The theory of fuzzy sets is presented in the sectionapplied mathematics, which is devoted to methods of analyzing uncertain data, describing the uncertainties of real events and processes using the notion of sets without clear boundaries.
Classical set theory determinesthe membership of a particular element of a particular aggregate. In this case, concepts are accepted under membership in a binary expression, i.e. there is a clear condition: the element in question either belongs to or does not belong to the set.
The theory of sets with respect to fuzzinessprovides a graded understanding of the belonging of the element in question to a particular set, and the degree of its belonging is to be described by means of the corresponding function. In other words, the transition from belonging to a given set of certain elements to non-membership does not occur abruptly, but gradually using the probabilistic approach.
Sufficient experience of foreign and domesticresearchers testifies to the unreliability and inadequacy of the probabilistic approach used as a tool for solving problems of a weakly structured type. The use of statistical methods in solving this type of problem leads to a significant distortion of the initial statement of the problem. It is the shortcomings and limitations associated with the use of classical methods for solving problems of a weakly structured form that are the consequence of the "incompatibility principle", which is formulated in the theory of fuzzy sets developed by LA. Zade.
Therefore, some foreign and domesticresearchers have developed methods for assessing the risk of investment projects and efficiency using tools of the theory of fuzzy sets. In them, the probability distribution was replaced by the distribution of possibilities, which is described by the fuzzy-type membership function.
Fundamentals of set theory are based ontools that are relevant to decision-making methods in uncertain conditions. When they are used, formalization of the initial parameters and target efficiency indicators is assumed as a vector of fuzzy interval (interval values). The hit in each such interval can be characterized by the degree of uncertainty.
Using arithmetic when working with suchfuzzy intervals, experts can be obtained as a result of a fuzzy interval for a specific target. Based on the initial information, experience and intuition, experts can give qualitative and quantitative characteristics of the boundaries (intervals) of possible values of the region and the parameters of their possible values.
The theory of sets can be actively usedin practice and in the theory of management systems, in finance and economics for solving problems under the condition of uncertainty of the main indicators. For example, such techniques as cameras and some washing machines are equipped with fuzzy controllers.
In mathematics the theory of sets, proposed by L.А.Zadeh, allows you to describe fuzzy knowledge and concepts, to operate on them and make vague conclusions. Thanks to the methods based on this theory for constructing fuzzy systems with the help of computer technologies, the areas of application of computers are significantly expanding. Recently, the management of fuzzy sets is one of the effective areas of research. The usefulness of fuzzy control manifests itself in a certain complexity of technological processes from the position of analysis using quantitative methods. Also, the management of fuzzy sets is used for qualitative interpretation of various information sources.