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Fuzzy number
About: Fuzzy number is a research topic. Over the lifetime, 35606 publications have been published within this topic receiving 972544 citations.
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01 Aug 1996
TL;DR: The calculus of fuzzy restrictions is concerned with translation of propositions of various types into relational assignment equations, and the study of transformations of fuzzy Restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations.
Abstract: A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable In terms of such restrictions, the meaning of a proposition of the form “x is P,” where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x)) = P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R For example, “Stella is young ,” where young is a fuzzy subset of the real line, translates into R(Age(Stella))= young The calculus of fuzzy restrictions is concerned, in the main, with (a) translation of propositions of various types into relational assignment equations, and (b) the study of transformations of fuzzy restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations An important application of the calculus of fuzzy restrictions relates to what might be called approximate reasoning , that is, a type of reasoning which is neither very exact nor very inexact The main ideas behind this application are outlined and illustrated by examples
579 citations
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TL;DR: The complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number, and a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership.
Abstract: The objective of this paper is to investigate the innovative concept of complex fuzzy sets. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. The inherent difficulty in acquiring intuition for the concept of complex-valued membership presents a significant obstacle to the realization of its full potential. Consequently, a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership. Examples of possible applications, which demonstrate the new concept, include a complex fuzzy representation of solar activity (via measurements of the sunspot number), and a signal processing application. A comprehensive study of the mathematical properties of the complex fuzzy set is presented. Basic set theoretic operations on complex fuzzy sets, such as complex fuzzy complement, union, and intersection, are discussed at length. Two novel operations, namely set rotation and set reflection, are introduced. Complex fuzzy relations are also considered. Index Terms-Complex fuzzy intersection, complex fuzzy relations, complex fuzzy sets, complex fuzzy union, complex-valued grades of membership, fuzzy complex numbers.
579 citations
TL;DR: Multiattribute decision making using intuitionistic fuzzy sets is investigated, in which multiple criteria are explicitly considered, several linear programming models are constructed to generate optimal weights for attributes, and the corresponding decision-making methods have also been proposed.
577 citations
TL;DR: Fuzzy linear programming belongs to goal programming in the sense that implicitly or explicitly aspiration levels have to be defined at which the membership functions of the fuzzy sets reach their maximum or minimum.
574 citations
01 Dec 1992-International Journal of Human-computer Studies \/ International Journal of Man-machine Studies
TL;DR: This paper shows that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained, and can derive both the algebraic and probabilistic rough set approximations.
Abstract: This paper explores the implications of approximating a concept based on the Bayesian decision procedure, which provides a plausible unification of the fuzzy set and rough set approaches for approximating a concept. We show that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained. On the other hand, if a given concept is approximated by two sets, we can derive both the algebraic and probabilistic rough set approximations. Moreover, based on the well known principle of maximum (minimum) entropy, we give a useful interpretation of fuzzy intersection and union. Our results enhance the understanding and broaden the applications of both fuzzy and rough sets.
572 citations