Topic
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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TL;DR: It will be shown in a following publication that contrary to the results obtained up to now, the Tychonoff-product theorem is safeguarded with fuzzy compactness.
894 citations
TL;DR: A related topic of the paper is to introduce an alternative representation of fuzzy measures, called the interaction representation, which sets and extends in a common framework the Shapley value and the interaction index proposed by Murofushi and Soneda.
891 citations
01 Jan 1993
TL;DR: The concept of fuzzy sets (precisely speaking, fuzzy subsets of an ordinary set) is nothing but an extended concept of ordinary sets, and the concept of probabilities is absolutely different from that of sets.
Abstract: As is well-known in recent years, there are two kinds of uncertainities, randomness and fuzziness, which can be both dealt with from a mathematical point of view. We know the concept of probabilities with respect to randomness and also that of fuzzy sets with respect to fuzziness. This fact tempts us to discuss fuzzy sets in comparison with probabilities. However, such a direct comparison must fail. The concept of fuzzy sets (precisely speaking, fuzzy subsets of an ordinary set) is nothing but an extended concept of ordinary sets. We have to notice that the concept of probabilities is absolutely different from that of sets. To discuss our problem in detail, let us consider probabilities for the time being. There are a number of interpretations for probabilities: classical probabilities (originated by Laplace); measure theoretical probabilities (by Kolmogorov); subjective probabilities in Bayesian statistics; probabilities as logics and so on.
889 citations
Book•
01 Aug 1996
TL;DR: Fuzzy sets as mentioned in this paper are a class of classes in which there may be grades of membership intermediate between full membership and non-membership, i.e., a fuzzy set is characterized by a membership function which assigns to each object its grade of membership.
Abstract: The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...
885 citations
TL;DR: This paper recapitulates the definition given by Atanassov (1983) of intuitionistic fuzzy sets as well as the definition of vague sets given by Gau and Byehrer (1993) and sees that both definitions coincide.
880 citations