Knowledge that Transforms

In this note we study dynamic inventory problem for a follow-on provisioning in which program length is subject to uncertainty with a known distribution. It is shown that under rather general cost conditions, optimal policy is of (S, s) type. This is true whether or not there exists a time lag in delivery provided that excess demand is always backlogged. The case of an infinite program horizon is also briefly discussed. MODERN INVENTORY theory has been a relatively recent development, but its brief existence has proceeded at least along two fronts: theory and applications. The earliest work falling under this theory is that by Masse [6], followed by those of Arrow, Harris, and Marschak [1], Dvoretzky, Kiefer and Wolfowitz [5], Bellman, Glicksberg and Gross [4], and Modigliani and Hohn [7]. An excellent account of historical back ground of this theory was given by Arrow [2], and its applications to a great variety of economic and business may be found in many journals in such fields as operations research, management science, and production control. A detailed discussion of the nature and structure of inventory problems was given by Arrow, Karlin, and Scarf [2, Chap. 2], and a simple mathematical exposition of theory may also be found in Bellman [3, Chap. 5]. Briefly, problem involves determination of (optimal) stock levels for inventories which extend over a sequence of time periods and are subject to fluctuating demand in each such period. Such may arise in a number of ways, e.g., in scheduling production or determining distribution of commodities over certain markets, in finding replacement policy for aged equipment, or in combinations of some or all of these features. The treatment of demand in modern inventory theory is usually handled in two ways: (1) time periods are regularly spaced, and demand in each period is a random variable with a known probability distribution; or (2) size of each demand is fixed but times at which successive demands occur are random variables. We study here an inventory problem which is in some sense a hybrid between two and which occurs frequently in involving follow-on provisioning. (Follow-on provisioning is a subsequent provisioning of same item from same supply source.) Here, time periods are equally spaced (corresponding to budget cycles) and demand in each period is a random variable subject to a known