Optimal selection of customers for a last-minute offer

We analyze a short-term revenue optimization problem involving the targeting of customers for a promotion in which a finite number of perishable items are sold on a last-minute offer. The goal is to select the subset of customers to whom the offer will be made available in order to maximize the expected return. Each client replies with a certain probability and reports a specific value that might depend on the customer type, so that the selected subset has to balance the risk of not selling all items with the risk of assigning an item to a low value customer.