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We study an optimal process control problem with multiple assignable causes.The process is initially in-control but is subject to random transition to oneof multiple out-of-control states due to assignable causes. The objective is tofind an optimal stopping rule under partial observation that maximizes thetotal expected reward in infinite horizon. The problem is formulated as apartially observable Markov decision process (POMDP) with the belief spaceconsisting of state probability vectors. New observations are obtained at fixedsampling interval to update the belief vector using Bayes' theorem. Understandard assumptions, we show that a conditional control limit policy isoptimal and that there exists a convex, non-increasing control limit thatpartitions the belief space into two individually connected control regions: astopping region and a continuation region. We further derive the analyticalbounds for the control limit. An algorithm is devised based on structuralresults, which considerably reduces the computation. We also shed light on theselection of optimal fixed sampling interval.

The Bayesian process control with multiple assignable causes