Pre-positioning and Deployment of Reserved Inventories in a Supply Network: Structural Properties

We study a two-stage decision problem, namely, the allocation and deployment of reserved inventories (RIs) in a supply network with random demand surges. The demand surge follows a time-dependent stochastic process and our objective is to minimize the expected total unmet demand in the presence of positive transshipment lead times. We first solve the optimal deployment problem given that the demand surges have occurred at some locations. We show that the optimal deployment policy is a “nested” policy with respect to the shadow price at each location, where a shadow price represents the marginal reduction of the expected total unmet demand due to a marginal increase of RIs. Specifically, locations with higher shadow prices have higher priority in inventory allocation. We then consider the optimal allocation problem in the pre-positioning stage. We show that under certain conditions the optimal allocation is increasing in the total amount of RIs. We introduce a new stochastic order for distributions defined on sets called the first-order stochastic dominance and use it to show that the expected total unmet demand is higher when one of the following is true: the demand surges tend to occur simultaneously at more locations, the post-surge delivery takes a longer time, more demand arrives earlier, or the demand has a higher volatility.