In this paper we consider a set of origin-destination pairs in a mixed model in which a network embedded in the plane represents an alternative high-speed transportation system, and study a trip covering problem which consists on locating two points in the network which maximize the number of covered pairs, that is, the number of pairs which use the network by acceding and exiting through such points. To deal with the absence of convexity of this mixed distance function we propose a decomposition method based on formulating a collection of subproblems and solving each of them via discretization of the solution set.The case of locating only one facility regarding transfer facilities already located in the network is a particular case of that dealt with in this paper and can be categorized as a conditional location problem. Thus, adding the extra demand captured by both pairs consisting of a new facility and an already located one, and those consisting of two new facilities, the global contribution of the two new transfer points is computed.The paper is organized as follows. After this introduction the problem is formulated in Section 2. The necessary definitions and results on the distance on networks are summarized in Section 3. Section 4 is devoted to solve the two cases arisen from the properties of the distance. The paper ends with some conclusions and further research.
The problemHereinafter we will use the notation introduced by [5]. Let A = {A i = (a i , b i ), i = 1, . . . , n} ⊂ IR 2 be a set of existing facilities on the plane. We assume that travel time distances between two points in the plane can be estimated by the Euclidean metric.Let T = (t ij ) ∈ IR n×n be an origin-destination (O/D) matrix in which trip patterns are codified, i.e., t ij is the weight of the ordered pair (A i , A j ) (or (i, j), if there is no confusion). This matrix is known a priori: for example, in a transportation context each t ij can be viewed as the number of trips from an origin A i to a destination A j , and in a telecommunication setting it could represent the amount of data transfered from server i to server j.In order to formulate the problem with a mixed mode of transportation, we consider an embedded network N (V, E) representing a high-speed system, with |V | vertices and |E| edges. Each vertex v ∈ V represents a junction or a node, and we assume that each undirected edge e ∈ E has a length l e and it can be modeled as a straight-line segment. The embedding of N in the Euclidean plane as well as the coordinates of the nodes in N will allow us to compute the travel distances between each pair of points. Let N be the continuum set of points on the edges. The edge lengths induce a distance function d on N , such that for any two points x, y ∈ N , d(x, y) is the length of any shortest path connecting x and y. Moreover, if x and y are on the same edge, d(x, y) = ||x − y||.For any two points x, y ∈ N , let us define the high-speed distance d N (x, y) = αd(x, y), with α ∈ (0, 1). Parameter α is a speed factor. It is straig...