Optimal preventive replacement last policy for a successive random works system with random lead time

Most systems deteriorate with age and use, and eventually, fail from either or both causes in random environment. It is reasonable in commercial industries that only one spare item is available for replacement. Selecting a cost efficient and effective replacement strategy to improve system reliability and inventory of spare parts is crucial work. In this paper, we consider an operating system which works at successive random times and models an imperfect maintenance action under two types of age-dependent failures, repairable minor failures (R) and unrepairable catastrophic failures (U). R failures are followed by a minimal repair and U failures are removed by a corrective replacement. A notation of preventive replacement last policy is considered in which the system is replaced before any U failures at age T or at number N of working times, whichever occurs last. A spare unit for replacement can be delivered upon order and is available only when the random lead time is finished. The main objective is to determine an optimal schedule (T, N) of that minimizes the mean cost rate function in a finite time horizon. The existence and uniqueness of optimal PRL policy are derived analytically and computed numerically.