Optimal Control of Brownian Inventory Models with Convex Inventory Cost: Discounted Cost Case

TL;DR

This paper analyzes a Brownian inventory model with convex costs, establishing the optimal control band policy using a three-step lower-bound approach and solving a free boundary problem for discounted costs.

Contribution

It introduces a novel three-step lower-bound method and proves the optimality of a four-parameter control band policy for the discounted cost case.

Findings

Optimal control band policy is proven to be optimal among all feasible policies.

Existence and uniqueness of the solution to the free boundary problem are established.

An algorithm for computing the parameters of the optimal policy is provided.

Abstract

We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy.