Solving of the maxisum location problem on network with a restriction on transport costs

We consider the problem of the optimal location of a facility on an undirected weighted network. A positive weight is assigned to each edge. Two positive parameters are assigned to the vertices. The first parameter reflects the requirement to place the facility as close to the vertex as possible, and the second — as far as possible. There is a limit on the total weighted distance from the facility to the vertices, taking into account the first parameter. It is necessary to find acceptable locations of the facility on the edges of the network with the maximum sum of weighted distances from them to the vertices, taking into account the second parameter (local extremes). A polynomial algorithm is proposed to find all local extremums at the edges of the network.