A weak convergence approach to inventory control using a long-term average criterion. (29th November 2018)

• Record Type: Journal Article
• Authors: Helmes, K. L.; Stockbridge, R. H.; Zhu, C.
• Is Part Of: Advances in applied probability. Volume 50:Number 4(2018)
• Page Start: 1032
• Page End: 1074
• Publication Date: 2018-11-29
• Subjects: Inventory, -- impulse control, -- long-term average cost, -- general diffusion model, -- (𝑠, 𝑆) policy, -- weak convergence; Primary 93E20, -- Secondary 90B05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.50
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: In this paper we continue the examination of inventory control in which the inventory is modeled by a diffusion process and a long-term average cost criterion is used to make decisions. The class of such models under consideration has general drift and diffusion coefficients, and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. The conditions on the cost functions are greatly relaxed from those in Helmes et al. (2017). Characterization of the cost of a general ( s, S ) policy as a function 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 under very weak conditions; nonexistence of an optimizing pair is also discussed. Using average expected occupation and ordering measures and weak convergence arguments, weak conditions are given for the optimality of the ( s *, S * ) ordering policy in the general class of admissible policies. The analysis involves an auxiliary function that is globally C 2 and which, together with the infimal cost, solves a particular system of linear equations and inequalities related to but different from the long-term average Hamilton‒Jacobi‒Bellman equation. This approach provides an analytical solution to the problem rather than a solution involving intricate analysis of the stochastic processes. The range of applicability of these results is illustrated on a drifted Brownian motion inventory model, both unconstrained and reflected, and on a geometric Brownian motion inventory model under two different cost structures.