Joint pricing and inventory control for a stochastic inventory system with Brownian motion demand

TL;DR

This paper develops an optimal joint pricing and inventory control policy for a stochastic inventory system with demand modeled as Brownian motion, balancing revenue and costs to maximize long-term profit.

Contribution

It introduces an $(s^*,S^*,p^*)$ policy framework for joint control, deriving equations for optimal parameters and analyzing price dependence on inventory levels.

Findings

Optimal policy is of $(s^*,S^*,p^*)$ form.

Optimal price depends on current inventory level.

Price increases with inventory in a certain range, decreases beyond a threshold.

Abstract

In this paper, we consider an infinite horizon, continuous-review, stochastic inventory system in which cumulative customers' demand is price-dependent and is modeled as a Brownian motion. Excess demand is backlogged. The revenue is earned by selling products and the costs are incurred by holding/shortage and ordering, the latter consists of a fixed cost and a proportional cost. Our objective is to simultaneously determine a pricing strategy and an inventory control strategy to maximize the expected long-run average profit. Specifically, the pricing strategy provides the price $p_t$ for any time $t\geq0$ and the inventory control strategy characterizes when and how much we need to order. We show that an $(s^*,S^*,p^*)$ policy is optimal and obtain the equations of optimal policy parameters, where $p^*=\{p_t^*:t\geq 0\}$. Furthermore, we find that at each time $t$, the optimal price $p_t^*$ depends on the current inventory level $z$, and it is increasing in $[s^*,z^*]$ and is decreasing in $[z^*,\infty)$, where $z^*$ is a negative level.