZHANG, 1998). Very often multiple quality characteristics are used to evaluate the performance of the process and in general they are correlated (MASON; YOUNG, 2002); in these situations, a common procedure is to evaluate the process capability considering each variable separatedly discarding the information of the possible correlation among them. An alternative is to use multivariate capability indexes. The univariate specification interval is then replaced by a specification region and capability indexes are generated according to the joint probability distribution of the variables. In general the multivariate normal distribution is used. Although the multivariate case is very common most of the existing papers in the literature deal with univariate process capability indexes. See Koltz and Johnson (2002) for a good review in the capability subject. Some multivariate PCI's were proposed by Chen (1994), Shahriari et al. (1995) and Taam et al. (1993;1998) using classical statistical estimation procedures. These indexes were compared by Wang et al. (2000) considering some particular examples. Niverthi and Dey (2000) extended the univariate Cp and Cpk indexes for the case were p quality characteristics are measured in each sample unit; Veever's (1998) introduced a viability index for multiresponse process and Wang (2005) proposed a capability index based upon principal components analysis for short run production. Other interesting references are: Polansky (2001), Foster et al. (2005), Wang (2006), Pearn and Wu (2006) and Pearn et al. (2007). Some PCI's indexes derived under the Bayesian framework are found in Cheng and Spiring (1989), Bernardo and Irony (1996) and Niverthi and Dey (2000) who used Gibbs sampler.

The index proposed by Chen (1994) is quite interesting and it depends on the value of the cumulative distribution function of the maximum coordinate of the random vector X with a p-variate normal distribution. However, some analytical or numerical resolution of equations are needed to obtain its value (WANG et al., 2000). By using some ideas suggested in Hayter and Tsui's paper (1994) for correction of control limits in multivariate control charts, Mingoti and Glória (2003) introduced a method which allowed to obtain the numerical value of Chen's capability index using a simulation procedure. This paper presents a comparison of Mingoti and Glória's (2003) with Niverthi and Dey's (2000) indexes by using Monte Carlo simulation. Some confidence intervals for the true capability indexes were generated by using bootstrap methodology.

 

2. UNIVARIATE CAPABILITY INDEXES

Let X be the quality characteristic of interest with normal distribution with parameters µ and σ. Let LSL and USL be the lower and upper specifications limits respectively. The well known capability indexes Cp andCpk are defined as

Basically, they represent the relationship between the process and the clients (or project) specification limits. Some references values, such as 1.33 or 2, are used to classify the process as being capable or not. When p random variables, p > 1, are monitored at the same time there is a need to build up multivariate indexes. A simple extention of the indexes defined in (1) to the multivariare case is to take the geometric mean of theCpi and Cpki values obtained for each quality characteristic Xi, i=1,2,…,p. However, this procedure does not take into consideration the relationship that might exist among the variables. Niverthi and Dey (2000) extended the univariate capability indexes, Cp and Cpk, for the multivariate case taking into account the correlation among the variables. Their indexes are basicaly linear combinations of the upper and lower specifications limits of the quality characteristics being the coefficients of the linear combinations related to the covariance matrix of the process. Another alternative was proposed by Chen in 1994 and modified by Mingoti and Glória (2003) who used Hayter and Tsui's (1994) multivariate control limits to build new capability indexes. In the next section Hayter and Tsui's methodology is introduced followed by Chen's, Mingoti and Glória's, Niverthi and Dey's indexes in sections 4, 5 and 6, respectively.

 

3. HAYTER AND TSUI CONTROL LIMITS CORRECTION

Let X = (X1 X2 ...Xp)' be the random vector with the quality characteristics of interest such that X has a p-variate normal distribution with vector mean µ0 = (µ01 µ02 ...µ0p) covariance and correlation matrices given by Σpxp and Ppxp, respectively. According to Hayter and Tsui (1994) for each variable Xi the control limits of (1 – α) 100% , 0 < α < 1, are obtained by choosing a constant CRα which satisfies (2):

i.e, the probability that the interval [ Xi ± σiCRα] contains the true value of µ0i for each i, i=1,2,…,p, is equal to (1 α). The choice of the critical value CRα depends upon the correlation matrix Ppxp and therefore, the correlation structure of X affects all the intervals simultaneously. The process will be considered out of control if

The equation (3) is the maximum of the coordinates of the vector Z which is the vector X standardized. The CRα value is obtained by using a procedure that involves a simulation of samples from a p-variate normal distribution with zero mean vector and covariance matrix Ppxp. In practice the matrix Ppxp is estimated by the sample correlation matrix Rpxp of X (JOHNSON; WICHERN, 2002). The steps of the simulation algorithm used to obtain the constant CRα is given as below.

Step 1. Generate a large number N of vectors of observations from a p-variate normal distribution with mean vector zero and correlation matrix Ppxp. The generated vectors are denoted by Z1, Z2,...ZN.

Step 2. Calculate the statistic M for each one of the generated vectors Zi = (Zi1, Zi2,...,Zip)' from step 1, i.e, for the every i=1,2,…,N, calculate the value of

Step 3. Find the value corresponding to the percentil of order (1 – α) of the sample (M1, M2, ...,MN) and use the obtained value as the critical value CRα, 0 < α < 1.

This algorithm was also used by Kaldonga and Kulkarni (2004) in control charts for autocorrelated multivariate normal processes. Hayter and Tsui (1994) suggested that a total of N=100000 simulations should be performed in order to obtain the value of CRαwith high precision. However, Mingoti and Glória (2003) showed that only N=10000 is necessary (see also MINGOTI; GLÓRIA, 2005). Hayter and Tsui (1994) also showed that the confidence intervals derived by using the CRα were better than the intervals derived by Bonferroni's method.

 

4. CHEN'S MULTIVARIATE CAPABILITY INDEX

Let V be the specification region of the process defined as

where µ0i is the specification mean value for the variable Xi and ri are the specification constants of the process, i = 1, 2,…,p. Chen's multivariate capability index (1994) is defined as

where r is such that

The process is considered capable if the value of MCp is larger than 1 and incapable otherwise. Other reference values could be used in place of 1. The value of r is obtained by using the accumulated distribution function FH of the random variable H which is defined as

Therefore, for a certain probability α, 0 < α < 1, the value of r is such that r = FH-1 (1 – α). Then, if MCp it larger than 1 the process will be considered capable with a certain confidence coefficient (1 – α) 100%. Mingoti and Glória (2003) introduced a modification in Chen's capability index. Instead of using some numerical procedure to find the constant r considering the theoretical distribution of the variable H and equation (5) they proposed to obtain a solution by using the simulation procedure described by the algorithm presented in section 3. In the next section Mingoti and Glória's modified multivariate MCp index will be presented.

 

5. MINGOTI AND GLÓRIA'S MULTIVARIATE CAPABILITY INDICES

In this section Mingoti and Glória's index will be presented for situations where the process and the nominal mean vectors are equal (section 5.1), the process is not centered in the nominal mean vector (section 5.3) and the specification limits are not centered in the nominal mean vector (section 5.2).

5.1 Processes Centered in the Nominal Vector Mean

Considering the specification region V defined as in (4) and by using the algorithm described in section 3, for a fixed value of α, 0 < α < 1, one can find the constant CRα such that

Therefore the process will be considered capable if for all i=1,2,…,p,

or equivalently

Thus, the multivariate capability index of the process can be defined as

or equivalently,

The process is considered capable if is smaller or equal to 1, by definition (9), or equivalently if is higher or equal to 1, by definition (10). The interesting part in this procedure is that there is no need to find the probability distribution of the random variable Y=max(Z) analytically since the constant CRα is obtained by using a simple simulation routine.

In this paper we will considered the definition (10) for .

The procedure described in this section can be implemented in situations where the specification area V is more complex (WANG et al., 2000) and also can be modified for situations where the process does not have the mean vector centered in the nominal value or when the specification limits are not centered in the nominal mean vector as it will be presented in sections 5.3 and 5.2, respectively.

5.2 More general case: specifications limits not centered in the nominal mean

Let LSLi and USLi be the lower and upper specification limits for the quality characteristic Xi, i = 1, 2,...,p. The multivariate capability index is then defined as

where

and σi is the standard deviation of Xi. The process is considered capable for higher or equal to 1. If for each variable Xi, i = 1, 2,...,p , the specification limits are centered in the nominal mean value then the equation (11) and (10) are equal since (USLi – LSLi) = 2ri.

5.3 Processes not centered in the nominal mean

In many situations the process is in statistical control but is not centered in the specification mean vector. The defined in sections 5.1 and 5.2 are not sensible to changes in the process vector mean and need to be modified. A similar approach as in the derivation of the Cpk index in the univariate case can be adopted to define a multivariate coefficient .

Let LSLi and USLi be defined as in section 5.2 and let µ0i and σi be the process mean and standard deviation of the variable Xi. Then the multivariate coefficient is defined as

Considering that LSLi = µisri1 and USLi = µis + ri1 where µis is the specification mean of Xi, and r1i and r2i are constants, the equation (12) reduces to

and therefore it takes into account possible deviations from the process means to the nominal means values. When for each variable Xi = 1, 2,..., p, the process is centered in the specification mean value, the is equal to the value obtained by equation (10) if the specification limits are centered in the nominal means or it is equal to (11) if they are not.

It is important to point out that the indexes and are the minimum (or maximum) of a vector that has p coordinates each one representing the capability index related to the quality characteristic Xi = 1, 2,..., p. Therefore, Mingoti and Glória's (2003) indexes quantify the global capability as well as the capability of the process for each quality characteristic individually. If the researcher wants to know which variables are responsible for the global non-capability of the process it will be enough to observe the individual indexes looking for those that are smaller than 1 if and are defined as a minimum or higher than 1, if and are defined as a maximum.

 

6. NIVERTHI AND DEY'S MULTIVARIATE PROCESS CAPABILITY INDEXES

Niverthi and Dey (2000) proposed an extension of the univariate Cp, Cpk, for the multivariate case as follows. Let X = (X1 X2 ... Xp)' be the vector containing the quality characteristics with a p-variate normal distribution with parameters µ0 = (µ01 µ02 ...µ0p)' and Σpxp. Let USL = (USL1 USL2 ... USLp)' and LSL = (LSL1 LSL2 ... LSLp)' be the upper and lower specification vectors, LSLi and USLi as defined in section 5.2, i=1,2,…,p. The Niverthi and Dey's multivariate versions of univariate Cp and Cpk are linear combinations of the upper and lower specifications limits of the p variables and are defined as

In this case a capability value is generated for each quality characteristic, since and are (px1) dimensional vectors. The value of the constant k is based on the univariate standard normal distribution. Niverthi and Dey (2000) used k=3 in (13) which corresponds to an area of 99.73% or a significance level of α = 0.0027.

 

7. EXAMPLE

For this example we will use p=4 variables of the aircraft data set presented in Niverthi and Dey's paper (2000) which originally has n=50 observations related to measurement (in centimeters) on 10 different aircraft features from a component hub which is part of the engine. The production of these parts is made with high degree of precision. The 4 variables presented in this example, according to the original notation of Niverthi and Dey's paper are: MQI128, MQI444, MQI519 and MQI514. The vectors with the specification limits given by Niverthi and Dey (2000) are

The sample covariance and correlation matrices are respectively given by

The Niverthi and Dey's (2000) estimated multivariate for α = 0.0027 and α = 0.05 are given respectively by

Due to the fact that the process is centered in the nominal mean (see Table 1) for each α the estimated value of is equal to . There is a difficulty to use to decide if the process is capable or not because there is no reference values to which the vector could be compare to. One possibility is to use the usual univariate Cp reference value for each variable separatedly. Another one is to define the global capability process estimate as the minimum value of the vector which for this example is 1.085 (for α = 0.0027) and 1.627 (for α = 0.05). It is important to clarify that the values showed in Niverthi and Dey's paper (p.677) are different than the values in (17) since in their example they used the complete random vector with p=10 variables to generate the capability estimation and therefore their covariance and correlation matrices are different than (15) and (16), respectively.

 

 

To obtain the value of the Mingoti and Glória's multivariate PCI proposed in section 5.1 () it is necessary to calculate the constant CRα. Table 2 presents the values of CRα, for α = 0.05, obtained by applying the simulation algorithm described in section 3, for N=1000, 10000 and 100000, considering a standard multivariate normal distribution with correlation matrix equals to (16). The corresponding values of (according to equation (10)) are also presented. As one can see the values of CRα for N=10000 and N=100000 are very similar indicating that in fact, there is no need to perform 100000 simulations as suggested by Hayter and Tsui (1994). For this example the value of CRα was considered as 3.327 for α = 0.0027 and 2.487 for α = 0.05. The multivariate process capability is estimated as

 

 

 

indicating that the process is incapable at 99.73% and capable at 95%. However, for this confidence level the estimated is very close to 1 which gives a warning signal.

 

8. A SIMULATION STUDY

In this section we present the results of a simulation study. Two processes were considered with parameters and respective specifications limits as given in sections 8.1 and 8.2.

8.1. Process 1 – centered in the specification vector mean

Let the mean vector, the specification limits, the covariance and correlation matrices be given as

mean vector: µ0 = [40 80]' = µS, covariance matrix:

correlation matrix: ; USL = [60 98]'; LSL = [20 62]'.

It represents a situation where there is a high correlation between the two quality characteristics. For α = 0.0027 the true univariate capability indexes are: Cp1 = 1.66 and Cp2 = 1.5 and the geometric mean is equal to 1.578. The multivariate Niverthi and Dey's is given by:

with minimum equals to 0.885. For α = 0.0027 the constant CRα is 3.149 and Mingoti and Glória's index is given by

For α = 0.05 the values of the univariate capability indexes are: Cp1 = 2.5, Cp2 = 2.25 and the geometric mean is equal to 2.37. The multivariate Niverthi and Dey's is given by

with minimum equals to 1.328. The constant CRα for α = 0.05 is 2.092 and Mingoti and Glória's is given by

Figure 1 shows the specification and confidence regions for 99.73 and 95% considering a bivariate normal distribution of process 1. It is very clear that this process is capable. However, for 99.73% Niverthi and Dey's suggests that the process is not capable in the second variable. The geometric means indicate that the process is capable for both α values. Comparing to Niverthi and Dey's in this example, Mingoti and Glória's index represented better the global capability of the process.

8.2. Process 2 – not centered in the specification mean vector

Let the mean vector, the specification limits, the covariance and correlation matrices be given as

mean vector: µ0 = [50 85]; covariance matrix: ; µs = [40 80]';

correlation matrix: ; USL = [60 90] ; LSL = [20 70]' .

It represents a situation where there is a moderate correlation between the two quality characteristics. For α = 0.0027 the true univariate capability indexes are: Cp1 = 1.66, Cp2 = 0.833 and the geometric mean is equal to 1.176. The multivariate Niverthi and Dey's is given by the vector

with minimum equals to 0.431. For α = 0.0027 the constant CRα is 3.195 and Mingoti and Glória's capability index is given by

For α = 0.05 the values of the univariate capability indexes are: Cp1 = 2.49, Cp2 = 1.249 and the geometric mean is equal to 1.764. The multivariate Niverthi and Dey's is given by the vector

with minimum equals to 0.647. For α = 0.05 the constant CRα is 2.198 and Mingoti and Glória's capability is given by

Figure 2 shows the specification and confidence regions for 99.73 and 95% considering the bivariate normal distribution of process 2. It is clear that this process is not capable for 99.73% as both multivariate capability indexes indicated. For 95% Niverthi and Dey's suggests that the process is not capable in the second variable but the confidence region is still inside the specification region although very close to the upper specification limit for the second variable. Mingoti and Glória's index indicated a warning signal since it resulted in a value very close to 1. By the geometric means the process is considered capable for both confidence levels although for 99.73% the estimated value is very close to 1.

It is important to point out that in both situations, process 1 and 2, the geometric mean resulted in higher values than both multivariate indexes and since it does not take into consideration the correlation between the two quality characteristics and therefore, it has the tendency of overestimate the capability of the process. On the other hand, Niverthi and Dey's has the tendency of underestimate the capability. Mingoti and Glória's resulted in values in between both and it was able to describe more properly the true capability of the processes considered. Also in all cases the values of the capability multivariate indexes are functions of the confidence level (1 – α), 0 < α < 1. Depending of the choice of α the process might be considered capable or not. Therefore, the choice of α is very important to evaluate the process capability.

8.3. Simulation

A total of k=100 random samples of size n=100 were generated for each simulated process. Niverthi and Dey's multivariate PCI and Mingoti and Glória's index were calculated for each sample considering α = 0.0027. Table 3 presents the average and the standard deviation of the PCI's estimates for process 1. The estimated Niverthi and Dey's PCI resulted in coefficients that are very similar to their corresponding theoretical values for each variable and with small standard deviations. The same occurred with Mingoti and Glória's index estimates which were similar to the theoretical values. The results for the multivariate Niverthi and Dey's and Mingoti and Glória's will not be shown because they were very similar to those obtained for and since the process 1 is centered in the specification mean vector. Table 4 presents the and average values for the process 2 for α = 0.0027. The fit was also good as expected. The resulted in smaller standard deviations in both cases. Therefore, this simulation study indicated that the estimators of the multivariate Niverthi and Dey's (2000) and Mingoti and Glória's (2003) indexes described well the true theoretical values of the multivariate process capability corresponding to each methodology. Mingoti and Glória's had better performance since it presented smaller mean error and standard deviation in both processes.

 

 

 

 

9. CONFIDENCE INTERVALS FOR CAPABILITY INDEXES USING BOOTSTRAP METHODOLOGY

The bootstrap methodology (EFRON; TIBSHIRANI,1993) can be used to generate confidence intervals for the true process capability indexes for each methodology. Given a sample of size n of the process m random samples with replacement are selected from this sample, called bootstrap samples. For each bootstrap sample the , , and are estimated and their sample distribution are obtained. A confidence interval for the true capability values can be obtained by using methods such as percentile, the accelerated bias-corrected, the bias-corrected percentile and t-bootstrap methods (GARTHWAITE et al.,1995). As an illustration we will return to the aircraft example presented in section 7. Considering the original sample of size n=50 presented in Niverthi and Dey's paper (2000) for those p=4 variables of the example, a total of m=500 bootstrap samples were selected with replacement. For each bootstrap sample the estimates and calculated using α =0.0027, were compared to the values of the vector given in (17) and to = 0.783 value given in (19). Table 5 shows the Mean Error (ME) and the Squared Mean Error (SME) resulted from this comparison. For Niverthi and Dey the ME and SME values are averages of the corresponding values calculated for each of the 4 variables. As we can see the errors were larger for Niverthi and Dey estimates than for Mingoti and Glória's index which had a very good fit and was practically unbiased. The 95% confidence limits obtained for Niverthi and Dey's and Mingoti and Glória's PCI true process indexes are given in Table 6. If the univariate capability reference values were considered for a comparison we could conclude that the process is capable for the first, third and fourth variables but might be incapable for the second variable since the confidence interval includes values lower than 1 according to Niverthi and Dey's. The confidence interval obtained according to Mingoti and Glória's capability index indicates that the process is incapable. Also, the confidence interval using Mingoti and Glória's index resulted in smaller range than Niverthi and Dey's.

 

 

 

 

In this section only the confidence intervals for the true Mingoti and Glória global capability index was presented which corresponds to the variable with lower capability. However, by using the bootstrap methodology it is possible to obtain confidence intervals for Mingoti and Glória's true capability of each quality characteristic individually.

 

10. FINAL REMARKS

The examples presented in this paper show that the Mingoti and Glória's capability index ( ), which is a modification of Chen's capability coefficient (1994), is more precise than Niverthi and Dey's () and less biased. The produces a capability value for each variable and a global capability index differently than Niverthi and Dey which was originally proposed to give only a capability coefficient for each variable separatedly. In this paper we introduced the idea of measuring the global capability by taking Niverthi and Dey's vector minimum value. By using Hayter and Tsui (1994) methodology, the calculation of Mingoti and Glória's capability index is more feasible for any number p of variables. This is because the calculation will depend only of a simple simulation procedure used to obtain the constant CRα related to the distribution of the maximum of the coordinates of a random vector with p-variate normal distribution. If the distribution is not multivariate normal the capability index still can be used since the constant CRα can be obtained by a non-parametric procedure as suggested in Hayter and Tsui (1994) or by Kernel methodology ( POLANSKY; BAKER, 2000; Glória, 2006). The same is not true for Niverthi and Dey's indexes. It is also important to point out that in the examples presented in this paper the value of the geometric mean was always higher than Mingoti and Glória's and Niverthi and Dey's overestimating the true process capability. On the other hand, Niverthi and Dey's penalizes the process more than Mingoti and Glória's indicating sometimes that the process is not capable when it really is. For the examples presented in this paper Mingoti and Glória's described more properly the true capability of the processes. Finally, the bootstrap methodology is an interesting alternative to produce confidence intervals for the true capability indexes of multivariate processes.

 

ACKNOWLEDGMENT

The authors are very grateful to the unknown referees for their helpful comments and suggestions.

The authors were partially supported by the Brazilian Council for Scientific and Technological Development (CNPq).

 

REFERENCES

BERNARDO, J. M.; IRONY, T. X. A general multivariate bayesian process capability index. Statistician, v. 45, n. 3, p. 487-502, 1996.         [ Links ]

CHEN, H. A multivariate process capability index over a rectangular solid tolerance zone, Statistica Sinica, v. 4, n. 2, p. 749-758, 1994.         [ Links ]

CHENG, S. W.; SPIRING, F. A. Assessing process capability: a bayesian approach. IEE Transactions, 21, p. 97-98, 1989.         [ Links ]

EFRON, B.; TIBSHIRANI, R. J. An Introduction to the Bootstrap. New York: Chapman and Hall, 1993.         [ Links ]

FOSTER, E. J.; BARTON, R. R.; GAUTAM, N.; TRUSS, L. T.; TEW, J. D. The process-oriented multivariate capability index. International Journal of Production Research, v. 43, n.10, p. 2135-2148, 2005.         [ Links ]

GARTHWAITE, P. H.; JOLLIFFE, I. T.; JONES, B. Statistical inference. New York: Prentice Hall, 1995.         [ Links ]

GLÓRIA, F. A. A. Uma avaliação do desempenho de núcleo-estimadores no controle de processos multivariados. Dissertação (Mestrado em Estatística). Universidade Federal de Minas Gerais, Departamento de Estatística. Belo Horizonte, 2006.         [ Links ]

HAYTER, A. J.; TSUI, K-L. Identification and quantification in multivariate quality control problems. Journal of Quality Technology, v. 26, n. 3, p. 197-208, 1994.         [ Links ]

JOHNSON, R. A.; WICHERN, D. W. Applied Multivariate Statistical Analysis. New Jersey: Prentice Hall, 2002.         [ Links ]

KALGONDA, A. A.; KULKARNI, S. R. Multivariate quality control chart for autocorrelated processes. Journal of Applied Statistics, v. 31,n. 3, p. 317-327, 2004.         [ Links ]

KOTZ, S.; JOHNSON, N. L. Process capability indexes-a review, 1992-2000. Journal of Quality Technology, v. 34, n. 1, p. 2-39, 2002.         [ Links ]

MASON, R. L.; YOUNG, J. C. Multivariate Statistical Process Control with Industrial Applications. Pennsylvania: Siam and ASA, 2002.         [ Links ]

MINGOTI, S. A.; GLÓRIA, F. A. A. Comparando os métodos paramétrico e não-paramétrico na determinação do valor crítico do teste estatístico de médias proposto por Hayter e Tsui. Produção, v. 15, n. 2, p. 251-262, 2005.         [ Links ]

MINGOTI, S. A.; GLÓRIA, F. A. A. Uma proposta de modificação do índice de capacidade multivariado de Chen. In Anais do XXIII ENEGEP, Ouro Preto, Minas Gerais, 2003 (em cd-rom).         [ Links ]

MONTGOMERY, D. C. Introduction to Statistical Quality Control. New York: John Wiley, 2001.         [ Links ]

NIVERTHI, M.; DEY, D. K. Multivariate process capability: a bayesian perspective. Communications in Statistics-Simulation and Computation, 29, p. 667-687, 2000.         [ Links ]

PEARN, W. L.; WANG, F. K.; YEN, C. H. Multivariate capability indices: distributional and inferential properties. Journal of Applied Statistics, v. 34, n. 8, p. 941-962, 2007.         [ Links ]

PEARN, W. L.; WU, C. W. Production quality and yield assurance for processes with multiple independent characteristics. European Journal of Operational Research, v. 173, n. 2, p. 637-647, 2006.         [ Links ]

POLANSKY, A. M. A smooth nonparametric approach to multivariate process capability. Technometrics, v. 43, n. 2, p. 199-211, 2001.         [ Links ]

POLANSKY, A. M.; BAKER, E. R. Multistage plug-in bandwidth selection for kernel distribution function estimates. Journal of Statistical Computation and Simulation, v. 65, n. 1, p. 63-80, 2000.         [ Links ]

SHAHRIARI, H.; HUBELE, N. F.; LAWRENCE, F. P. A multivariate process capability vector. In Proceedings of the 4th Industrial Engineering Research Conference, Institute of Industrial Engineers, p. 305-309, 1995.         [ Links ]

TAAN, W.; SUBBAIHA, P.; LIDDY, J. W. A note on multivariate capability indexes. Journal of Applied Statistics, v. 20, n. 3, p. 339-351, 1993.         [ Links ]

TAAN, P. F.; BARNETR, N. S. Capability indexes for multivariate processes. Technical report. Division of computation Mathematics and Science. Victoria University. Melbouurne, Australia, 1998.         [ Links ]

VEEVERS, A. Viability and capability indexes for multiresponse processes. Journal of Applied Statistics, v. 25, n. 4, p. 545-558, 1998.         [ Links ]

WANG, C. H. Constructing multivariate process capability indices for short-run production. International Journal of Advanced Manufacturing Technology, 26, p. 1306-1311, 2005.         [ Links ]

WANG, F. K. Quality evaluation of a manufactured product with multiple characteristics. Quality and Reliability Engineering International, v. 22, n. 2, p. 225-236, 2006.         [ Links ]

WANG, F. K.; HUBELE, N.F.; LAWRENCE, F. P.; MISKUKIN, J. O.; SHARIARI, H. Comparison of three multivariate process capability indexes. Journal of Quality Technology, v. 32, n. 3, p. 263-275, 2000.         [ Links ]

ZHANG, N. Z. Estimating process capability indexes for autocorrelated data. Journal of Applied Statistics, v. 25, n. 4, p. 559-574, 1998.         [ Links ]

 

 

Artigo recebido em 20/06/2007
Aprovado para publicação em 11/04/2008.

 

 

ABOUT THE AUTHORS

Sueli Aparecida Mingoti
Departamento de Estatística - ICEx
Universidade Federal de Minas Gerais
End.: Av. Antonio Carlos, 6627 - Campus Pampulha – CEP 30161-970 – Belo Horizonte - Minas Gerais
Tel.: (031) 3-4995948 - Fax: 3-499-5924
E-mail: sueliam@est.ufmg.br

Fernando Augusto Alves Glória
Nestle S/A
End.: Rua Henry Nestle, S/N – Vila Formosa – São José do Rio Pardo – SP – CEP 13720-000
E-mail: fernando.gloria@br.nestle.com

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License