A minimal repair replacement model with two types of failure and a safety constraint

Consider a system subject to two types of failures. If the failure is of type 1, the system is minimally repaired, and a cost C1 is incurred. If the failure is of type 2, the system is minimally repaired with probability p and replaced with probability 1−p  . The associated costs are C2,mC2,m and C2,rC2,r, respectively. Failures of type 2 are safety critical and to control the risk, management has specified a requirement that the probability of at least one such failure occurring in the interval [0, A] should not exceed a fixed probability limit ω. The problem is to determine an optimal planned replacement time T, minimizing the expected discounted costs under the safety constraint. A cost Cr is incurred whenever a planned replacement is performed. Conditions are established for when the safety constraint affects the optimal replacement time and causes increased costs.