Location Theory, Dominance, and Convexity

This paper explores the nature of optimal solutions to a plant-location problem on a plane under general distance measures. It develops conditions that guarantee an optimal location of a facility to lie in the convex hull of source and destination points. The effect of restricting the solution to some predetermined set is explored. The development is based on a generalization of Kuhn's characterization of a convex hull by dominance. When a “Manhattan” norm is employed, it is shown to be sufficient to consider, as optimal locations, the finite number of “intersection points” in the convex hull.