A discrete semi-Markov decision model to determine the optimal repair/replacement policy under general repairs

The state of a machine (system) that may experience failures is characterized by the real age of the machine and the number of failures incurred to date. On failure, the unit may undergo a repair which can partially reset the failure intensity of the unit. In this paper, it is assumed that repairs can, at most, shift back the failure intensity so as to remove the most recent run-time sojourn. The other alternative at a failure is to conduct a major overhaul that serves to refresh the failure intensity of the unit. General cost structures, depending upon both real age and number of failures, are permitted. The decision, on failure, to repair or renew is formulated as a discrete semi-Markov decision process with real age and number of failures as the state space. Optimal decisions are of the threshold type. That is, on the nth failure, if the real age is above a predetermined threshold value, refresh; otherwise conduct an imperfect repair.