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An adaptive Bayesian scheme for joint monitoring of process mean and variance. (English) Zbl 1348.62273

Summary: This paper presents a new model for the economic optimization of a process operation where two assignable causes may occur, one affecting the mean and the other the variance. The process may thus operate in statistical control, under the effect of either one of the assignable causes or under the effect of both assignable causes. The model employed uses the Bayes theorem to determine the probabilities of operating under the effect of each assignable cause. Based on these probabilities, and following an economic optimization criterion, decisions are made on the necessary actions (stop the process for investigation or not) as well as on the time of the next sampling instance and the size of the next sample. The superiority of the proposed model is estimated by comparing its economic outcome against the outcome of simpler approaches such as Fp (Fixed-parameter) and adaptive Vp (Variable-parameter) Shewhart charts for a number of cases. The numerical investigation indicates that the economic improvement of the new model may be significant.

MSC:

62P30 Applications of statistics in engineering and industry; control charts
62F15 Bayesian inference
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[1] Bai, D. S.; Lee, K. T., An economic design of variable sampling interval \(\overline{X}\) control charts, International Journal of Production Economics, 54, 1, 57-64 (1998)
[2] Bather, J. A., Control charts and minimization of costs, Journal of the Royal Statistical Society - Series B, 25, 49-80 (1963) · Zbl 0124.35404
[3] Calabrese, J., Bayesian process control for attributes, Management Science, 41, 4, 637-645 (1995) · Zbl 0836.90080
[4] Celano, G.; Costa, A.; Fichera, S.; Trovato, E., One-sided Bayesian \(S^2\) control charts for the control of process dispersion in finite production runs, International Journal of Reliability, Quality and Safety Engineering, 15, 4, 305-327 (2008)
[5] Celano, G.; De Magalhães, M. S.; Costa, A. F.B.; Fichera, S., A stochastic shift model for economically designed charts constrained by the process stage configuration, International Journal of Production Economics, 132, 2, 315-325 (2011)
[6] Costa, A. F.B., Joint economic design of \(\overline{X}\) and R control charts for processes subject to two independent assignable causes, IIE Transactions, 25, 6, 27-33 (1993)
[7] Costa, A. F.B., Joint X and R charts with variable parameters, IIE Transactions, 30, 6, 505-514 (1998)
[8] Costa, A. F.B., Joint X and R charts with variable sample sizes and sampling intervals, Journal of Quality Technology, 31, 4, 387-397 (1999)
[9] Costa, A. F.B.; Rahim, M. A., Economic design of \(\overline{X}\) charts with variable parameters: the Markov chain approach, Journal of Applied Statistics, 28, 7, 875-885 (2001) · Zbl 1155.62488
[10] Das, T. K.; Jain, V.; Gosavi, A., Economic design of dual-sampling-interval Policies for \(\overline{X}\) charts with and without run rules, IIE Transactions, 29, 6, 497-506 (1997)
[11] De Magalhães, M. S.; Costa, A. F.B; Neto, F. D.M., Adaptive control charts: a Markovian approach for processes subject to independent disturbances, International Journal of Production Economics, 99, 1-2, 236-246 (2006)
[12] De Magalhães, M. S.; Epprecht, E. K.; Costa, A. F.B., Economic design of a Vp \(\overline{X}\) chart, International Journal of Production Economics, 74, 1-3, 191-200 (2001)
[13] De Magalhães, M. S.; Neto, F. D.M., Joint economic model for totally adaptive X and R charts, European Journal of Operational Research, 161, 1, 148-161 (2005) · Zbl 1067.90031
[14] Duncan, A. J., The economic design of \(\overline{X}\) charts used to maintain current control of a process, Journal of the American Statistical Association, 51, 274, 228-242 (1956) · Zbl 0071.13703
[15] Duncan, A. J., The economic design of \(\overline{X}\) charts when there is a multiplicity of assignable causes, American Statistical Association Journal, 66, 333, 107-121 (1971) · Zbl 0221.62034
[16] Girshick, M. A.; Rubin, H., A Bayes approach to a quality control model, Annals of Mathematical Statistics, 23, 1, 114-125 (1952) · Zbl 0046.35405
[17] Goel, A. L.; Jain, S. C.; Wu, S. M., An algorithm for the determination of the economic design of \(\overline{X}\) charts based on Duncan’s model, American Statistical Association Journal, 63, 321, 304-320 (1968)
[18] Knappenberger, H. A.; Gra3ndage, A. H., Minimum cost quality control tests, AIIE Transactions I, 24-32 (1969)
[19] Lorenzen, T. J.; Vance, L. C., The economic design of control charts: a unified approach, Technometrics, 28, 1, 3-10 (1986) · Zbl 0597.62106
[20] Makis, V., Multivariate Bayesian control charts, Operations Research, 56, 2, 487-496 (2008) · Zbl 1167.90471
[21] Nenes, G., A new approach for the economic design of fully adaptive control charts, International Journal of Production Economics, 131, 2, 631-642 (2011)
[22] Nenes, G., Optimization of fully adaptive Bayesian \(\overline{X}\) charts for infinite-horizon processes, International Journal of Systems Science, 44, 2, 289-305 (2013) · Zbl 1307.93462
[23] Nenes, G.; Tagaras, G., The economically designed two-sided Bayesian \(\overline{X}\) control charts, European Journal of Operational Research, 183, 1, 263-277 (2007) · Zbl 1128.62133
[24] Nikolaidis, Y.; Rigas, G.; Tagaras, G., Using economically designed Shewhart and Adaptive \(\overline{X} \) - charts for monitoring the quality of tiles, Quality and Reliability Engineering International, 23, 2, 233-245 (2006)
[25] Park, C.; Reynolds, M. R., Economic design of a variable sample size \(\overline{X}\) chart, Communications in Statistics - Simulation and Computation, 23, 2, 467-483 (1994) · Zbl 0825.62040
[26] Porteus, E. L.; Angelus, A., Opportunities for improved statistical process control, Management Science, 43, 9, 1214-1228 (1997) · Zbl 1043.90510
[27] Reynolds, M. R.; Amin, R. W.; Arnold, J. C.; Nachlas, J. A., \( \overline{X}\) charts with variable sampling intervals, Technometrics, 30, 2, 181-192 (1988)
[28] Saniga, E. M., Joint economically optimal design of X and R control charts, Management Science, 24, 4, 420-431 (1977) · Zbl 0371.62143
[29] Saniga, E. M., Joint economic design of X and R control charts with alternate process models, AIIE Transactions, 11, 3, 254-260 (1979)
[30] Saniga, E. M.; Montgomery, D. C., Economical quality control policies for a single cause system, AIIE Transactions, 13, 3, 258-264 (1981)
[31] Stoumbos, Z. G.; Reynolds, M. R., Economic statistical design of adaptive control schemes for monitoring the mean and variance: an application to analyzers, Nonlinear Analysis: RealWorld Applications, 6, 5, 817-844 (2005) · Zbl 1074.62078
[32] Rahim, M. A.; Costa, A. F.B., Joint economic design of \(\overline{X}\) and R charts under Weibull shock models, International Journal of Production Research, 38, 13, 2871-2889 (2000)
[33] Tagaras, G., A dynamic programming approach to the economic design of X-charts, IIE Transactions, 26, 3, 48-56 (1994)
[34] Tagaras, G., Dynamic control charts for finite production runs, European Journal of Operational Research, 91, 1, 38-55 (1996) · Zbl 0947.90515
[36] Tagaras, G.; Nikolaidis, Y., Comparing the effectiveness of various Bayesian \(\overline{X}\) control charts, Operations Research, 50, 5, 878-888 (2002) · Zbl 1163.62354
[37] Tasias, K. A.; Nenes, G., Joint monitoring of process mean and variance using variable parameter Shewhart charts, Computers and Industrial Engineering, 63, 4, 1154-1170 (2012)
[38] Taylor, H. M., Markovian sequential replacement processes, Annals of Mathematical Statistics, 36, 6, 1677-1694 (1965) · Zbl 0139.37802
[39] Taylor, H. M., Statistical control of a Gaussian process, Technometrics, 9, 1, 29-41 (1967) · Zbl 0147.16101
[40] von Collani, E., Economic control of continuously monitored production processes, Reports of Statistical Application Research, 34, 2, 1-18 (1987)
[41] von Collani, E., A unified approach to optimal process control, Metrika, 35, 1, 145-159 (1988) · Zbl 0709.90547
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