Topic

# Defuzzification

About: Defuzzification is a research topic. Over the lifetime, 25081 publications have been published within this topic receiving 768367 citations.

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01 Aug 1996

TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

52,705 citations

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01 Jan 1985TL;DR: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented and two applications of the method to industrial processes are discussed: a water cleaning process and a converter in a steel-making process.

Abstract: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented. The premise of an implication is the description of fuzzy subspace of inputs and its consequence is a linear input-output relation. The method of identification of a system using its input-output data is then shown. Two applications of the method to industrial processes are also discussed: a water cleaning process and a converter in a steel-making process.

18,803 citations

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TL;DR: Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.

13,376 citations

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01 Jan 1973TL;DR: By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

Abstract: The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

8,547 citations

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01 Jan 1995TL;DR: Fuzzy Sets and Fuzzy Logic is a true magnum opus; it addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic.

Abstract: Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,

7,131 citations