Inverse of Backup 2-Median Problems with Variable Edge Lengths and Vertex Weight on Trees and Variable Coordinates on the Plane

In this paper we consider the inverse of backup 2-median problem. In this problem, a set of weighted points are given and we should change some parameters of the problem such as weights of vertices and edges and coordinates of points such that the two given points be the backup 2-median. We present mathematical models for inverse backup 2-median problems on graphs. In the case that the underlying network is a tree, linear models are presented for the problem with variable edges and weight of vertices. We also consider the continuous case of the problem with variable coordinates of vertices on the plane. In this case, we solve the model by PSO and a hybrid improved PSO methods. Computational results are compared for the varying amounts of parameters.   The inverse and backup location facility problems are two important branches of location theory that have been interested by many researchers in the recent decades.   Let n weighted points be given in the plane or on a graph. The inverse median models investigate to change some parameters of problem such as coordinates, edge lengths and vertex weights such that the given facilities be the median points. For more information about inverse location problems see Burkard et al. (2004). On the other hand, in the backup median problems supposed that some facilities may failed. Therefore the other facilities should serve the clients. The backup 2-median problem on trees has been considered by Wang et al. (2009). Fathali (2014) investigated the backup multi-facility location problem on the plane.
In this paper we consider the combination of inverse location and backup facility location problems. We want to change coordinates, weight of vertices or length of edges with minimum cost such that the given facilities be backup median facilities.  2. inverse Backup 2-Median On Trees: Let T= (V, E)  be a tree with n vertices. Each vertex  has a nonnegative weight. Let  be the distance between two points  and,  and  be the two given vertices in T which are assumed the location of facilities.   Each facility may fail with a probability. For any vertex, suppose that the cost of increasing and decreasing per unit of  is  and, respectively. Let  and  be the amounts by which the weight is increased and decreased, respectively. Then, the model of inverse backup 2-median problem can be written as follows. In this paper we investigated the backup 2-median problem with variable edge lengths and vertices weights on trees. The problem with variable coordinates on the plane is also considered. The models of mentioned problems and computational results which obtained by two PSO methods are presented.