Algorithms for Constructing Optimal Networks for Transport Logistics Problems in the Case of a Heterogeneous Environment

We study the problem of constructing an optimal network of centers in the problem of transport logistics. It is believed that the network is designed to serve the area M of the territory with a complex terrain. The cost of moving a transport on an elementary section of the path in the vicinity of any point depends on the coordinates of the point. An optimal network S is one for which the cost of transporting from any given point in M to any given point in S is minimal. A non-Euclidean metric is proposed, the distance in which is equal to the minimum cost of transportation from one point to another along one of the routes. We have introduced a differential inclusion whose reachability sets coincide with circles in the new metric. The algorithms are based on partitioning the set M into areas of influence of current points from S and finding for each area a point that provides minimal costs. At the same time, the coordinates of new logistics centers are calculated based on the optimal trajectories connecting the current center with the points farthest from it in the non-Euclidean metric from its area of influence. We also have created a software package that implements the developed algorithms for building a network of centers, with the function of visualizing the areas of influence of network nodes S. An example of solving the problem for an area in which the relief defines the function of transportation costs, the level lines of which are ellipses with a common center, is considered.