Pricing and Production Planning Under Supply Uncertainty

This paper addresses the problem of determining the sale price and the production quantity under supply uncertainty. The problem is commonly observed in the agricultural industry. We consider a firm that initially leases farm space in order to grow fruit. The realized amount of fruit supply fluctuates due to weather conditions, diseases, etc. At the end of the growing season, the firm makes two production decisions: the amount of realized supply to be converted to finished product and the amount of additional supply to purchase from other growers. However, the second opportunity to purchase from other growers occurs at a unit cost that depends on the realized supply. This is defined as the yield-dependent purchasing cost. Specifically, the lower the realized supply, the higher the unit purchasing cost. Two modeling approaches are presented for the price-setting behavior. In the Early Pricing model, the sale price is determined when the leasing agreement is made, and in the Postponed Pricing model, it is set after observing the realized supply. The presence of the yield-dependent purchasing cost differentiates this work from studies that feature price-setting behavior under supply uncertainty. In traditional models where the unit purchasing cost is a constant, the firm experiences either the case of “complete backlogging” (all demand is satisfied) or “lost sales” (some of the demand is not satisfied), but not both. The yield-dependent cost, however, forces the firm to experience both cases simultaneously in this problem. The paper first identifies the optimal sale price and production decisions for the two variants of the problem with complete backlogging and lost sales, and later shows how these decisions differ when the firm utilizes the purchasing alternative with the yield-dependent cost. In the Early Pricing model, it proves that 1) the optimal stocking level is greater than or equal to its equivalent developed for the complete backlogging and lost sales variants, 2) the optimal sale price is greater than or equal to that of the lost sales variant, and is less than or equal to that of the complete backlogging variant, and 3) the purchasing alternative does not always reduce the firm’s initial investment in the leased farm space. In the Postponed Pricing model, it shows that the optimal amount of farm space to be leased can be uniquely determined and that the contribution of the purchasing alternative is always positive.