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Multivariate Bayesian Control Chart

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8
makis{at}mie.utoronto.ca

A multivariate Bayesian control chart for monitoring process mean under the assumption that the vector of process observations follows a multivariate normal distribution is considered. Traditional control charts such as Hotelling's T², EWMA, and CUSUM charts have been applied to control industrial processes characterized by several measurable variables. It is well known that these traditional, non-Bayesian process control techniques are not optimal, but very few results regarding the structure of the Bayesian control policy have been reported in the literature, all dealing with the univariate, finite-horizon case. In this paper, we formulate the multivariate Bayesian process control problem in the optimal stopping framework. The objective is to find a stopping rule under partial observations, minimizing the long-run expected average cost per unit time for a given sample size and sampling interval. Under standard operating and cost assumptions, it is proved that a control limit policy is optimal, and an algorithm is presented to find the optimal control limit and the minimum average cost.

Subject classifications: reliability; quality control; replacement/renewal; dynamic programming/optimal control; models; statistics; Bayesian.
History: Received September 2005; revision received June 2007; accepted September 2007.