### 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.

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