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

In this article, we consider an infinite horizon, continuous-review, stochastic inventory system in which the 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 >= 0 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 >= 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 decreasing in [z*,infinity), where z* is a negative level.