On the Multiplicity of Optimal Trajectories in Logistics Problems in Heterogeneous Mediums

The paper addresses the problem of constructing an optimal route on a plane in the case of a heterogeneous environment and identifying logistics zones. The optimality criterion is the cost of cargo transportation. Descriptions of optimal trajectories in the form of broken lines are obtained. The case of a heterogeneous medium, which consists of two half-planes, in each of which the costs of transportation along a section of unit length are constant, is considered. At the same time, the properties of the half-planes differ from one another, which corresponds to situations encountered in transport logistics when two areas with different characteristics border. A theorem on the structure of optimal trajectories connecting two points is proven. It has been established that if two points lie in different half-planes, then the trajectory has the form of a broken line of two segments that form an angle with a given formula defined by Snell’s law. A singular set has been identified - a set of points on the plane from which two or more optimal trajectories emanate. An example of constructing logistics service areas is given and a graph of a function is constructed that determines the minimum cost of transporting cargo to the base. To simulate the example, a software package was used, based on optical-geometric analogies when constructing wave fronts in an inhomogeneous medium. When visualizing the results in the form of a map of level lines of the price function of transporting cargo to the base, two branches of a singular set are constructed, on which these lines lose smoothness. On the boundary of the base set, characteristic points (pseudo-vertices) are found that are responsible for the origin of branches of the singular set.