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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 504713, 23 pages
http://dx.doi.org/10.1155/2012/504713
Research Article

An Exact Algorithm for Bilevel 0-1 Knapsack Problems

1Centro de Investigação Algoritmi da Universidade do Minho, Escola de Engenharia, Universidade do Minho, 4710-057 Braga, Portugal
2Departamento de Produção e Sistemas, Universidade do Minho, 4710-057 Braga, Portugal
3LAMIH-SIADE, UMR 8530, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 Valenciennes Cedex 9, France

Received 5 August 2011; Revised 26 October 2011; Accepted 31 October 2011

Academic Editor: Piermarco Cannarsa

Copyright © 2012 Raid Mansi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a new exact method for solving bilevel 0-1 knapsack problems. A bilevel problem models a hierarchical decision process that involves two decision makers called the leader and the follower. In these processes, the leader takes his decision by considering explicitly the reaction of the follower. From an optimization standpoint, these are problems in which a subset of the variables must be the optimal solution of another (parametric) optimization problem. These problems have various applications in the field of transportation and revenue management, for example. Our approach relies on different components. We describe a polynomial time procedure to solve the linear relaxation of the bilevel 0-1 knapsack problem. Using the information provided by the solutions generated by this procedure, we compute a feasible solution (and hence a lower bound) for the problem. This bound is used together with an upper bound to reduce the size of the original problem. The optimal integer solution of the original problem is computed using dynamic programming. We report on computational experiments which are compared with the results achieved with other state-of-the-art approaches. The results attest the performance of our approach.