Multivariate Bayesian process control for a finite production run

• Published in: European journal of operational research Vol. 194; no. 3; pp. 795 - 806
• Main Author: Makis, Viliam
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.05.2009; Elsevier; Elsevier Sequoia S.A
• Series: European Journal of Operational Research
• Subjects: Applied sciences; Bayesian analysis; Comparative analysis; Control charts; Correlation analysis; Cost control; Decision theory. Utility theory; Dynamic programming; Exact sciences and technology; Finite horizon; Inventory control, production control. Distribution; Maintenance; Manufacturing; Markov analysis; Multivariate analysis; Multivariate quality control; Multivariate quality control Maintenance Partially observable Markov decision process Dynamic programming Finite horizon; Operational research and scientific management; Operational research. Management science; Partially observable Markov decision process; Process controls; Random variables; Studies; Finite horizon; Multivariate quality control; Maintenance; Dynamic programming; Partially observable Markov decision process; Markov process; Industrial product; Markov decision; Production quality; Sample size; Optimal policy; Control synthesis; Random vector; Multivariate analysis; Economic sciences; Optimization; Optimal design; Sampling; Random variable; Bayes estimation; Statistical process control; Multivariate process; Hotelling test; Control chart; Statistical analysis; Critical value; Process control; Operating cost; Cost control; Optimal control; Quality control
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2008.01.001

Most industrial products and processes are characterized by several, typically correlated measurable variables, which jointly describe the product or process quality. Various control charts such as Hotelling’s T 2, EWMA and CUSUM charts have been developed for multivariate quality control, where the values of the chart parameters, namely the sample size, sampling interval and the control limits are determined to satisfy given economic and/or statistical requirements. It is well known that this traditional non-Bayesian approach to a control chart design is not optimal, but very few results regarding the form of the optimal Bayesian control policy have appeared in the literature, all limited to a univariate chart design. In this paper, we consider a multivariate Bayesian process mean control problem for a finite production run under the assumption that the observations are values of independent, normally distributed vectors of random variables. The problem is formulated in the POMDP (partially observable Markov decision process) framework and the objective is to determine a control policy minimizing the total expected cost. It is proved that under standard operating and cost assumptions the control limit policy is optimal. Cost comparisons with the benchmark chi-squared chart and the MEWMA chart show that the Bayesian chart is highly cost effective, the savings are larger for smaller values of the critical Mahalanobis distance between the in-control and out-of-control process mean.