Renewal Decision Problem-Random Horizon

A system must operate for T units of time. T is a random variable with known distribution function F. A certain component is essential for the system’s operation and when it fails must be replaced with a new component. There are n possible types of replacements. An unlimited supply of each type is assumed. A type i replacement costs c_i (c_i > 0) and functions independently of T for an exponentially distributed length of time with rate λ_i. The problem is to assign the replacements from among the various possible types so as to minimize the expected total cost of providing an operative component for the entire life of the system. The principal result of this paper, generalizing previous work where T was assumed to have a degenerate or truncated exponential distribution, is that if F is an increasing failure rate function (IFR) the optimal replacement policy has a simple intuitive interval structure. An algorithm for finding the optimal policy is indicated. Some results are obtained for the case where component life distributions are not exponential.