The Nonstationary Infinite Horizon Inventory Problem

By allowing disposal in each period, a sequence of upper and lower bounds on the infinite horizon nonstationary periodic review inventory problem is obtained. The nth bounds depend only on knowledge of the demand distributions in the first n periods, giving planning horizon results. In general, calculation of any bound requires solution of an ordinary n period problem, although for the proportional cost case the solution for n = 1 and good approximations for n = 2 may easily be given. The upper and lower relative cost and policy bounds are monotonic in the horizon length and, under mild conditions, converge to a unique infinite horizon solution. An analytic expression is given for the asymptotic convergence rate, which is geometric even for the undiscounted case. The formulation given here is a concrete example of the general planning horizon formulation for infinite horizon dynamic programming models set forth in [Morton, T. 1975. Infinite horizon dynamic programming models--a planning horizon formulation. Carnegie-Mellon University Management Sciences Research Report No. 372, August (to appear in Operations Res.)].