Multistate Bayesian Control Chart Over a Finite Horizon

We study a multistate partially observable process control model with a general state transition structure. The process is initially in control and subject to Markovian deterioration that can bring it to out‐of‐control states. The process may continue making transitions among the out‐of‐control states, or even back to the in‐control state until it reaches an absorbing state. We assume that at least one out‐of‐control state is absorbing. The objective is to minimize the expected total cost over a finite horizon. By transforming the standard Cartesian belief space into the spherical coordinate system, we show that the optimal policy has a simple control‐limit structure. We also examine two specialized models. The first is the phase‐type transition time model, in which we develop an algorithm whose complexity is not affected by the number of phases. The second is a model with multiple absorbing out‐of‐control states, by which we show that certain out‐of‐control states may incur less total cost than the in‐control state, a phenomenon never occurs in the two‐state models. We conclude that there are fundamental differences between multistate models and two‐state models, and that the spherical coordinate transformation offers significant analytical and computational benefits.