Greedy Heuristics and Weight-Coded EAs for Multidimensional Knapsack Problems and Multi-Unit Combinatorial Auctions

Abstract

The multidimensional knapsack problem (MDKP) is a generalized variant of the $\mathcal{N}\mathcal{P}$-complete knapsack problem (KP). The MDKP assumes one knapsack being packed with a number of items x _j so that the total profit Σ_pj of the selected items is maximized. In contrast to the standard KP, each item has m different properties (dimensions) r _ij (i = 1, ...,m; j = 1, ..., n) consuming c _i of the knapsack:

$$ maximize{\text{ }}\sum\limits_{j{\text{ = 1}}}^n {p_j x_j } $$

(1)

$$ \begin{gathered} subject\ to \sum\limits_{j = 1}^n {r_{ij} x_j} \leqslant c_i ,i = 1,...,m \\ with\; x_j \in \{0,1\} ,j = 1,...,n, p_j ,c_i \in \mathbb{N}, r_{ij} \in \mathbb{N}_0 \end{gathered} $$

(2)

A number of relevant real-world problems can be modelled as MDKPs such as allocation problems, logistics problems, or cutting stock problems [6]. Recently [4], it has been noticed that also the winner determination problem (WDP) in the context of multi-unit combinatorial auctions (MUCA) can be modelled as MDKP. MUCAs are combinatorial auctions (CA) where multiple copies of each good are available. In CAs, bidding is allowed on bundles of goods, which allows bidders to express synergies between those goods they want to obtain. First, the agents submit their bids and then, the auctioneer allocates the goods to the agents so that his revenue is maximized. The revenue is the sum of all submitted bids which are accepted by the auctioneer. This allocation problem is called the WDP¹.

The profit p _j used in the MDKP corresponds to the price of bid j, while the resource consumption r _ij corresponds to the number of units of good i requested in bid j. The decision variable x _j denotes whether bid j wins (is accepted by the auctioneer) or looses (is not accepted by the auctioneer).