The ($\textsc{k}, t_p$) Replacement Policy for the System subject to Two Types of Failure

In this paper, we consider a new preventive replacement policy for the system which deteriorates while it is in operation with an increasing failure rate. The system is subject to two types of failure. A type 1 failure is repairable while a type 2 failure is not repairable. In the new policy, a system is replaced at the age of $t_p$ or at the instant the$\textsc{k}^{th}$ type 1 failure occurs, whichever comes first. However, if a type 2 failure occurs before a preventive replacement is performed, a failure replacement should be made. We assume that a type 1 failure can be rectified with a minimal repair. We also assume that a replacement takes a non-negligible amount of time while a minimal repair takes a negligible amount of time. Under a cost structure which includes a preventive replacement cost, a failure replacement cost and a minimal repair cost, we develop a model to find the optimal ($\textsc{k},t_p$) policy which minimizes the expected cost per unit time in the long run while satisfying a system availability constraint.