A maximum trip covering location problem with an alternative mode of transportation on tree networks and segments

In this paper the following facility location problem in a mixed planar-network space is considered: We assume that traveling along a given network is faster than traveling within the plane according to the Euclidean distance. A pair of points (A _i ,A _j ) is called covered if the time to access the network from A _i plus the time for traveling along the network plus the time for reaching A _j is lower than, or equal to, a given acceptance level related to the travel time without using the network. The objective is to find facilities (i.e. entry and exit points) on the network that maximize the number of covered pairs. We present a reformulation of the problem using convex covering sets and use this formulation to derive a finite dominating set and an algorithm for locating two facilities on a tree network. Moreover, we adapt a geometric branch and bound approach to the discrete nature of the problem and suggest a procedure for locating more than two facilities on a single line, which is evaluated numerically.