POLYINTERVAL MATHEMATICS AND OPTIMIZATION IN CONDITIONS OF UNCERTAINTY

V.I. Levin

Abstract


Contex. In recent decades, in the civil and military spheres new information technologies are increasingly encountered based on new
approaches to describing various types of uncertainty. These technologies are widely used in engineering, economics, social sphere. To support them, new fairly powerful mathematical models and methods are needed. In this regard, this article devoted to the development of a new model of uncertainty (polyinterval) and mathematical methods and models for its study with regard to solving optimization problems under
uncertainty is very relevant.
Objective. The aim of the article is to elaborate a new mathematical model of uncertainty – a polyinterval which is a sequence of a finite
number of independent intervals of uncertainty in order to optimize various technical, economic, social and other systems with polyinterval
parameters.
Method. To achieve this goal, it is proposed to extend the method of introducing operations on intervals in the form of a set-theoretical
generalization of the corresponding operations over real numbers to the study of optimal operations over polyintervals.
Result. In the article a new mathematical model of non-definiteness is developed in detail – polyinterval. The optimal operations (max,
min) over the polyintervals have been determined and the rules for their implementation have been derived. The necessary and sufficient
conditions for the existence of these operations are established, i.e. the conditions for the comparability of polyintervals over the relations
“more” and “less”. An example of using the results obtained for making the optimal economic decision on choosing the best place of work by
the criterion “the highest salary” is given. It is shown that the polyinterval, which is a more complex model of uncertainty than the interval, allows one to investigate uncertain systems with the same time costs.
Conclusions. The scientific novelty of this work consists in the proposed by the author new mathematical model of uncertainty of various systems in the form of polyintervals, in conjunction with a mathematical apparatus that allows performing optimal operations on polyintervals and thereby enabling the optimization of technical, economic, social and other systems with polyinterval parameters.

Keywords


interval value; polyinterval value; uncertainty; algebra of polyinterval values.

References


Zade L. A. Ponjatie lingvisticheskoj peremennoj i ego primenenie

k prinjatiju priblizhennyh reshenij. Moscow, Mir, 1976, 176 p.

Levin V.I. Nepreryvnaya Logika. Penza, Penzа State Technological

Academy, 2008, 496 p.

Gorban’ I. I. Fenomen Statisticheskoy Ustoychivosti. Kiev,

Naukova Dumka, 2014, 370 p.

Alefeld G., Herzberger J. Introduction to Interval Computation.

N.Y., Academic Press, 1983, 352 p.

Levin V. I. Intervalnaya Matematika i Issledovanie Sistem v

Usloviyah Neopredelennosti. Penza, Penza Technological

Institute Publishing, 1998, 55 p.

Levin V.I. Poliintervaly, ih Ischislenie i Primenenie, Sistemy

upravleniya, svyazi i bezopasnosti, 2016, No. 3, pp. 239–246.

Levin V. I. Metody Optimizacii Sistem v Usloviyah Intervalnoy

Neopredelennosti Parametrov, Informacionnye tehnologii, 2012,

No. 4, pp. 52–59.

Voschinin A. P., Sotirov G. R. Optimizaciya v Usloviyah

Neopredelennosti. Moscow, MEI, Sofiya, Tehnika, 1989, 226 p.

Tsoukias A., Vincke P. A Characterization of PQI Interval Order,

Discrete Applied Mathematics, 2003, No. 127 (2), pp. 387–397.

Aschepkov L. T., Davydov D. V. Reductions of Interval

Noncooperative Games, Computational Mathematics and

Mathematical Physics, 2006, Vol. 46, No. 11, pp. 1910–1917.

Davydov D.V. Identification of Parameters of Linear Interval

Controllable Systems with Interval Observation, Journal of

Computer and Systems Sciences International, 2008, Vol. 48,

No. 6, pp. 861–865.

Gorban’ I. I. Sluchainost’ i Gipersluchainost’. Kiev, Naukova

Dumka, 2016, 290 p.

Ventcel’ E.S. Teoriya Veroyatnostey. Moscow, Vysshaya Shkola,

, 575 p.


GOST Style Citations


1. Заде Л. А. Понятие лингвистической переменной и его применение к принятию приближенных решений / Л. А. Заде. – М. : Мир. – 1976. – 176 с.
2. Левин В. И. Непрерывная логика / В. И. Левин. – Пенза :
ПензГТА, 2008. – 496 с.
3. Горбань И. И. Феномен статистической устойчивости /
И. И. Горбань. – Киев : Наукова Думка, 2014. – 370 с.
4. Алефельд Г. Введение в интервальные вычисления / Г. Але
фельд, Ю. Херцбергер. – М. : Мир, 1987. – 370 с.
5. Левин В. И. Интервальная математика и исследование систем в условиях неопределенности / В. И. Левин. – Пенза : Изд-во Пензенского технологического ин-та, 1998. – 55 с.
6. Левин В. И. Полиинтервалы, их исчисление и применение / В. И. Левин // Системы управления, связи и безопасности. – 2016. – № 3. – С. 239–246.
7. Левин В. И. Методы оптимизации систем в условиях интер-
вальной неопределенности параметров / В. И. Левин // Ин-
формационные технологии. – 2012. – № 4. – С. 52–59.
8. Вощинин А.П. Оптимизация в условиях неопределенности / А. П. Вощинин, Г.Р. Сотиров. – М. : МЭИ, София : Техника, 1989. – 226 с.
9. Tsoukias A. Characterization of PQI Interval Order / A. Tsoukias, P. A Vincke // Discrete Applied Mathematics. – 2003. – №127 (2). – P. 387–397.
10. Ащепков Л. Т. Редукции интервальных бескоалиционных игр / Л. Т. Ащепков, Д. В. Давыдов // Журнал вычислительной математики и математической физики. – 2006. – Т. 46, № 11. – С. 2001–2008.
11. Давыдов Д. В. Идентификация параметров линейных интервальных управляемых систем с интервальным наблюдением / Д. В. Давыдов // Известия РАН. Теория и системы управления. – 2008. – № 6. – С. 25–29.
12. Горбань И. И. Случайность и гиперслучайность / И. И. Горбань. – Киев : Наукова Думка, 2016. – 290 с.




DOI: https://doi.org/10.15588/1607-3274-2018-1-7



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