Gráfico de controle de VMAX para o monitoramento da matriz de covariâncias
The VMAX control chart for monitoring the covariance matrix
Machado, Marcela Aparecida G.; Magalhães, Maysa Sacramento de; Costa, Antonio Fernando B.
http://dx.doi.org/10.1590/S0103-65132008000200003
Prod, vol.18, n2, p.222-239, 2008
Resumo
Neste artigo é proposto, para o monitoramento de processos normais bivariados, um gráfico de controle baseado nas variâncias amostrais de duas características de qualidade. Os pontos plotados no gráfico correspondem ao valor da maior variância amostral. O gráfico proposto, denominado gráfico de VMAX, tem um desempenho superior ao do gráfico da variância amostral generalizada |S| e, além disso, tem uma melhor capacidade de diagnóstico, ou seja, com ele é mais fácil identificar a variável que teve sua variabilidade alterada pela ocorrência da causa especial. Quando a amostragem dupla está em uso o gráfico proposto também tem um desempenho superior ao do gráfico de |S|, exceto em alguns casos em que o tamanho da segunda amostra é muito grande.
Palavras-chave
Gráficos de controle de VMAX, matriz de covariâncias, processos bivariados, amostragem dupla.
Abstract
In this paper a control chart based on sample variances from two quality characteristics for monitoring bivariate normal processes is proposed. The points plotted on the chart are the maximum of the values of these two statistics. The proposed chart (VMAX chart) detects process disturbances faster than the generalized variance |S| chart and has a better diagnostic feature, that is, with the VMAX chart it is easier to relate an out-of-control signal to the variable whose variability has moved away from its in-control state. When the double sampling scheme is used the proposed chart has also a better performance, except in a few cases in which the size of the samples at the second stage is very large.
Keywords
VMAX control chart, covariance matrix, bivariate processes, double sampling
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. Communications 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. Communications in Statistics - Simulation and Computation, v. 30, p. 931-948, 2001.
CAROT, V.; JABALOYES, J.M.; CAROT, T. Combined double sampling and variable sampling interval chart. International Journal of Production Research, v. 40, p. 2175-2186, 2002.
CHANG, S. I.; ZHANG, K. Statistical process control for variance shift detections of multivariate autocorrelated processes. Quality Technology of Quantitative Management, v. 4, p. 413-435, 2007.
CHEN, G.; CHENG, S. W.; XIE, H. Monitoring process mean and variability with one EWMA chart. Journal of Quality Technology, v. 33, p. 223-233, 2001.
COSTA, A. F. B. charts with variable sample size. Journal of Quality Technology, v. 26, p. 155-163, 1994.
COSTA, A. F. B. Joint and R charts with variable sample sizes and sampling intervals. Journal of Quality Technology, v. 31, p. 387-397, 1999.
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.; RAHIM, M. A. A single EWMA chart for monitoring process mean and process variance. Quality Technology & Quantitative Management, v. 3, p. 295-305, 2006.
COSTA, A.F.B.; EPPRECHT E.K.; CARPINETTI, L.C.R. Controle Estatístico de Qualidade. 2. ed. São Paulo: Editora Atlas, 2005. 334 p.
COSTA, A.F.B.; De MAGALHÃES, M.S.; EPPRECHT E.K. The non-central chi-square chart with double sampling. Brazilian Journal of Operation and Production Management, v. 2, p. 57-80, 2005a.
CROASDALE, R. Control charts for a double sampling scheme based on average production run length. International Journal of Production Research, v. 12, p. 585-592, 1974.
DAUDIN, J. J. Double sampling charts. Journal of Quality Technology, v. 24, p. 78-87, 1992.
De MAGALHÃES, M. S.; COSTA, A. F. B.; EPPRECHT, E. K. Constrained optimization model for the design of an adaptive chart. International Journal of Production Research, v. 40, p. 3199-3218, 2002.
De MAGALHÃES, M. S.; COSTA, A. F. B.; MOURA NETO, F. D. Adaptive control charts: a Markovian approach for processes subject to independent out-of-control disturbances. International Journal of Production Economics, v. 99, p. 236-246, 2006.
DJAUHARI, M. A. Improved monitoring of multivariate process variability. Journal of Quality Technology, v. 37, p. 32-39, 2005.
DOMANGUE, R.; PATCH, SC. Some omnibus exponentially weighted moving average statistical process monitoring schemes. Technometrics, v. 33, p. 299-313, 1991.
GRIGORYAN, A.; HE, D. Multivariate double sampling |S| charts for controlling process variability. International Journal of Production Research, v. 43, p. 715-730, 2005.
HAWKINS, D. M.; OLWELL, D. H. Cumulative sum charts and charting for quality improvement. New York: Springer-Verlag. 1998.
HE, D; GRIGORYAN, A. Construction of double sampling S- control charts for agile manufacturing. Quality and Reliability Engineering International, v. 18, p. 343-355, 2002.
HE, D; GRIGORYAN, A. Multivariate multiple sampling charts. IIE Transactions, v. 37, p. 509-521, 2005.
HE, D; GRIGORYAN, A. Joint statistical design of double sampling and S charts. European Journal of Operational Research, v. 168, p. 122-142, 2006.
HE, D; GRIGORYAN, A; SIGH, M. Design of double and triple sampling control charts using genetic algorithms. International Journal of Production Research, v. 40, p. 1387-1404, 2002.
IRIANTO, D.; SHINOZAKI, N. An optimal double sampling control chart. International Journal of Industrial Engineering - Applications and Practice, v. 5, p. 226-234, 1998.
LUCAS, J. M.; SACCUCCI, M. S. Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, v. 32, p. 1-16, 1990.
MACHADO, M. A. G. Estudo das propriedades dos gráficos de controle bivariados com amostragem dupla. Dissertação (Mestrado em Engenharia Mecânica) - Faculdade de Engenharia do Campus de Guaratinguetá, Universidade Estadual Paulista, Guaratinguetá, 129f., 2006.
MASON, R. L.; YOUNG, J. C. Multivariate Statistical Process Control with Industrial Applications. Philadelphia: ASA - SIAM. 2002.
MICROSOFT FORTRAN POWER STATION 4.0. Professional Edition with Microsoft IMSL Mathematical and Statistical Libraries, Microsoft Corporation, 1995.
MONTGOMERY, D. C. Introduction to Statistical Quality Control. New York: John Wiley & Sons, Inc. 2004.
PAGE, E. S. Continuous inspection schemes. Biometrika, v. 41, p. 100-115, 1954.
REYNOLDS, M. R., Jr.; STOUMBOS, Z. G. Should exponentially weighted moving average and cumulative sum charts be used with Shewhart limits? Technometrics, v. 47, p. 409-424, 2005.
SEREL, D. A.; MOSKOWITZ, H.; TANG, J. Univariate control charts for individual characteristics in a multinormal model. IIE Transactions, v. 32, p. 1115-1125, 2000.
SOUZA, A. M.; RIGÃO, M. H. Identificação de variáveis fora de controle em processos produtivos multivariados. Revista Produção, v. 15, p. 74-86, 2005.
SURTIHADI, J.; RAGHAVACHARI, M.; RUNGER, G. Multivariate control charts for process dispersion. International Journal of Production Research, v. 42, p. 2993-3009, 2004.
YEH, A. B.; LIN, D. K. J.; ZHOU, H.; VENKATARAMANI, C. A multivariate exponentially weighted moving average control chart for monitoring process variability. Journal of Applied Statistics, v. 30, p. 507-536, 2003.
YEH, A. B.; HUWANG, L.; WU, Y. F. A likelihood-ratio-based EWMA control chart for monitoring variability of multivariate normal processes. II Transactions, v. 36, p. 865-879, 2004.
YEH, A. B.; HUWANG, L.; WU, C. W. A multivariate EWMA control chart for monitoring process variability with individual observations. II Transactions, v. 37, p. 1023-1035, 2005.
YEH, A. B.; LIN, D. K. J.; MCGRATH, R. N. Multivariate control charts for monitoring covariance matrix: a review. Quality Technology and Quantitative Management, v. 3, p. 415-436, 2006.
Facebook
Google+
Twitter
LinkedIn
Mendeley
StumbleUpon
CiteULike
Reddit
Email