Optimal Inventory Policies when Purchase Price and Demand Are Stochastic

In this paper we consider the problem of a firm that faces a stochastic (Poisson) demand and must replenish from a market in which prices fluctuate, such as a commodity market. We describe the price evolution as a continuous stochastic process and we focus on commonly used processes suggested by the financial literature, such as the geometric Brownian motion and the Ornstein‐Uhlenbeck process. It is well known that under variable purchase price, a price‐dependent base‐stock policy is optimal. Using the single‐unit decomposition approach, we explicitly characterize the optimal base‐stock level using a series of threshold prices. We show that the base‐stock level is first increasing and then decreasing in the current purchase price. We provide a procedure for calculating the thresholds, which yields closed‐form solutions when price follows a geometric Brownian motion and implicit solutions under the Ornstein‐Uhlenbeck price model. In addition, our numerical study shows that the optimal policy performs much better than inventory policies that ignore future price evolution, because it tends to place larger orders when prices are expected to increase.