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June 2017 Continuous inventory models of diffusion type: Long-term average cost criterion
Kurt L. Helmes, Richard H. Stockbridge, Chao Zhu
Ann. Appl. Probab. 27(3): 1831-1885 (June 2017). DOI: 10.1214/16-AAP1247

Abstract

This paper establishes conditions for optimality of an $(s,S)$ ordering policy for the minimization of the long-term average cost of one-dimensional diffusion inventory models. The class of such models under consideration have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. Characterization of the cost of a general $(s,S)$ policy as a function $F$ of two variables naturally leads to a nonlinear optimization problem over the ordering levels $s$ and $S$. Existence of an optimizing pair $(s_{*},S_{*})$ is established for these models. Using the minimal value $F_{*}$ of $F$, along with $(s_{*},S_{*})$, a function $G$ is identified which is proven to be a solution of a quasi-variational inequality provided a simple condition holds. At this level of generality, optimality of the $(s_{*},S_{*})$ ordering policy is established within a large class of ordering policies for which local martingale and transversality conditions involving $G$ hold. For specific models, optimality of an $(s,S)$ policy in the general class of admissible policies can be established using comparison results. This most general optimality result is shown for the classical drifted Brownian motion inventory model with holding and fixed plus proportional ordering costs. Optimality of an $(s,S)$ ordering policy is also extended to the general class of admissible policies for a geometric Brownian motion inventory model with fixed plus level-dependent ordering costs. However, for a drifted Brownian motion process with reflection at $\{0\}$, a new class of non-Markovian policies is introduced which have lower costs than the $(s,S)$ policies. In addition, interpreting reflection at $\{0\}$ as “just-in-time” ordering, a necessary and sufficient condition is given that determines when just-in-time ordering is better than traditional $(s,S)$ policies.

Citation

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Kurt L. Helmes. Richard H. Stockbridge. Chao Zhu. "Continuous inventory models of diffusion type: Long-term average cost criterion." Ann. Appl. Probab. 27 (3) 1831 - 1885, June 2017. https://doi.org/10.1214/16-AAP1247

Information

Received: 1 October 2015; Revised: 1 September 2016; Published: June 2017
First available in Project Euclid: 19 July 2017

zbMATH: 1369.93720
MathSciNet: MR3678486
Digital Object Identifier: 10.1214/16-AAP1247

Subjects:
Primary: 60H30 , 90B05 , 93E20
Secondary: 60H30 , 90B05

Keywords: $(s,S)$ policies , general diffusion models , impulse control , inventory , long-term average cost

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 3 • June 2017
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