Finite horizon semi‐Markov decision processes with application to maintenance systems

This paper investigates finite horizon semi‐Markov decision processes with denumerable states. The optimality is over the class of all randomized history‐dependent policies which include states and also planning horizons, and the cost rate function is assumed to be bounded below. Under suitable conditions, we show that the value function is a minimum nonnegative solution to the optimality equation and there exists an optimal policy. Moreover, we develop an effective algorithm for computing optimal policies, derive some properties of optimal policies, and in addition, illustrate our main results with a maintenance system.