Average Cost Optimality in Partially Observed Lost-Sales Inventory Systems

25 Pages Posted: 15 Jun 2021

See all articles by Xingyu Bai

Xingyu Bai

University of Illinois at Urbana-Champaign, Industrial Enterprise and Systems Engineering

Xin Chen

University of Illinois at Urbana-Champaign

Alexander Stolyar

University of Illinois at Urbana-Champaign

Date Written: June 12, 2021

Abstract

We consider a partially observable inventory system in which the inventory level can only be observed when it reaches zero, the unmet demand is lost, and replenishment orders must be decided so as to minimize the long-run average cost. This problem has an infinite-dimensional state space and is therefore difficult to analyze. We prove the existence of an average-cost optimal policy, using the so-called vanishing discount factor approach. As a main methodological contribution, we provide a way to verify the key condition -- the uniform boundedness of the relative discounted value function -- for the partially observed system. To accomplish that we construct -- for the partially observed system -- a valid policy which in a certain sense "copies" the actions of another policy for the process with a different initial state. To our best knowledge, this paper is the first dealing with inventory models with partially observable inventory levels under the long-run average cost criterion.

Keywords: Markov decision process, average cost, optimality inequality, partial observations, lost sales

Suggested Citation

Bai, Xingyu and Chen, Xin and Stolyar, Alexander, Average Cost Optimality in Partially Observed Lost-Sales Inventory Systems (June 12, 2021). Available at SSRN: https://ssrn.com/abstract=3865728 or http://dx.doi.org/10.2139/ssrn.3865728

Xingyu Bai (Contact Author)

University of Illinois at Urbana-Champaign, Industrial Enterprise and Systems Engineering ( email )

Urbana-Champaign, IL
United States
2177217269 (Phone)

Xin Chen

University of Illinois at Urbana-Champaign ( email )

601 E John St
Champaign, IL 61820
United States

Alexander Stolyar

University of Illinois at Urbana-Champaign

ISE Department and Coordinated Science Lab
1308 W. Main Street, 156CSL
Urbana, IL 61801
United States

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