Solving a Huff-like competitive location and design model for profit maximization in the plane

A chain wants to set up a single new facility in a planar market where similar facilities of competitors, and possibly of its own chain, are already present. Fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility, determined by the different perceived qualities of the facilities and the distances to them, through a gravitational or logit type model. Both the location and the quality (design) of the new facility are to be found so as to maximise the profit obtained for the chain. Several types of constraints and costs are considered.Two solution methods are developed and tested. The first is a repeated local optimisation heuristic, extending earlier proposals to the supplementary design question and the presence of locational constraints. The second is an exact global optimisation technique based on reliable computing using interval analysis, incorporating several novel features. An example and comparative computational results demonstrate that this difficult and very multi-modal problem can be solved by such techniques. The local optimisation method turns out not to be very robust in its results, even after numerous repetitions, whereas the global optimisation method yields very useful and complete information on guaranteed near to optimal solutions after an important but still quite acceptable computational effort.