Quadratic ordered median location problems

The criteria used in location analysis have to be chosen according to the character of the facility. The single facility location models addressed in this paper accommodate simultaneous multiple criteria in a continuous space in the framework of ordered median problems, which generate and unify many standard location problems. We demonstrate that tools of computational geometry such as Voronoi diagrams and arrangements of curves and lines, enable us to identify the entire set of Pareto-optimal locations, when the squared Euclidean distances between the facility and affected inhabitants are used. For two objectives this works for any type of ordered median objectives and any polygonally bounded feasible region. When more than two criteria are present the objectives and the feasible region have to be convex. For the analysis of this last case we extend several recent structural results for unconstrained convex vector optimization to a convex and compact constraint. Our findings are illustrated by several examples.