In this paper we deal with single facility location problems in a general normed space where the existing facilities are represented by sets. The criterion to be satis ed by the service facility is the minimization of an increasing function of the distances from the service to the closest point ofeach demand set. We obtain a geometrical characterization of the set of optimal solutions for this problem. Two remarkable cases - the classical Weber problem and the minmax problem with demand sets - are studied as particular instances of our problem. Finally, for the planar polyhedral case we give an algorithmic description of the solution set of the considered problems.
In this paper we address the question of how many objective functions are needed to decide whether a given point is a Pareto optimal solution for a multicriteria optimization problem. We extend earlier results showing that the set of weakly Pareto optimal points is the union of Pareto optimal sets of subproblems and show their limitations. We prove that for strictly quasi-convex problems in two variables Pareto optimality can be decided by consideration of at most three objectives at a time. Our results are based on a geometric characterization of Pareto, strict Pareto and weak Pareto solutions and Helly's Theorem. We also show that a generalization to quasi-convex objectives is not possible, and state a weaker result for this case. Furthermore, we show that a generalization to strictly Pareto optimal solutions is impossible, even in the convex case.
We examine the feasibility polyhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal delivery services. In particular we determine the dimension and derive some classes of facets of this polyhedron. We develop some general rules about lifting facets from the uncapacitated facility location (UFL) for UHL and projecting facets from UHL to UFL. By applying these rules we get a new class of facets for UHL which dominates the inequalities in the original formulation. Thus we get a new formulation of UHL whose constraints are all facet defining. We show its superior computational performance by benchmarking it on a well known data set.