Optimal maintenance procedure for multi-state deteriorated system with incomplete monitoring

The optimal replacement problem was investigated for a multi-state deteriorated system for which the true internal state cannot be observed directly except when the system breaks down completely. The internal state was assumed to be monitored incompletely by a monitor that gives information related to the true state of the system. The problem was formulated as a partially observable Markov decision process. The optimal procedure was found to be a monotone procedure with respect to stochastic increasing ordering of the state probability vectors under some assumptions. Limiting the optimal procedure to a monotone procedure would greatly reduce the tremendous amount of calculation time required to find the optimal procedure.