Optimal Replacement Policies for a Single Loaded Sliding Standby

The purpose of this paper is to define an optimal replacement policy for identical components performing different functions in a given system, when one spare part is available. It is assumed that the failure law of the component and of the spare part is known and that the mission of the system has a duration T, that the components cannot be permuted and that the breakage of a given component, i, produces a certain loss L i per unit of time it remains out of use. The optimal policy r*(x) is first computed for the two components--one spare part case by minimizing the expected loss over that set of policies consisting of replacing, if possible, the most expensive component, when it fails and waiting until a certain time \tau (x) for replacing the other component should it fail at time x. For exponential failure laws and with certain restrictions, concave laws, \tau *(x) = \tau * is independent of x for x \leqq r* and is equal to x for x > \tau *. The n components case is then considered in which the optimal policy is defined by a sequence \tau i *(x)(i = 1,..., n), where \tau i *(x) represents the time at which the replacement of component i by the spare part will occur, should component i break at time x and no other component, whose loss is more expensive, break before \tau i *(x).