Question

Exercise 2.16 (Combinatorial auctions). A company sets an auction for N objects. Bidders place their bids for some subsets of the N objects that they like. The auction house has received n bids, pamely bids b; for subset Sj, for j = 1,..., n. The auction house is faced with the problem of choosing the winning bids so that profit is maximized and each of the N objects is given to at most one bidder. Formulate the optimization problem faced by the auction house as a set packing problem.

Answer 1

Introduce a binary variable x_j for each bidder j= 1, . . . , n indicating whether the bidder was assigned his subset or not. The problem can then be solved by the following integer program:

  

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subject to a; 51 i = 1, ..., N jes

x_i <= 1 i = 1,.....N

x_{i >= 0 i = 1,......,N}

_{I EZ i= 1, ..., N}