Equity-efficiency bicriteria location with squared euclidean distances

A facility must be located within a given region taking two criteria of equity and efficiency into account. Equity is sought by minimizing the inequality in the facility-inhabitant distances, as measured by the sum of the absolute differences between all pairs of squared Euclidean distances from inhabitants to the facility. This measure meets the Pigou-Dalton condition of transfers and can easily be minimized. Efficiency is measured through optimizing the sum of squared inhabitant-facility distances, either to be minimized or maximized for an attracting or repellent facility, respectively. Geometric localization results are obtained for the whole set of Pareto-optimal solutions for each of the two resulting bicriteria problems within a convex polygonal region. A polynomial procedure is developed to obtain the full bicriteria plot, both trade-off curves, and the corresponding efficient sets.