Applied Soft Computing 11 (2011) 1858–1866 Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc An evolutionary-based approach for solving a capacitated hub location problem  Jozef Kratica a,∗ , Marija Milanovi ´ c b , Zorica Stanimirovi ´ c b , Duˇ san Toˇ si´ c b a Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11 000 Belgrade, Serbia b Faculty of Mathematics, University of Belgrade, Studentski trg 16/IV, 11 000 Belgrade, Serbia article info Article history: Received 5 October 2009 Received in revised form 17 May 2010 Accepted 30 May 2010 Available online 8 June 2010 Keywords: Genetic algorithms Network design Capacitated hub location problems Transportation and telecommunication networks abstract This paper addresses the capacitated hub location problem (CHLP), which is a variant of the classical capacitated hub problem. What is presented is a modified mixed integer linear programming (MILP) formulation for the CHLP. This modified formulation includes fewer variables and constraints compared to the existing problem formulations in the literature. We propose two evolutionary algorithms (EAs) that use binary encoding and standard genetic operators adapted to the problem. The overall performance of both EA implementations is improved by a caching technique. In order to solve large-scale instances within reasonable time, the second EA also uses a newly designed heuristic to approximate the objective function value. The presented computational study indicates that the first EA reaches optimal solutions for all smaller and medium-size problem instances. The second EA obtains high-quality solutions for larger problem dimensions and provides solutions for large-scale instances that have not been addressed in the literature so far. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The hub location problem is concerned with locating hub facil- ities and allocating demand nodes to hubs in order to route the traffic between origin-destination pairs. There is a given network of nodes on which the set of origins, destinations and potential hub locations are identified. The flow in the network (usually repre- senting passengers, mail, data, packages,...) is known. Hub location problems usually involve the following assumptions: (a) the hub network is complete with a link between every hub pair, (b) there are economies of scale incorporated by a discount factor ˛ for using inter-hub connections, (c) direct communication between two non-hub nodes is not allowed. Under the said assumptions, the goal in the classical hub location problems is to locate a set of hubs and to allocate demand nodes so that the total transportation cost in the network will be minimized (hub median problems), or the worst origin-destination cost will be  This research was partially supported by the Serbian Ministry of Science under grant no. 144007. ∗ Corresponding author. E-mail addresses: jkratica@mi.sanu.ac.rs, jkratica@gmail.com (J. Kratica), marija.milanovic@gmail.com (M. Milanovi ´ c), zoricast@mi.sanu.ac.rs (Z. Stanimirovi ´ c), tdusan@mi.sanu.ac.rs (D. Toˇ si´ c). URL: http://www.mi.sanu.ac.rs/∼jkratica (J. Kratica). minimized (hub center problems). Moreover, some additional con- straints may be involved: the number of hubs to be located may be fixed, the allocation of non-hub nodes may be either to a single hub (single allocation scheme) or to multiple hubs (multiple allo- cation scheme), different types of capacity restrictions on hubs or arcs may be assumed, costs for establishing hub facilities may be fixed, etc. Numerous hub location models have been considered in the lit- erature so far and they are mostly derived from practice. A detailed review of hub location problems and their classification is out of this paper’s scope and it can be found in [1–3]. Most of the hub location problems are proved to be NP-hard (see [4,5]). This paper addresses the capacitated hub location problem (CHLP), introduced in [6,7]. The problem considers a lot of practical situations where common assumptions in hub location problems do not hold. The goal of the problem is to decide which hubs are to be opened and how to allocate terminal nodes to open hubs in order to minimize the sum of transportation costs in the network and fixed costs for establishing hubs. It is not assumed that all the hubs are connected by an arc. Terminal nodes may be allocated to more than one open hub (multiple allocation scheme). Capacities on both arcs and hubs are considered. The CHLP is closely related to the well-known Multi Commodity Flow Problem in Network Design, but differs in the fact that its objective function is not piece-wise in the total flow traversing an arc. According to our knowledge, the papers [6,7] are unique in the literature dealing with the CHLP. The authors of these papers presented a Mixed Integer Linear Programming model of the prob- lem and designed two exact solution methods relying on Benders’ decomposition. The first method is a branch-and-cut algorithm 1568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2010.05.035