Search of optimal connection lines using the graph method

Abstract

The article analyzes the role of mathematical research of economic phenomena and processes. It has been established that many problems of the economy involving finding the optimal distances, time, volume of work, cost, number of objects are solved by the method of constructing graphs. These tasks are reduced to ordering some works, which leads to finding the optimal value of certain values. Comparatively new mathematical problems related to the theory of shortest lines and the theory of graphs, which have practical applications for economics, are considered. The purpose of the article is to apply the theory of shortest lines for solving economic problems. The basis of finding extremums is the method of constructing graphs, which is called porphyrian tree. The article describes the construction of a graph-tree. The properties of the shortest network of segments are defined. Based on the established two properties of the shortest grid, two basic rules for constructing the shortest grid are formulated. and methods of its construction, which is used to solve the optimal planning of production processes. The first of them is that an arbitrary isolated vertex must connect to its nearest neighbor. The second property states that an arbitrary isolated fragment must connect to its nearest neighbor shortest grid. By applying these two rules in any sequence, the shortest grid can be constructed. The paper proposes approximate methods for solving problems of finding optimal lines of various types of road connections. It has been established that each such problem requires a unique approach, which is based on the permutation method. The application of the sorting method to solving problems of sequence and permutations, which boil down to ordering the execution of some works by the method of graphs, is considered. Porphyrian tree presented a set of possible solutions to the problem of the wandering merchant. It has been established that the application of methods of constructing the shortest grid and tree will ensure the construction of some optimal plans of a production, trade and transport nature, will provide opportunities for modeling production processes.