Problem

Consider a single manufacturer and a single supplier. Six months before demand is realized, the manufacturer has to sign a supply contract with the supplier. Let D be a random variable representing demand and f(D) be the demand density function. Let p be the selling price, that is, the price at which the manufacturer sells products to consumers. The sequence of events is as follows. Procurement contracts are signed in February and demand is realized during a short period of 10 weeks that starts in August. Components are delivered from the supplier to the manufacturer at the beginning of August, and the manufacturer produces items to customer orders. Thus, we can ignore any inventory holding cost. We will assume that unsold items at the end of the 10-week selling period have zero value. Finally, assume that the manufacturer can also purchase additional items in the spot market. Let s be a random variable representing the per-unit spot market price and f(s) be its density function. The objective is to identify a procurement strategy so as to maximize expected profit. Assume the supplier offers an option contract in which the per-unit reservation price is v and the per-unit execution price is w. Given the existence of the spot market, how much capacity should the manufacturer reserve with the supplier when the contract is signed in February?