Optimal Ordering Policy for Inventory Systems with Quantity-Dependent Setup Costs

TL;DR

This paper develops an optimal ordering policy for a continuous-review inventory system with quantity-dependent setup costs, using a lower bound approach and a comparison theorem to handle technical challenges, including non-smooth costs.

Contribution

It introduces a novel lower bound approach and a comparison theorem to establish global optimality in inventory models with complex setup costs.

Findings

Derived the optimal policy minimizing long-run average costs.

Established a comparison theorem for policy optimality.

Proposed a method for computing policy parameters with step-function setup costs.

Abstract

We consider a continuous-review inventory system in which the setup cost of each order is a general function of the order quantity and the demand process is modeled as a Brownian motion with a positive drift. Assuming the holding and shortage cost to be a convex function of the inventory level, we obtain the optimal ordering policy that minimizes the long-run average cost by a lower bound approach. To tackle some technical issues in the lower bound approach under the quantity-dependent setup cost assumption, we establish a comparison theorem that enables one to prove the global optimality of a policy by examining a tractable subset of admissible policies. Since the smooth pasting technique does not apply to our Brownian inventory model, we also propose a selection procedure for computing the optimal policy parameters when the setup cost is a step function.