# Mathematical method of objective function calculation and routing problems solving

### Аuthors

Alliluyeva N. V.1*, Rudenko E. M.2**

1. Joint-stock company “Scientific and production enterprise “Radar-mms”, 37, Novosel'kovskaya str., lit. A, Sankt-Peterburg, 197375, Russia
2. Peter the Great Strategic Missile Troops Academy, 17, Brigadnaya str., Serpukhov, Moscow region, 142210, Russia

**e-mail: eduard5529@yandex.ru

### Abstract

A lot of tasks of unmanned aviation on observation of the objects, in the most general set up is reduced to route selection between the two benchmarks, which lead to mathematical problem of optimization on the graph. It is assumed herewith that the route is closed and passes through all ribs of the graph. The optimization task on the graph meant for calculating the set of closed routes of minimal length relates to the integer programming problem [1].

The goal of the work consists in solving the optimization problem on the graph leading to obtaining optimal closed routes (OCR). The authors suggest genetic algorithm (GA) as an optimization method. For its implementation the objective function of sevral variables was developed, which simplified significantly the program code and reduced the computation time. The objective function represents the sum of three functions. The first function depends on the edges of the graph, and penalizes edges and loops which do not belong to the graph. The second function depends on the Euler model of the original graph, i. e. the method of the graph reduction to the Euler type by adding multiple edges. The third addend depends on the vertices of the graph and accounts for their multiplicity. The minimum of the objective function is achieved on OCR only.

The GA application allows obtain limited quantity of OCRs from a large variety of variants during finite time, which can be reproduced several times by using the group of the graph’s automorphisms. The presence of the OCRs variety allows planning the order of graphs fly-offs afield while solo and group flights, promptly regroup monitoring routes in consecutive and dispersed flight over various ribs and without intersection at vertices.

### Keywords:

graph, reference point, optimal closed routes, automorphism, objective function, genetic algorithm

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