The Bayesian process control with multiple assignable causes

TL;DR

This paper develops an optimal process control strategy for systems with multiple potential failure causes, using a POMDP framework, and derives structural properties and bounds for the control policy.

Contribution

It introduces a convex, non-increasing control limit policy for multi-cause process control and provides analytical bounds and an efficient algorithm for implementation.

Findings

Optimal control limit policy is proven to be convex and non-increasing.

Structural results enable reduced computational complexity.

Guidelines for selecting optimal sampling intervals are provided.

Abstract

We study an optimal process control problem with multiple assignable causes. The process is initially in-control but is subject to random transition to one of multiple out-of-control states due to assignable causes. The objective is to find an optimal stopping rule under partial observation that maximizes the total expected reward in infinite horizon. The problem is formulated as a partially observable Markov decision process (POMDP) with the belief space consisting of state probability vectors. New observations are obtained at fixed sampling interval to update the belief vector using Bayes' theorem. Under standard assumptions, we show that a conditional control limit policy is optimal and that there exists a convex, non-increasing control limit that partitions the belief space into two individually connected control regions: a stopping region and a continuation region. We further derive the analytical bounds for the control limit. An algorithm is devised based on structural results, which considerably reduces the computation. We also shed light on the selection of optimal fixed sampling interval.