Optimal replacement of a system with imperfect repair

We consider a deteriorating system subject to random failure. At failure the system can be replaced or can undergo a repair with the following three possible outcomes: as good as new, returned to the functioning state just prior to failure or the system must be scrapped and replaced at additional replacement cost. We assume that the repair cost increases with the system's age and that both maintenance actions take negligible time. Viewing the repair/replacement process as a semi-Markov decision process, we prove that under certain assumptions the optimal replacement policy is an age-dependent policy proposed by Muth (5) for a system with constant repair cost and minimal repairs. The average cost function is then derived which can be minimized to obtain the optimal replacement time T. The generalizes previously obtained results for repairable systems with minimal repair.