AUTHOR COPY A discrete competitive facility location model with variable attractiveness H Ku¨c ¸u¨kaydın, N Aras  and _ IK Altınel Bog ˘azic¸i University, _ Istanbul, Turkey We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff ’s gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the loca- tions of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time. Journal of the Operational Research Society (2011) 62, 1726–1741. doi:10.1057/jors.2010.136 Published online 8 September 2010 Keywords: competitive facility location; variable facility attractiveness; mixed-integer nonlinear programming; Lagrangean heuristic; branch-and-bound 1. Introduction In competitive facility location (CFL) problems, a firm is concerned with installing new facilities to serve customers in a market where existing facilities with known locations and attractiveness levels compete for increasing their market share and profit. In some cases, the firm may be a new entrant with no already existing facilities, while in others the firm may own one or more existing facilities. The choice of the customers as to which facility to visit can be modelled using different approaches. For example, models can be formulated in which customers do not choose a facility solely on the basis of their proximity to its location, but they also take into account some of the characteristics of the facility. The first paper on CFL was by Hotelling (1929), in which he developed a model with two equally attractive ice cream sellers along a beach strip where customers patronize the closest one. This very first model was extended later for unequally attractive facilities, which is a more realistic assumption given the current situation in the market. It is possible to divide the CFL models into two cate- gories: deterministic utility models and random utility models. In both categories, the attractiveness level of a facility is determined by a function of its attributes, and customers’ attraction is modelled by a utility function. The main difference between the two types of models is that in the deterministic utility models the customers patronize only the facility with the highest utility for them, whereas in random utility models customers visit each facility with a certain probability. Among the examples of deterministic utility models, we can mention the work by Drezner (1994), and Plastria and Carrizosa (2004). Although both papers consider a single facility in continuous space, the former focuses only on the determination of an optimal location, whereas the latter addresses the best location as well as the best quality of the single facility simultaneously. The most widely used random utility model in the CFL literature is the gravity-based model introduced by Huff (1964, 1966). In this model, the probability that a customer patronizes a facility is proportional to the attractiveness of the facility and inversely proportional to a function of the distance between the customer and the facility. A variety of attributes can be used as a proxy for the attractiveness of the facility. For example, when the facility considered is Journal of the Operational Research Society (2011) 62, 1726–1741 © 2011 Operational Research Society Ltd. All rights reserved. 0160-5682/11 www.palgrave-journals.com/jors/  Correspondence: N Aras, Department of Industrial Engineering, Bog ˘azic¸i University, Istanbul 34342, Turkey. E-mail: arasn@boun.edu.tr