Abstract

In this paper we study a facility location problem in the plane in which a single point (median) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time of the clients to the facility, using the L-1 or Manhattan metric. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. We represent the highway by a line segment with fixed length and arbitrary orientation. This problem was introduced in [Computers & Operations Research 38(2) (2011) 525-538]. They gave both a characterization of the optimal solutions and an algorithm running in O(n(3)logn) time, where n represents the number of clients. In this paper we show that the previous characterization does not work in general. Moreover, we provide a complete characterization of the solutions and give an algorithm solving the problem in O(n(3)) time. (C) 2012 Elsevier B.V. All rights reserved.