A Little's result approach to the service constrained spares provisioning problem for repairable items

A set of identical machines are deployed to meet a known and constant demand. If a machine fails, a replacement part must be available before repair of the machine may be initiated. If the part is currently out of stock, it must be ordered. Once repaired at one of a finite number of repair stations, the machine serves as an ‘operational ready’ standby if demand is currently being met; otherwise, the machine is immediately deployed. Machine time-to-failure, ordering leadtimes, and repair times are all assumed to be exponentially distributed. The objective is to determine the number of machines and repair channels that minimize a cost function subject to the service constraint; i.e., on average, the number of machines operating should be at least some fraction of the demand. We present an algorithm that efficiently generates all the boundary points of the feasible region, from which the optimal solution is readily identified, for the special cases in which there is either zero or infinite stock. Use of the algorithm, which is based upon Little's result and first-passage time analysis, requires negligible storage and computational effort. In fact, for problems of moderate size, computation of optimal solutions via a nonprogrammable pocket calculator is feasible.