Computers & Operations Research 35 (2008) 2292 – 2306 www.elsevier.com/locate/cor Core problems in bi-criteria {0, 1}-knapsack problems Carlos Gomes da Silva b, c , ∗ , João Clímaco a, c , José Rui Figueira d , 1 a Faculdade de Economia da Universidade de Coimbra,Av. Dias da Silva, 165, 3004-512 Coimbra, Portugal b Escola Superior de Tecnologia e Gestão de Leiria, Morro do Lena, AltoVieiro, 2401-951 Leiria, Portugal c INESC-Coimbra, Rua Antero de Quental, 199, 3000-033 Coimbra, Portugal d CEG-IST, Center for Management Studies, Departamento de Engenharia e Gestão, Instituto Superior Técnico, Tagus Park, Av. Cavaco Silva, 2780-990 Porto Salvo, Portugal Available online 18 January 2007 Abstract The most efficient algorithms for solving the single-criterion {0, 1}-knapsack problem are based on the core concept (i.e., based on a small number of relevant variables). But this concept is not used in problems with more than one criterion. The main purpose of this paper is to validate the existence of such a set of variables in bi-criteria {0.1}-knapsack instances. Numerical experiments were performed on five types of {0, 1}-knapsack instances. The results are presented for the supported and non-supported solutions as well as for the entire set of efficient solutions. A description of an approximate and an exact method is also presented.  2006 Elsevier Ltd. All rights reserved. Keywords: Bi-criteria knapsack problem; Core problem; Combinatorial optimization 1. Introduction The {0, 1}-knapsack problem consists of selecting a set of items such that their total profit is maximized and their total weights does not exceed the capacity of the knapsack. The {0, 1}-knapsack problem can be formulated as follows: max z(x) = z(x 1 ,...,x j ,...,x n ) = n ∑ j =1 c j x j s.t . n ∑ j =1 w j x j  W, x j ∈{0, 1}, j = 1,...,n, (1) ∗ Corresponding author. Escola Superior de Tecnologia e Gestão de Leiria, Morro do Lena, Alto Vieiro, 2401-951 Leiria, Portugal. Tel.: +351 244 820 300; fax: +351 244 820 301. E-mail addresses: cgsilva@estg.ipleiria.pt (C. Gomes da Silva), jclimaco@inescc.pt (J. Clímaco), figueira@ist.utl.pt (J. Rui Figueira). 1 Visiting Researcher (January–February, 2004) at the Operational Research Department, London School of Economics, Houghton Street, London WC2A 2AE, UK. 0305-0548/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2006.11.001