A single-period stocking and pricing problem involving stochastic emergency supply

We develop a model for determining jointly optimal price and production policy of products with short life cycles subject to partially uncertain supply capacity. Within the price-setting newsvendor framework, we consider that demand in excess of the stocking quantity can be met by an emergency order. Due to the randomness of emergency supply capacity, it may not always be possible to satisfy the demand completely. Assuming that all customers are charged the same price, we explore the optimal sourcing and pricing decisions in the commonly used additive and multiplicative demand scenarios. The first-order conditions imply that the optimal price in the problem with random emergency supply is less than the optimal price associated with the deterministic demand problem when demand uncertainty is additive, and it is greater than the optimal price associated with the deterministic demand problem when demand uncertainty is multiplicative. We also find that the same type of comparative static applies in the multiplicative uncertainty case when there exists a dual source emergency system consisting of a perfectly reliable supplier and a second supplier with a stochastic capacity.