Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30477
##### Optimal Bayesian Control of the Proportion of Defectives in a Manufacturing Process

Abstract:

In this paper, we present a model and an algorithm for the calculation of the optimal control limit, average cost, sample size, and the sampling interval for an optimal Bayesian chart to control the proportion of defective items produced using a semi-Markov decision process approach. Traditional p-chart has been widely used for controlling the proportion of defectives in various kinds of production processes for many years. It is well known that traditional non-Bayesian charts are not optimal, but very few optimal Bayesian control charts have been developed in the literature, mostly considering finite horizon. The objective of this paper is to develop a fast computational algorithm to obtain the optimal parameters of a Bayesian p-chart. The decision problem is formulated in the partially observable framework and the developed algorithm is illustrated by a numerical example.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1127922

References:

[1] W.H. Woodall, “The use of control charts in healthcare and public health surveillance,” J. Quality Tech., vol. 38, 2006, pp. 89-104.
[2] M.J. Kim, R. Jiang, V. Makis, and C.G. Lee, “Optimal Bayesian fault prediction scheme for a partially observable system subject to random failure,” European Journal of Operational Research, vol.214, 2011, pp. 331-339.
[3] J.B. Jumah, R.P. Burt, and B. Buttram, “An exploration of quality control in banking and finance,” International Journal of Business and Social Science, vol.3, 2012, pp. 273-277.
[4] T.S. Vaughan, “Variable sampling interval np process control chart,” Comm. Statist.-Theory Meth., vol. 22, 1993, pp. 147-167.
[5] B. Sengupta,“The exponential riddle,” J. Appl. Probab., vol. 19, 1982, pp. 737-740.
[6] I. Kooli, and M. Limam, “Economic Design of an Attribute np Control Chart Using a Variable Sample Size,” Sequential Analysis, vol. 30, 2011, pp. 145-159.
[7] M.A. Girshik, and H. Rubin “A Bayes’ approach to a quality control model,” Ann.Math.Statist. , vol. 23, 1952, pp. 114-125.
[8] J.E. Eckles, “ Optimum maintenance with incomplete information”, Oper. Res, vol.16, 1968, pp. 1058-1067.
[9] S.M. Ross,“Quality control under Markovian deterioration,” Management Sci., vol. 17, 1971, pp. 587-596.
[10] C.C. White, “A Markov quality control process subject to partial observation,” Management Sci., vol. 23, pp. 843-852.
[11] H.M. Taylor, “ Markovian sequential replacement processes,” Ann. Math. Statist, vol. 36, 1965, pp. 1677-1694.
[12] H.M. Taylor, “Statistical control of a Gaussian process,” Technometrics, vol. 9, 1967, pp. 29-41.
[13] J.M. Calabrese, “Bayesian process control for attributes,” Management Sci., vol. 41, 1995, pp. 637-645.
[14] G. Tagaras, “A dynamic programming approach to the economic design of X-charts,” IIE Transaction, vol. 26, 1994, pp. 48-56.
[15] G. Tagaras, “Dynamic control charts for finite production runs,” European Journal of Operational Research, vol. 91, 1996, pp. 38-55.
[16] E.L. Porteus, and A. Angelus, “Opportunities for improved statistical process control,” Management Sci., vol. 43, 1997, pp. 1214-1229.
[17] G. Tagaras, and Y. Nikolaidis, “Comparing the effectiveness of various Bayesian X control charts,” Operations Research, vol. 50, 2002, pp. 878-888.
[18] V. Makis, “Multivariate Bayesian control chart,” Operations Research, vol. 56, 2008, pp. 487-496.
[19] V. Makis, ”Multivariate Bayesian process control for a finite production run,” European Journal of Operational Research, vol. 194, 2009, pp. 795-806.
[20] W.H. Woodall, “ Weakness of the economic design of control charts”, Technometrics, vol. 28, 1986, pp. 408409.
[21] E.M. Saniga,“ Economic statistical control chart designs with an application to ¯X and R control charts. Technometrics, vol. 31, 1989, pp. 313320.
[22] W.E. Molnau, D.C. Montgomery, G.C. Runger, “Statistically constrained economic design of the multivariate exponentially weighted moving average control chart,” Qual. Reliab. Engrg. Int., vol. 17, 2001, 39-49.
[23] G. Tagaras, “ A survey of recent developments in the design of adaptive control charts”, J. Quality Tech, vol. 30, 1998, pp. 212231.
[24] V. Makis, L. Jafari, and F. Naderkhani ZG, “Optimal Bayesian Control Chart for the Proportion of Defectives,” Proceedings of the 13th Viennese Conference on Optimal Control and Dynamic Games, Rome, Italy, April 2015.
[25] H.C. Tijms, “Stochastic models- an algorithmic approach”, John Wiley & Sons, 1994.