On the Formal Equivalence of some Simple Facility Location Models

This paper shows the equivalence among some different formulations of a simple location-allocation problem. A feature shared by all the formulations is an allocation subproblem (the distribution of users among the facilities) based on spatial-interaction theory (gravity models). The initial mathematical programming formulation, useless for computation, is shown to be equivalent to some much simpler mathematical programs, built up by suitably widening the feasible region. Finally, a duality relationship is shown to hold between the location models where accessibility is maximized (having the location and size of the facilities as control variables) and those where travel cost is minimized (having the allocation of users to the facilities as control variables). The last ones are shown to tend to the usual location-allocation models with linear cost function as the distance decay effect increases