# TENSOR NETWORK MODEL OF LOGISTICS SYSTEM

Petrov Andrey Evgenyevich, Professor of Department of Computer aided design and engineering design, Ph.D &D.Sc., Professor, National University of Science and Technology MISIS (NUST MISIS), Department of Computer aided design and engineering design

Abstract

The network model of the logistics system is developed on the basis of the tensor method of dual networks. The network consists of branches of producers, consumers and their connecting routes, along which the flow of products from points of departure passes along routes to destinations. Producer and consumer flows are specified. Branch resistances set the rates for storing and transporting products. The network is considered as a tensor, the projections of which in the coordinate system are various connections of branches. Coordinates in the space of the network are closed and open paths. A new solution is obtained using the path transformation matrix when the structure changes. The producer and consumer branches define the basis of the open paths. Route branches define the basis of closed paths. The flows of producers and consumers are specified by voltage sources. The calculation of the mesh network with these sources yields currents in the branches representing part of the distribution of product flows along routes. To supplement the complete product flows, voltage sources are introduced in the route branches that define the base circuits. The currents in the input and output branches of the sources in the routes are equal to the differences in the currents in the simplest and connected network. They produce additional currents in the branches of the routes according to Kirchhoff’s law, and only for routes whose number is equal to the number of basic open paths without one. For route branches of the base circuits beyond this number, the values of the complement currents should be selected. The sum of the currents from all sources gives values of product flows from suppliers to consumers, solving the problem of logistics. The tensor method of network models allows you to calculate flows when the structure changes, including decomposition and calculation of parts of complex networks.

KEYWORDS: network model, tensor method, path transformation matrices, duality invariant, logistics system, product transportation.