A General Failure Model: Optimal Replacement with State Dependent Replacement and Failure Costs

Abstract

We shall consider the optimal replacement of a unit which is subject to random failure. It is assumed that there exists a hazard rate process rt, adapted to the history of the unit and, for a fixed t, denoting the conditional probability density of failure at t, given that the unit has not failed until t. If the unit is functioning at t, it can be replaced by a new one and a random replacement cost ct depending on the history of the unit up to t is incurred. If the unit fails at t, then it is replaced, and in addition to the replacement cost, a failure cost Kt, depending on the history of the unit, is incurred. The optimality criterion is the minimization of the total long run average cost per unit time. Our model generalizes the model of Bergman (Bergman, B. 1978. Optimal replacement under a general failure model. Adv. Appl. Probab. 10 431–444.) by allowing the state-dependence of the replacement and failure costs. The main result of the present paper states that there exists an optimal replacement rule, and that it is a threshold rule for the process γt = dct/dt + Ktrt. This means that there exists a* < ∞, such that an optimal replacement rule is to replace at the time when the process rt (assumed to be nondecreasing) reaches the level a*, or if failure occurs before this, to replace at failure. An iterative procedure is given for the determination of the threshold value a*.