Capacity, pricing and production under supply and demand uncertainties with an application in agriculture

We consider capacitated joint lot-sizing and pricing problems when supply and a price-sensitive demand are uncertain. In cases where there is excess capacity, the decision maker has the option to rent her capacity. We model the case where price and production quantity are determined before the yield is realized (joint decisions) as well as the case where the price is determined after the yield is known (sequential decisions). In the joint decisions case, we introduce the concept of expected demand fill rate elasticity and characterize the conditions in which a one-sided production and pricing policy is optimal. We extend the results to the case when there is a fixed production cost and show that a two-sided production and pricing policy is optimal. We also investigate the conditions under which it is optimal for the decision maker to rent her capacity. In addition, we look at the case in which the decision maker is risk averse and analyze the effect of the risk attitude on the optimal price and production quantity. We extend our model to the multi-period case and the case where there is a lead time to acquire new capacity. We show that a one-sided production, pricing and capacity planning policy is optimal in this situation as well. We apply our model to the agricultural sector and present numerical examples using data from the California almond industry.