Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not ‘as good as new’. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long‐run average cost rate of the system. We give the explicit expressions for the limiting availability and the long‐run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long‐run average cost rate C(N), we theoretically obtain the optimal replacement policies N ^* in both cases. Finally, some numerical simulations are presented to verify the theoretical results.