FUNCTION OF CONCENTRATION P. LEVY AND ITS APPLICATIONS IN THE THEORY OF FUZZY SETS L. ZADEH 

Smagin Vladimir Aleksandrovich, doctor of technical sciences, professor, sub-department of metrological support of weapons, military and special equipment, Military Space Academy named after A. F. Mozhaysky (13 Zhdanovskaya street, St. Petersburg, Russia), E-mail: va_smagin@mail.ru
Novikov Alexander Nikolaevich, candidate of technical sciences, associate professor, sub-department of metrological support of weapons, military and special equipment, Military Space Academy named after A. F. Mozhaysky (13 Zhdanovskaya street, St. Petersburg, Russia), E-mail: alnovikov80@mail.ru 

519.218 

10.21685/2227-8486-2020-3-9 

Subject and goals. This article presents the results of the study of the concept of "Concentration function", its semantic content. The authors aim to expand the possibilities of using the concentration function as one of the characteristics of a random variable to the area of solution of applied problems associated with the need to use not only smooth functions, but also functions with discontinuities, which is typical in cases of solving problems of constructing decision-making models with inaccurate initial information by means of the theory of fuzzy sets and represent an attempt to simplify the solution to the problem of "concentration" on a simpler mathematical level.
Methods. The possibilities of applying the principles of constructing the concentration function to the membership functions of L. Zadeh's theory of fuzzy sets have been investigated by the example of constructing concentration functions to the trapezoidal and triangular membership functions.
Results and conclusions. The article shows that the application of the principles of constructing a concentration function to membership functions is possible provided that membership functions are reduced to probability density functions and then, based on the probability distribution functions, the concentration functions of the membership functions are found. Moreover, any discontinuous membership functions can be used to find the necessary concentration functions inherent in them. It follows from the examples considered in the work that the concentration function for any probability distribution is a new, nested distribution function of a random variable of concentration for the considered basic distribution. The article contains recommendations for the application of the concentration function in various applied fields of science. In addition, the work shows that the concentration value can be determined based on the value of the “reliability resource” proposed by professor N. M. Sedyakin in 1965. The proposed approach to the construction of the membership function based on the principles of modeling the concentration function can find application in solving many problems of a different nature, including, for example, in the development of expert systems in modern metrology. 

concentration function, distribution of probabilities, function of accessories