Weber problems with high-speed lines (Q611010)

The paper deals with a generalization of the Weber problem. The original version of the Weber problem searches for such location in a planar Euclidean space, which minimizes the sum of weighted distances from the location to a finite set of demand points. The presented generalization introduces a set of one-dimensional objects in the considered space, so called curves, so that a move along these curves is faster than a move in the rest of the space. Then the generalized Weber problem is formulated as the task of finding a location, which minimizes the sum of weighted time distances. This extension gives the possibility to model real-world situations like highway networks. The authors confine their study to the case, where a polyhedral gauge, e.g. a Minkowski gauge, gives a distance in the considered space. They make use of special properties of this gauge and prove that there is only a finite dominating set of points, which has to contain an optimal location of the generalized Weber problem. The authors give a way of obtaining these points. The suggested approach to this problem is based on scanning this dominating set of points.