Heuristic Solutions for General Concave Minimum Cost Network Flow Problems Dalila B. M. M. Fontes and José Fernando Gonçalves Faculdade de Economia da Universidade do Porto and LIACC Rua Dr. Roberto Frias, 4200-464 Porto, Portugal We address the single-source uncapacitated minimum cost network flow problem with general concave cost functions. Exact methods to solve this class of prob- lems in their full generality are only able to address small to medium size instances, since this class of prob- lems is known to be NP-Hard. Therefore, approximate methods are more suitable. In this work, we present a hybrid approach combining a genetic algorithm with a local search. Randomly generated test problems have been used to test the computational performance of the algorithm. The results obtained for these test prob- lems are compared to optimal solutions obtained by a dynamic programming method for the smaller prob- lem instances and to upper bounds obtained by a local search method for the larger problem instances. From the results reported it can be shown that the hybrid method- ology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 67–76 2007 Keywords: Network flow; heuristics; genetic algorithms; local search; concave-cost optimization; combinatorial optimization 1. INTRODUCTION In this work, we focus on the Single Source Uncapaci- tated (SSU) concave Minimum Cost Network Flow Problem (MCNFP), which can be used to address a broader class of MCNFPs. General nonlinear MCNFPs can be transformed into concave MCNFPs on an expanded network [27]. In addi- tion, multiple source and capacitated networks can be trans- formed into single source and uncapacitated networks [35]. Many practical situations where a product is routed through a network exist. This can be seen from the wide and diverse application areas in which network flow problems Received April 2005; accepted July 2006 Correspondence to: D. B. M. M. Fontes; e-mail: fontes@fep.up.pt Contract grant sponsor: FCT; Contract grant number: POCTI/MAT/61842/ 2004 DOI 10.1002/net.20167 Published online in Wiley InterScience (www.interscience.wiley. com). © 2007 Wiley Periodicals, Inc. have been utilized (see e.g., [18] and the references therein). Concave cost functions in network flow problems arise in practice as a consequence of taking into account eco- nomic considerations; for example, fixed costs may arise and economies of scale often lead to a decrease in marginal costs. The complexity of concave MCNFPs arises from min- imizing a concave function over a convex feasible region, defined by the network constraints, which implies that a local optimum is not necessarily a global optimum. Furthermore, there is no simple criterion for deciding whether a local min- imum is also a global minimum. The main feature defining the complexity of MCNFPs is the type of cost function for each arc. A discussion of other parameters affecting problem complexity can be found in [7]. Most of the work developed on concave MCNFPs consid- ers SSU problems with fixed-charge cost functions, that is, functions having a fixed cost component and a linear rout- ing cost component. This problem is a particular case of the more general SSU concave MCNFP, although NP-Hard itself. Recent work on MCNFPs with fixed-charge costs is given in [30], where the authors also discuss other fixed-charge prob- lems. Other works [19, 20, 32] considering routing concave costs do not include fixed costs. MCNFPs involving both concave routing costs and fixed costs on each arc have been addressed only by Burkard et al. [4] and Fontes et al. [8, 9, 11]. 2. OVERVIEW OF EXISTING METHODS Existing algorithms for concave MCNFPs can be char- acterized in terms of the type of problems they solve and whether the solution provided is exact (a global optimum) or an approximation (a bound). The problem types include restrictions on both the objective function and on the under- lying network. For example, a special case arises from considering a fixed number of nonlinear arcs (the remaining arcs being linear) for which polynomial-time or even strongly polynomial-time algorithms have been developed, see [34]. Another special case of MCNFPs, for which numer- ous methods have been proposed, is the MCNFP with fixed-charge costs. Recent works have been reported, for example, by Kim and Hooker [25] that have developed a NETWORKS—2007—DOI 10.1002/net