An Inclusive Solution to the Inventory Problem

This paper gives a unified approach to a set of inventory problems subsumed under the classical inventory problem proposed by Arrow, Harris, and Marschak. This set includes those problems with periodic reordering, proportional costs and revenues, fixed time lags in receipt of an order, and a static criterion of optimality. Within this class shortages may be either back ordered or forgotten, the time lag can be any fixed integral number of periods, including zero, and demands in the various periods may follow any known joint distribution. The method yields a solution for the forgotten shortages case with time lag, as well as the familiar results for the instantaneous case and the back-order lag case. The approach can be modified to allow for a setup cost in purchasing.