Aggregation without Loss of Optimality in Competitive Location Models * Frank Plastria & Lieselot Vanhaverbeke MOSI - Dept. of Math. , O.R. , Stat. and Inf. Syst. for Management, Vrije Universiteit Brussel Pleinlaan 2, B 1050 Brussels, Belgium e-mail: {Frank.Plastria, Lieselot.Vanhaverbeke}@vub.ac.be June 23, 2006 Abstract In the context of competitive facility location problems demand points often have to be ag- gregated due to computational intractability. However, usually this spatial aggregation biases the value of the objective function and the optimality of the solution cannot be guaranteed for the original model. We present a preprocessing aggregation method to reduce the number of demand points which prevents this loss of information, and therefore avoids the possible loss of optimality. It is particularly effective in the frequent situation with a large number of demand points and a comparatively low number of potential facility sites, and coverage defined by spatial nearness. It is applicable to any spatial consumer behaviour model of covering type. This aggregation approach is applied in particular to a Competitive Maximal Covering Location Problem and to a recently developed von Stackelberg model. Some empirical results are presented, showing that the approach may be quite effective. Keywords: demand point aggregation, competitive location, consumer behaviour, von Stack- elberg. 1 Introduction In their recent survey on aggregation errors in location models, Francis, Lowe Rayco and Tamir [6] argue that aggregation often decreases data collection cost, modeling cost, computing cost, confidentiality concerns and data statistical uncertainty. And most of the aggregation studies have focused on evaluating the errors incurred when solving location models to optimality using aggregated demand data instead of the unaggregated data, and to develop methods for aggregation aimed at reducing this error. For competitive location models to yield trustworthy results the spatial distribution of demand should be as detailed as possible. Ultimately every possible consumer represents a source of demand. Such precise data is nowadays quite readily available, at least for the static part of demand, e.g. by online phone directories including address information which, in combination with GIS, yield very exact positions. As an example see the Palm-Beach Country data [5]. Inclusion of such detailed demand information leads to very large scale models, since every single demand point corresponds to a term in the objective function, and/or one or more variables and constraints. The models considered in this paper are discrete and of deterministic covering type (see Plastria [11] for a survey), and our aim is to solve them to optimality, which may easily become infeasible for the demand data sizes we are thinking of. Therefore aggregation of demand is necessary. However, we would like to be able to aggregate without any loss of ∗ This research was partially supported by the projects OZR1067 and SEJ2005-06273ECON 1