J Intell Manuf (2013) 24:45–54 DOI 10.1007/s10845-011-0536-2 Ant colony optimization algorithm for the Euclidean location-allocation problem with unknown number of facilities Jean-Paul Arnaout Received: 11 October 2010 / Accepted: 6 April 2011 / Published online: 20 April 2011 © Springer Science+Business Media, LLC 2011 Abstract This paper addresses the Euclidean location- allocation problem with an unknown number of facilities, and an objective of minimizing the fixed and transportation costs. This is a NP-hard problem and in this paper, a three- stage ant colony optimization (ACO) algorithm is introduced and its performance is evaluated by comparing its solutions to the solutions of genetic algorithms (GA). The results show that ACO outperformed GA and reached better solutions in a faster computational time. Furthermore, ACO was tested on the relaxed version of the problem where the number of facilities is known, and compared to existing methods in the literature. The results again confirmed the superiority of the proposed algorithm. Keywords Ant colony optimization · Euclidean location- allocation Problem · Design of experiments Introduction Facility location is a critical aspect of strategic planning for a broad spectrum of public and private firms. Whether a retail chain sitting a new outlet, a manufacturer choosing where to position a warehouse, or a city planner selecting locations for fire stations, strategic planners are often challenged by difficult spatial resource allocation decisions. In fact, the lit- erature reports that most of the models developed to solve the facility location problem are very hard to solve to optimal- ity, as most problems are classified as NP-hard (Owen and Daskin 1998). J.-P. Arnaout (B ) Industrial & Mechanical Engineering Department, Lebanese American University, P.O. Box 36, Byblos 101-F, Lebanon e-mail: jparnaout@lau.edu.lb One of the toughest facility location problems is the location-allocation problem, which comprises of two ele- ments (Ghosh and Rushton 1987): Location: where to locate the central facilities; and Allocation: which subsets of the demand should be served from each facility. The Euclidean uncapacitated location allocation problem (also known as Uncapacitated Multisource Weber Problem (MWP)) involves generating m new facilities to be located in R 2 , that will serve n fixed demand points with the objective of minimizing the transportation costs. Supply centers such as plants and warehouses may constitute the facilities while retailers and dealers may be considered as demand points (Aras et al. 2008). The first formulation of the problem was given by Cooper (1963); the latter attributed the difficulty of solving the problem to the nonconvex objective function, and stated that the problem may have a very large number of local minima. Eilon et al. (1971) considered the well- known 50 customer problem, and using 200 random initial solutions, they obtained 61 local optima for 5 facilities, with around 40% deviation of the worst solution from the best one. Furthermore, Megiddo and Supowit (1984) represented the problem as an enumeration of the Voronoi partitions of the customer set and proved its NP-hardness. Exact methods for the location-allocation problem have been scarce in the literature. Kuenne and Soland (1972) were able to solve problems of the size of 2 facilities and 30 customers or 4 facilities and 15 customers with branch- and-bound algorithms. Nevertheless, new tools have helped increase the problem sizes significantly. As an example, Krau (1997) obtained exact solutions for 2 to 100 facilities and 287 customers using a column generation approach combined with global optimization and branch-and-bound. Since practical scenarios often require a large number of facilities and customers, heuristics for this problem became popular in the literature. Ohlemuller (1997) implemented a 123