Optimal Repair Time Limit and Replacement Age for a System with Multiple Types of Failures

In industry, it is often needed to make a decision to repair or replace a system. For this problem, many maintenance models have been proposed in academia. However, for the sake of simplicity, most of these models assumed that the repair time is negligible, which is often inconsistent with reality. In this paper, considering non-zero repair time, we propose a new bivariate model of maintenance policy for a system with possible minor and catastrophic failures. In this model, the repair or replacement decision is made based on the system’s age and repair time for failure. Under this policy, the corrective replacement is implemented at a catastrophic failure or a minor failure whose repair time exceeds the limit; and the preventive replacement is conducted at age T. Only one replacement whichever occurs first will be implemented. The failure probabilities are age-varying, and the repair and failure costs depend on repair time. A realistic and generalized long-run expected cost rate model is developed, and we provide an application in the warranty field. To minimize the cost rate, the optimal maintenance policy is obtained through the Nelder–Mead method. A numerical example is given to illustrate its existence and uniqueness.