Locating line segments with vertical distances

One generalization of the Weber problem is to locate not a single point, but dimensional facilities such as lines, line segments, or paths. In this paper we deal with locating line segments in the plane. Given a set of existing facilities in the plane, we are looking for a line segment with given length such that the weighted sum of the vertical distances between the existing facilities and the line segment is minimized. An efficient algorithm to solve this problem will be given. Furthermore we show a dual interpretation of the line-location problem by which the line segment problem can be solved and give an example.