The Nonstationary Infinite Horizon Inventory Problem

Thomas E. Morton

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Supply Chain and Inventory Management Management and Optimization Techniques Optimization and Mathematical Programming

Abstract

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.)].

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References

1960Dynamic Inventory Policy with Varying Stochastic Demands 1966On the Opimality of $( {s,S} )$ Inventory Policies: New Conditions and a New Proof 1969Cash Balance and Simple Dynamic Portfolio Problems with Proportional Costs 1971The Near-Myopic Nature of the Lagged-Proportional-Cost Inventory Problem with Lost Sales 1960OPTIMAL POLICY FOR DYNAMIC INVENTORY PROCESS WITH NON-STATIONARY STOCHASTIC DEMANDS 1960Optimal Policy for Dynamic Inventory Process with Stochastic Demands Subject to Seasonal Variations

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