A Continuous-time Inventory Model with Procurement from Spot Market

Abstract Not only the amount of product demand but also the prices 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 of the spot price be available, the manufacturer wishes to buy the product from the spot market if profitable. The purpose of this paper is to find an optimal procurement policy so as to minimize the total expected discounted costs over an infinite planning horizon. We extend Sulem(1986) model into the one in which not only demand but also the market price of the product follows geometric Brownian motions. Then we obtain the optimal cost as the solution of Quasivariational inequality, and show that there exists an optimal procurement policy as (s, S) policy. We shall clarify the dependence of such optimal (s, S) policy on the spot price at the procurement epoch. These values of (s, S) policy can be used and revised in the succeeding ordering cycles. Finally, some numerical examples are provided to investigate analytical properties of the expected cost function as well as of the optimal policy.