Optimal replacement period with cumulative repair cost limit under a cumulative damage model

In this paper, a periodic replacement policy for a system under a cumulative shock damage process is presented; this policy includes the concept of a cumulative repair cost limit. When an external shock occurs, this shock can be divided into two types. Each type-1 shock causes some damage to the system; these damages are additive, and the system goes into serious failure when the total damage has exceeded a failure level K. A type-2 shock causes the system to experience a minor failure that can be corrected by minimal repair. When a minor failure occurs, the repair cost will be evaluated, and the minimal repair is executed if the accumulated repair cost is less than a predetermined limit L. The system is replaced anticipatively at scheduled time T, or at the jth minor failure (j < n) at which the accumulated repair cost exceeds a predetermined limit L, or any serious failure. The long-term expected cost per unit time is formulated by introducing relative costs. By minimising that cost, the optimal period T* is also verified finite and unique under certain conditions.