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STOCHASTIC SETUP-COST INVENTORY MODEL WITH BACKORDERS AND QUASICONVEX COST FUNCTIONS

Published online by Cambridge University Press:  04 April 2019

Eugene A. Feinberg Open the ORCID record for Eugene A. Feinberg [Opens in a new window]  and
Yan Liang
Eugene A. Feinberg
Affiliation:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY11794, USA E-mail: [email protected], [email protected]
Yan Liang
Affiliation:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY11794, USA E-mail: [email protected], [email protected]
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Abstract

This paper studies a periodic-review single-commodity setup-cost inventory model with backorders and holding/backlog costs satisfying quasiconvexity assumptions. We show that the Markov decision process for this inventory model satisfies the assumptions that lead to the validity of optimality equations for discounted and average-cost problems and to the existence of optimal (s, S) policies. In particular, we prove the equicontinuity of the family of discounted value functions and the convergence of optimal discounted lower thresholds to the optimal average-cost lower threshold for some sequence of discount factors converging to 1. If an arbitrary nonnegative amount of inventory can be ordered, we establish stronger convergence properties: (i) the optimal discounted lower thresholds converge to optimal average-cost lower threshold; and (ii) the discounted relative value functions converge to average-cost relative value function. These convergence results previously were known only for subsequences of discount factors even for problems with convex holding/backlog costs. The results of this paper also hold for problems with fixed lead times.

Keywords

average-cost optimality equationsinventory controlrelative value functions(s, S) policies
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Special Issue 3: Analytic Methods in Health Care and in Clinical Trials In Honor and Memory of Lycourgos (Lee) Papayanopoulos , July 2020 , pp. 429 - 468
Copyright
Copyright © Cambridge University Press 2019

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