Discrete Optimization Solving multiobjective, multiconstraint knapsack problems using mathematical programming and evolutionary algorithms Kostas Florios, George Mavrotas * , Danae Diakoulaki Laboratory of Industrial and Energy Economics, National Technical University of Athens, Zographou Campus, 15780 Athens, Greece article info Article history: Received 7 November 2007 Accepted 26 June 2009 Available online 1 July 2009 Keywords: Branch and bound Knapsack problem Multiobjective Evolutionary algorithms abstract In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the mul- ticriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive e-con- straint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932–942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive e -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Par- eto front, as measured by the coverage metric (especially for the largest instance). Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The knapsack problem is a widely-studied combinatorial opti- mization problem that has applications in many fields (Martello and Toth, 1990). Mathematical Programming, Dynamic Program- ming and Metaheuristics are the most common tools for solving such problems. In the last decade, the multicriteria formulation of the knapsack problem (multiobjective knapsack problem, MOKP) and the construction of the corresponding Pareto front have attracted significant attention from the Operational Research and the Computational Science community. In the present paper we will deal with the most complicated case where multiple con- straints are present, giving rise to the multiobjective multicon- straint knapsack problems – MOMCKP (Jaszkiewicz, 2004; Erlebach et al., 2002; Zitzler and Thiele, 1999; Klamroth and Wie- cek, 2000). In the previous definition, the term ‘‘multiconstraint” may also be found as ‘‘multidimensional”. Specifically, we are interested in solving: max Px st Wx  c; x ¼ðx 1 ; :::; x n Þ T 2f0; 1g n ; P 2 R kn ; W 2 R mn ; c ¼ðc 1 ; :::; c m Þ T 2 R m : ð1Þ A solution x 0 is Pareto optimal (nondominated, efficient) if and only if it is feasible and there is no other feasible x such that p i x P p i x 0 for i = 1, 2, ..., k with at least one strict inequality. The set of the Pareto optimal solutions is coined as the Pareto set (in the decision variable space). In the case of MOMCKP it is actually the set of the nondom- inated binary vectors x whose corresponding images Px into R k com- prise the Pareto front(in the criteria space). Multiple constraints and multiple objectives are degenerated to the conventional knapsack problem if the W and P matrices are simply n dimensional line-vec- tors, namely, if m = 1 and k = 1, accordingly. In this paper, we deal with the case where k = 3, m = 3 and n is varying from 10 to 50 according to the data sets available in the literature (Laumanns et al., 2005, 2006). We solve the problem exactly as well as approximately. Although the approximate solution of multiobjective combinato- rial problems using metaheuristics is the main trend nowadays, the usefulness of exact algorithms is also undoubted. One of the basic reasons is the necessity for benchmarks for the approximate algorithms. The quality of the Pareto front approximation cannot be evaluated properly if the Pareto front is not exhaustively 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.06.024 * Corresponding author. E-mail address: mavrotas@chemeng.ntua.gr (G. Mavrotas). URL: http://liee.ntua.gr/gm (G. Mavrotas). European Journal of Operational Research 203 (2010) 14–21 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor