On the optimality of repair-cost-limit policies

An optimal repair/replacement problem for a single-unit repairable system with minimal repair and random repair cost is considered. The existence of the optimal policy is established using results of the optimal stopping theory, and it is shown that the optimal policy is a ‘repair-cost-limit’ policy, that is, there is a series of repair-cost-limit functionsgn(t),n= 1, 2,…, such that a unit of agetis replaced at thenth failure if and only if the repair costC(n,t) ≥gn(t); otherwise it is minimally repaired. If the repair cost does not depend onn, then there is a single repair cost limit functiong(t), which is uniquely determined by a first-order differential equation with a boundary condition.