Production
https://prod.org.br/doi/10.1590/S0103-65132011005000029
Production
Article

Monitoring the mean vector and the covariance matrix of multivariate processes with sample means and sample ranges

Monitoramento do vetor de médias e da matriz de covariâncias de processos multivariados baseado nas médias e nas amplitudes amostrais

Costa, Antonio Fernando B.; Machado, Marcela Aparecida G.

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Abstract

The joint X and R charts and the joint X and S2 charts are the most common charts used for monitoring the process mean and dispersion. With the usual sample sizes of 4 and 5, the joint X and R charts are slightly inferior to the joint X and S2 charts in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the charts based on the standardized sample means and sample ranges (MRMAX chart) or on the standardized sample means and sample variances (MVMAX chart) are similar in terms of efficiency in detecting shifts in the mean vector and/or in the covariance matrix. User’s familiarity with the computation of sample ranges is a point in favor of the MRMAX chart. An example is presented to illustrate the application of the proposed chart.

Keywords

Control charts. Mean vector. Covariance matrix. Multivariate processes.

Resumo

Os gráficos conjuntos de X e R e X e S2 são os mais utilizados para o monitoramento da média e da dispersão do processo. Com os tamanhos de amostra usuais de 4 e 5, os gráficos de X e R em uso conjunto são ligeiramente inferior aos gráficos de X e S2 em uso conjunto em termos da eficiência em detectar alterações no processo. Neste artigo, mostra-se que para o caso multivariado, os gráficos baseados nas médias amostrais padronizadas e amplitudes amostrais (gráfico MRMAX) ou nas médias amostrais padronizadas e variâncias amostrais (gráfico MVMAX) são similares em termos da eficiência em detectar alterações no vetor de médias e/ou na matriz de covariâncias. A familiaridade do usuário com o cálculo de amplitudes amostrais é um aspecto favorável do gráfico MRMAX. Um exemplo é apresentado para ilustrar a aplicação do gráfico proposto.

Palavras-chave

Gráficos de controle. Vetor de médias. Matriz de covariância. Processos multivariados.

References



ALT, F. B. Multivariate quality control. In: KOTZ, S.; JOHNSON, N. L. (Ed.). Encyclopedia of Statistical Sciences. Wiley, 1985.

ANDERSON, T. W. An introduction to multivariate statistical analysis. Stanford: John Wiley, 2003.

APARISI, F.; JABALOYES, J.; CARRIÓN, A. Statistical properties of the |S| multivariate control chart. Communication in Statistics – Theory and Methods, v. 28, p. 2671-2686, 1999.

APARISI, F.; JABALOYES, J.; CARRIÓN, A. Generalized variance chart design with adaptive sample sizes. The bivariate case. Communication in Statistics – Simulation and Computation, v. 30, p. 931-948, 2001.

CALZADA, M. E.; SCARIANO, S. M. The robustness of the synthetic control chart to non-normality. Communications in Statistics: Simulation and Computation, v. 30, p. 311‑326, 2001.

CHEN, G.; CHENG, S. W.; XIE, H. A new multivariate control chart for monitoring both location and dispersion. Communications in Statistics-Simulation and Computation, v. 34, p. 203-217, 2005.

CHOU, C. Y. et al. Economic-statistical design of multivariate control charts using quality loss function. International Journal of Advanced Manufacturing Technology, v. 20, p. 916-924, 2002.

COSTA, A. F. B.; MACHADO, M. A. G. Synthetic control chart with two-stage sampling for monitoring bivariate processes. Pesquisa Operacional, v. 27, p. 117-130, 2007.

COSTA, A. F. B.; MACHADO, M. A. G. A new chart for monitoring the covariance matrix of bivariate processes. Communications in Statistics – Simulation and Computation, v. 37, p. 1453-1465, 2008a.

COSTA, A. F. B.; MACHADO, M. A. G. Bivariate control charts with double sampling. Journal of Applied Statistics, v. 35, p. 809-822, 2008b.

COSTA, A. F. B.; MACHADO, M. A. G. A new chart based on the sample variances for monitoring the covariance matrix of multivariate processes. International Journal of Advanced Manufacturing Technology, v. 41, p. 770-779, 2009.

COSTA, A. F. B.; RAHIM, M. A. A synthetic control chart for monitoring the process mean and variance. Journal of Quality In Maintenance Engineering, v. 12, p. 81-88, 2006.

COSTA, A. F. B.; EPPRECHT E. K.; CARPINETTI, L. C. R. Controle estatístico de qualidade. 2. ed. São Paulo: Atlas, 2005. 334 p.

COSTA, A. F. B.; DE MAGALHÃES, M. S; EPPRECHT, E. K. Monitoring the process mean and variance using synthetic control chart with two-stage testing. International Journal of Production Research, http://dx.doi.org/10.1080/00207540802047098, 2008.

DAVIS, R. B.; WOODALL, W. H. Evaluating and improving the synthetic control chart. Journal of Quality Technology, v. 34, p. 200-208, 2002.

HOTELLING, H. Multivariate quality control, illustrated by the air testing of sample bombsights. Techniques of Statistical Analysis. New York: McGraw Hill, 1947. p. 111‑184.

KHOO, M. B. C. A new bivariate control chart to monitor the multivariate process mean and variance simultaneously. Quality Engineering, v. 17, p. 109-118, 2005.

MACHADO, M. A. G; COSTA, A. F. B. The double sampling and the EWMA charts based on the sample variances. International Journal of Production Economics, v. 114, p. 134-148, 2008a.

MACHADO, M. A. G; COSTA, A. F. B. The use of principal components and simultaneous univariate charts to control multivariate processes. Pesquisa Operacional, v. 28, p. 173-196, 2008b.

MACHADO, M. A. G.; DE MAGALHÃES, M. S; COSTA, A. F. B. Gráfico de controle de VMAX para o monitoramento da matriz de covariâncias. Revista Produção, v. 18, p. 222‑239, 2008.

MACHADO, M. A. G.; COSTA, A. F. B.; RAHIM, M. A. The synthetic control chart based on two sample variances for monitoring the covariance matrix. Quality and Reliability Engineering International, v. 25, p. 595-606, 2008.

MACHADO, M. A. G.; COSTA, A. F. B.; MARINS, F. A. S. Control charts for monitoring the mean vector and the covariance matrix of bivariate processes. International Journal of Advanced Manufacturing Technology, v. 45, p. 772-785, 2009.

MACHADO, M. A. G.; COSTA, A. F. B.; CLARO, F. A. E. Monitoring bivariate processes. Pesquisa Operacional, v. 29, p. 547-562, 2009.

MICROSOFT FORTRAN POWER STATION 4.0. Professional edition with Microsoft IMSL Mathematical and Statistical Libraries, Microsoft Corporation, 1995.

MOOD, A. M.; GRAYBILL, F. A.; BOES, D. C Introduction to the theory of statistics. McGraw-Hill, 1974.

TAKEMOTO, Y.; ARIZONO, I. A study of multivariate (X, S) control chart based on Kullback-Leibler information. International Journal of Advanced Manufacturing Technology, v. 25, p. 1205-1210, 2005.

SEREL, D. A.; MOSKOWITZ, H.; TANG, J. Univariate X control charts for individual characteristics in a multinormal model. IIE Transactions, v. 32, p. 1115-1125, 2000.

WU, Z.; SPEDDING, T. A. Implementing synthetic control charts. Journal of Quality Technology, v. 32, p. 75-78, 2000a.

WU, Z.; SPEDDING, T. A. A synthetic control chart for detecting small shifts in the process mean. Journal of Quality Technology, v. 32, p. 32-38, 2000b.

WU, Z.; YEO, S. H.; SPEDDING, T. A. A synthetic control chart for detecting fraction nonconforming increases. Journal of Quality Technology, v. 33, p. 104-111, 2001.

WU, Z.; ZHANG, X.; YEO, S. H. Design of sum-of-conforming-run-length control charts. European Journal of Operational research, v. 132, p. 187-196, 2001.

ZHANG, G.; CHANG, S. I. Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis. International Journal of Production Research, v. 46, p. 6855-6881, 2008.

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