Multi-Unit Combinatorial Auctions: an Exact Approach Renata Mansini Department of Electronics for Automation, via Branze 38, 25123 Brescia, Italy University of Brescia fax.: +39 – 030 – 380014,
e.mail:
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M. Grazia Speranza Department of Quantitative Methods, C. da S.Chiara 48/B, 25122 Brescia, Italy University of Brescia e.mail:
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Introduction
In recent times combinatorial auctions have become an interesting domain for theoretical and applied developments of the electronic market technology. A combinatorial auction allows the simultaneous selling of several items and each participant to the auction may bid for bundles of items. When items in a bid are all different the auction is said a Single Unit Combinatorial Auction (SUCA), while when a bid may specify a number of units of a given item we call it a Multi-Unit Combinatorial Auction (MUCA). The bid taker’s objective is to find that set of bids which maximizes his total revenue. It has been shown that the problem, known as Winner Determination Problem, also for the special case in which each item is available in single units, is NP-hard ([4]). Up to now the bulk of research work on this subject has been focused on SUCA while a limited number of papers are available on the problem of Winner Determination in a MUCA (see, for instance, [2], [3] and [5]). For a complete survey on combinatorial auctions the reader is referred to [1]. From a practical point of view, the impact of auction increased importance is evident on the B2C (Business to Consumer) market (people taking part to on-line auctions organized by Ebay.com are increasing day by day). Less evident, but definitely not less important, is the impact of auctions development on the B2B (Business to Business) market, especially in the domain of supply logistics. We will focus our attention on a distribution problem which can be formulated as a Multi-Unit Combinatorial Auction. A transportation company provides a service of goods transshipment from some origins to different destinations. Each origin/destination pair is identified as a delivery route. For each route a given number of vehicles are made available for 1
the service. Production companies are usually interested in the transportation of products from some origins (e.g. the production sites) to some destinations (e.g. the retailers). The transportation company requires that if a route is acquired by a producer at least one vehicle has to be used, i.e. the use of a vehicle by different companies is not allowed. Thus a production company can submit a bid on bundles of routes and may require multiple copies (i.e. multiple vehicles) of the same delivery route. The problem of finding the winner allocation, i.e. the subset of bids on delivery routes which maximizes the transportation company revenue can be formulated as the Winner Determination Problem in a Multi-Unit Combinatorial Auction. In this case the transportation company plays the role of an auctioneer (the seller) of a set of m different delivery routes (items). Multiple copies for each route (item) are available according to the availability of more than one vehicle for each delivery route. The participants to the auction, namely the production companies interested in the transportation service, may offer a price for any combination of routes. A bid is feasible when the number of copies of any item (i.e. the number of vehicles required for each route) included in the bid does not exceed the number of available vehicles. The bid taker wants to find that allocation, i.e. that subset of feasible bids, which guarantees the maximum revenue. The problem is known to be strongly NP-hard.
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The solution approach and the computational results
A combinatorial auction is a mechanism for simultaneously selling different items by allowing participants to the auction to bid on bundles (combinations) of them. The Winner Determination Problem in a MUCA can be stated as follows. Let M = {1, 2, ..., m} be the set of different items (delivery routes) to be sold in the auction. A number ui , ui ∈ Z + , of copies are available for each item i ∈ M. We define as combination a subset of items on which it is possible to make a bid. A bid j is defined by the pair (Bj , pj ) where pj is the price offered for combination Bj and Bj , j ∈ N = {1, ..., n}, is defined as hw1,j , w2,j , ..., wm,j i, with wij ≤ ui equal to the number of requested units of item i in combination j. Only non void combinations of items are allowed. We assume that pj is the highest price offered for the combination Bj if more than one offer exists for that combination of items. Bids on combination Bj and lower price are rejected. There may be combination of items on which no bid is made. We define as allocation any set π ⊂ B, where B = {B1 , ..., Bn }. An allocation is feasible if the total number of copies it contains for each item i ∈ M does not exceed ui . The problem of determining a winning allocation (Winner Determination Problem) is that of finding an allocation which is optimal with respect to the maximization of the revenue and can be defined for a MUCA as a 0-1 multi-dimensional knapsack problem. In the worst case the transportation company (the bid taker) offering m different delivery routes could receive an exponential number of bids on different combinations. This makes
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the problem of finding an optimal solution in a reasonable amount of time a hard task. We propose an exact enumerative algorithm based on the generation of binary search trees. More precisely the method generates a sequence of binary trees each of which includes all the allocations having a given cardinality (given number of selected bids). The binary structure of each search tree is given by the inclusion/exclusion of one bid in a partial allocation. Thus, a son is generated from a node by adding a new bid or by refusing it. The first two sons are generated from the node root selecting or excluding bid 1, i.e. setting x1 = 1 (left son) and x1 = 0 (right son), respectively. This implies that each level of the tree is associated with a given bid, for a total number of levels equal to n. We assume that bids are analyzed in the tree according to the non-increasing value of their prices pj , j = 1, ..., n. The choice is consistent with the introduced bounding operations. Since each generated binary tree allows the search of allocations with a given cardinality, we have exploited such characteristic to introduce a bid dominance relation which has proven to be an effective nodes fathoming rule. The computation of a good initial lower bound on the problem through the solution of a greedy knapsack-type heuristic is an important part of our procedure: it is used to determine the cardinality of the first binary tree which the method will analyze. An upper bound on the maximum number of bids in a feasible allocation is also computed to determine the last tree cardinality and guarantees algorithm finiteness and correctness. Computational results are in progress. However, the size of the already solved instances is larger than that of the instances solved for the same problem by means of Branch and Bound approaches (see [2]). At present the algorithm has been tested on uniformly random generated instances as well as on the test problems described in [5]. We think that our algorithm implementation can be further improved. Nevertheless, in its present form, the algorithm comes out to be already very promising by getting, on large instances, better results than CPLEX 7.0.
References [1] de Vries S., Vohra, R. V., (2000), Combinatorial auction: a brief survey. Unpublished manuscript. [2] Gonen R., Lehmann, D., (2000), Optimal solution for multi-unit combinatorial auctions: Branch and Bound heuristics. In Proceeding ACM-EC, 13-20 Minneapolis, MN. [3] Leyton-Brown, K., Shoham, Y., Tennenholtz, M., (2000), An algorithm for multi-unit combinatorial auctions, American Association for Artificial Intelligence, www.aaai.org. [4] Rothkopf, M., Pekec. A., Harstad, R., (1998), Computationally manageable combinatorial auctions, Mng. Sci., 44(8), 1131-1147. [5] Sandholm, T., (1999), An algorithm for optimal winner determination in combinatorial auctions, Proceedings of the Sixteenth IJCAI, 542-547, Stockholm, Sweden. 3
