Simpson points in planar problems with locational constraints. The polyhedral-gauge case

In this paper we address the problem of finding Simpson points in planar models with locational constraints when distances are measured by polyhedral gauges. Making use of the results we have stated in a previous paper, we show here the existence of a finite set of points in the plane, independent of the weights associated with the users, that contains at least a Simpson point. Connection between Simpson points (as a result of a voting process) and Weber points (as the outcome of a planning process) are explored. It is shown by means of an example that the existing relations for the unconstrained case are no longer true when locational constraints are imposed. In order to reconcile both voting and planning processes, a biobjective problem is described, for which we construct a finite dominating set.