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A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems

Published online by Cambridge University Press:  14 July 2016

Yu-Sheng Zheng
Yu-Sheng Zheng*
Affiliation:
University of Pennsylvania
*
∗ Postal address: Decision Sciences Department, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA.
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Abstract

The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.

Keywords

DISCOUNTED COSTAVERAGE COSTUNBOUNDED ONE-STEP COST
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 802 - 810
Copyright
Copyright © Applied Probability Trust 1991 

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