Geometric process model for a system with inspections and preventive repair

• Modeling a system with periodic inspections and preventive repair under GP repair.
• Proved the existence and uniqueness of the optimal inspection/replacement policy.
• An algorithm is provided to find the optimal bivariate policy (T∗, N∗).
• Some sensitive analyses are provided about some parameters.

In this paper, the optimal repair–replacement problem is studied for a simple repairable system whose failures are detected only by periodic inspections. When a failure of the system is detected, it will be repaired by a repairman, and the lifetimes of the system after repair are geometrically decreasing. If the system is detected to be in working state, it will be preventively repaired, and the preventive repair brings the system to the state just as that of the beginning of the working cycle. A bivariate policy (T,N) is considered, where T is the time interval between inspections while N   is the number of failures at which the system is replaced. The objective is to determine the optimal bivariate policy (T∗,N∗) such that the long-run average cost rate C(T,N) is minimized. Sufficient conditions for the existence and uniqueness of the optimal policies N∗N∗ and T∗T∗ are derived theoretically. An algorithm is also presented to find the optimal policy (T∗,N∗). Finally, a numerical example is given to validate the developed theoretical model. Some sensitivity analysis are provided to show the influence of the varying of some parameters on the optimal policy (T∗,N∗).