A continuous review inventory model with stochastic prices procured in the spot market

Not only the amount of product demanded, but also the price of the product have a strong impact on a manufacturer's revenue. In this paper we consider a continuous-time inventory model where the spot price of the product stochastically fluctuates according to a Brownian motion. Should information on the spot price be available, the manufacturer would wish to buy the product on the spot market when profitable. The purpose of this paper is to find an optimal procurement policy so as to minimize total expected discounted costs over an infinite planning horizon. We extend the Sulem (1986) model into one in which the market price of the product follows a geometric Brownian motion. By applying this, we obtain the optimal cost as a solution to a quasi-variational inequality, and show that there exists an optimal procurement solution as an (s,S) policy. We clarify the dependence of the optimal ( s,S) policy on the spot price at the procurement epoch. These values of the (s,S) policy can be used and revised in the following ordering cycles. Finally, some numerical examples are provided to investigate the analytical properties of the expected cost function as well as of the optimal policy.