Joint Production Planning and Product Delivery Commitments with Random Yield

We consider a producer who turns a raw material into a product. Before embarking upon production, the producer has to consider the quantity of raw material to order and the finished product delivery commitments; the actual amount produced is a random multiple of the amount of raw material ordered. A concave expected profit function is introduced which gives rise to simple formulas for determining the optimal quantities to order and to commit for delivery. We also analyze the relations between the optimal quantities to order and to commit, the expected amount received and production capacity. We show that among several vendors of the raw material, there exists a preferred one, no matter what the producer's cost parameters, if and only if the random multiple associated with that vendor is dominant in the sense of the second-degree stochastic order.