Mathematics / 5. Mathematical modeling

C.t.s. Konovalov O.A., c.t.s., Associate Prof. Malykov K.A., c.t.s. Kaberov S.R.

Military Educational Scientific Center of the Air Force «Air Force Academy Professor N.E. Zhukovsky and Y.A. Gagarin» (Voronezh), Russia

RATIONAL PLANNING OF TECHNICAL OPERATION
OF MEANS OF COMMUNICATION

 

One of the ways of increasing the effectiveness of complex technical systems, which include modern communications equipment, radio systems and automated control systems for aircraft, is to improve aircraft maintenance and repair programmes. In additional, an important task is to define the list of preventive work (PW) q* carried out on stables on the aircraft, which provide the maximum value of the effectiveness of PW Wq:

                                  , ,                                      (1)

where YPWk is a mathematical model of the k-th PW, k=;

Rn – status of resources and maintenance unit, n=.

Mathematical models represent the functional dependence of duration of the
k-th PW YPWk on Rn value (the number of systems to be on technical service and repair, the number of specialists in the maintenance and repair Brigade and their level of training, verification equipment, simply, spare parts and accessories, exchange fleet, etc.):

                                      YPWk = F(Rn), k=, n=.                                         (2)

Method of obtaining (2) is presented in [1]. Mathematical models are constructed by one of the methods of the theory of the experiment planning as a set of algebraic polynomials to describe a broad class of dependencies and which involve the possibility of broad application:

        (3)

Representation of the equations (3) in the form of a system and its solution in order to obtain an optimal execution plan PW (q*), is impossible, because for various maintenance Rn can take different values for different limitations.

In this case, the appropriate approach to the method of search engine optimization is using penalty function. This method stands out from the simple implementation and has good convergence property [2].

The essence of the method is to construct such a fitness function, the minimum of which is a solution of this problem.

According to (1) take , where  – is the function of effective execution of sequence of PW. Then the quality function has the form:

                                         ,                                             (4)

where Fpen.(q) – is a non-negative penalty function for violation of the restrictions, which are a weighted sum of partially differentiable function ¦pen. i(q), which have the possible positive value if there is a restrictions violation (6):

                                         ,                                             (5)

where the coefficients ai – importance of partially differentiable function ¦pen.i(q), and

                                                                               (6)

"Fines" are constructed so that all :

                                        .                                           (7)

Then the task that is equivalent to the original, will appear as:

                             , .                                (8)

If the operation of taking the minimum and limit is permutable, thus we obtained of a sequence of common tasks of absolute minimization:

                                     , .                                         (9)

The point of minimum of fitness function  on the sets of Gq will be solution limit of this task when i®¥.

Thus, a set of PW q* is defined by step-by-step search through all possible q within the scope of the Gq definition, which calculation of Fpen.(q) values on each stage. Set q which the minimum value of the penalty function Fpen.(q), q* will be required.

Application of the method of penalty functions for optimization of the mathematical models at a maintenance body, when carring out prevention work in order to build preliminary plan of work and rational allocation of resource units conducting maintenance and repair, allows to design the process of maintenance planning on modern communications equipment, complexes of radio engineering and automated control systems for aircraft and increase the effectiveness of their implementation.

 

Literature

 

1. Zyryanov Y.T., Malykov K.A. Prevention management in organizational-technical systems: monograph / under general supervision Y.T. Zyryanova – M.: AST-PRESS BOOK, 2005. –160 p.

2. Moiseev N.N., Ivanilov Y.P., Stolyarova E.M. Optimisation methods. M.: Nauka, 1978.

3. Konovalov O.A. Konovalchuk E.V., Malykov K.A., Kaberov S.R. Modeling of the distribution of resources based on uncertain factors in the operation of aircraft // Materiały VIII Międzynaro-dowej naukowi-praktycznej konferencji «Dynamika naykowych badań-2012», 07-15 lipca 2012 roku. – Przemyśl: Sp. z o.o. «Nauka i studia», 2012. – Volume 21. Matematyka. – Str. 52-55.