Branch-and-cut algorithms for winner determination in discount auctions

Discount auction is a procurement mechanism for buying M indivisible heterogeneous items. The bidders are suppliers and a bid consists of individual cost for each of the items and a nondecreasing discount function defined over the number of items. The winner determination problem faced by the buyer is to determine the winning suppliers and their corresponding winning items that minimizes the total procurement cost, subject to the supply, demand, and discount constraints. We show that this problem is NP-hard upon reduction from the set covering problem. An integer programming formulation is presented and valid inequalities are derived, which serve as cuts to the linear relaxation. A suite of branch-and-cut algorithms are developed with different cut addition techniques and branching strategies. The performance of the proposed algorithms for different problem types are studied with extensive computational experiments.