Analysis of supply contracts with total minimum commitment

In this paper we analyze a supply contract for a single product that specifies that the cumulative orders placed by a buyer, over a finite horizon, be at least as large as a (contracted upon) given quantity. We assume that the demand for the product is uncertain, and the buyer places orders periodically. We derive the optimal purchase policy for the buyer for a given total minimum quantity commitment and a discounted price. We show that the policy is characterized by the order-up-to levels of the corresponding finite horizon and a single-period standard newsboy problem with no commitment but with discounted price. We show that this policy can be computed easily. We can use this to evaluate any discount schedule characterized by a set of (price discount, minimum commitment) pairs.