Dynamic Inventory Models and Stochastic Programming

A wide class of single-product, dynamic inventory problems with convex cost functions and a finite horizon is investigated as a stochastic programming problem. When demands have finite discrete distribution functions, we show that the problem can be substantially reduced in size to a linear program with upper-bounded variables. Moreover,, we show that the reduced problem has a network representation; thus network flow theory can be used for solving this class of problems. A consequence of this result is that, if we are dealing with an indivisible commodity, an integer solution of the dynamic inventory problem exists. This approach can be computationally attractive if the demands in different periods are correlated or if ordering cost is a function of demand. Introduction The purpose of this paper is twofold, to formulate a wide class of dynamic inventory models as a stochastic programming problem that can be reduced to a linear program with upper-bounded variables and to show that the special structure of the problem allows the use of network flow solution techniques in preference to the simplex method of linear programming. To keep the exposition in terms of one particular model we restrict our initial discussion to the single-commodity, multiperiod inventory model with no back orders and with a finite horizon. Later, attention is turned to other models to which our approach applies. The model to be considered in detail is described as follows: A decision maker can procure a single item in any of a finite number of time periods. We assume that procurements are accomplished immediately and that the distribution of the demand for the item is known for each period. The decision maker balances the discrepancy between stock on hand and the actual demand either by holding the item in inventory or by emergency purchasing. His economic problem arises because he can expect to accrue savings by buying in one period, while facing the demand uncertainty, and then holding the item in inventory for future periods. The problem is formulated in the next section as a stochastic program and assumptions are made about the different cost elements. Then we focus attention on the discrete demand-distribution case and formulate the