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Showing papers on "Membership function published in 1982"


Journal ArticleDOI
TL;DR: Using Zadeh's extension principle, the integral of fuzzy mappings over a crisp interval is defined, and provided a special analytical representation of the fuzzy mapping, the practical computation of such an integral is shown to be easy.

260 citations


Journal ArticleDOI

253 citations


Journal ArticleDOI
TL;DR: Using some concepts from fuzzy set theory, a measure for the anxiety induced by a given decision function is suggested and some implications of this measure are investigated.
Abstract: The selection of one alternative from a set of alternatives as the one to implement is in general an anxiety provoking situation. Using some concepts from fuzzy set theory described in the paper, a measure for the anxiety induced by a given decision function is suggested. Some implications of this measure are investigated.

181 citations


Journal ArticleDOI
Ronald R. Yager1
TL;DR: Under the condition of Bayesian belief structure a whole family of possible definitions for the probability of fuzzy sets is obtained and a procedure for including belief qualification in pruf is suggested.

166 citations


Journal ArticleDOI
TL;DR: Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described, under the restrictive assumption that for any x in X, the membership function of Rχ has a ‘complementary function’.

145 citations


Journal ArticleDOI
TL;DR: The paper considers the determination of optimal inventory policy of firms from a global viewpoint of top management, represented as a fuzzy conditional statement equated with a fuzzy relation which is the firm's optimal fuzzy replenishment rule.

132 citations


Journal ArticleDOI
TL;DR: The present article seeks to show that these inadequacies are the consequence of embracing fuzzy-set theory within prototype theory, and proposes an alternative, more adequate version of prototype theory in which concep! gal gradedness is represented ordinally rather than by fuzzy sets.

104 citations


Journal ArticleDOI
TL;DR: Axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by Zadeh based on the postulate that the sum of the conditional probability and its complement given any fuzzy event adds to one or on soft independence for orthogonal sets with independent constitutive elements.

57 citations


Journal ArticleDOI
TL;DR: Tools for applying fuzzy set concepts to the social and behavioral sciences are presented, and examples of their uses are taken from empirical research into individuals' cognitive representations of helping behavior.

48 citations


Journal ArticleDOI
TL;DR: It is shown how such probabilities may be estimated from fuzzy data and a basic relation between joint, conditional and marginal fuzzy probabilities is established.

40 citations


Journal ArticleDOI
TL;DR: A general framework for a theory is presented that encompasses both statistical uncertainty and nonstatistical uncertainty, which relates to the concept of a fuzzy set and possibility theory, and the possibility of using this theory in Bayesian estimation with fuzzy prior information.

Journal ArticleDOI
TL;DR: This relation is used to prove some properties of λ-additive fuzzy measures and it is shown that some measures have properties similar to those of positive fuzzy measures.

Journal ArticleDOI
TL;DR: The main stresses are that the approach to the quantification treatment of such qualitative data is based on fuzzy set theory, that this is different from Hayashi's approach employed in his quantification theory type I which deals with crisp cases of the authors' and that the method can be solved by linear programming.
Abstract: From fuzzy set-theoretical points of view, this paper deals with a method to model quantitative external criterion by using qualitative multivariate data which are obtained in vague and ambiguous circumstances. The problem is to explain quantitative change in the external criterion by using qualitative values of multivariate data which are given by subjective recognition and judgement. In this paper those qualitative values are assumed to be fuzzy degree of membership in qualitative categories which are type II fuzzy sets and quantitative change in the external criterion is given as fuzzy numbers. The main stresses are that our approach to the quantification treatment of such qualitative data is based on fuzzy set theory, that this is different from Hayashi's approach employed in his quantification theory type I which deals with crisp cases of ours and that our method can be solved by linear programming.

Journal ArticleDOI
TL;DR: The main result shows that under suitable hypotheses, the integral on a fuzzy set equals the integral over some level set.

Journal ArticleDOI
TL;DR: The referents of metaphor are interpreted as members of fuzzy sets defined by semantic features, which allows metaphors to express both analogies and disanalogies and a range of truth values defined by membership functions.

Journal ArticleDOI
TL;DR: An application of the fuzzy approach is proposed, where the specification of the grade of membership functions for linguistic values of the linguistic variables allows the use of a fuzzy model.
Abstract: A classical quantitative analysis is not appropriate for some ill-defined or/and very complex chemical engineering problems. An application of the fuzzy approach is therefore proposed. It is possible to build up a linguistic model. The specification of the grade of membership functions for linguistic values of the linguistic variables allows the use of a fuzzy model.

01 Jan 1982
TL;DR: The effects that the usual set theoretic and arithmetic operations with fuzzy sets and fuzzy numbers have with respect to the energies and entropies of the fuzzy sets connected and of the resulting fuzzy sets are discussed.
Abstract: We discuss the effects that the usual set theoretic and arithmetic operations with fuzzy sets and fuzzy numbers have with respect to the energies and entropies of the fuzzy sets connected and of the resulting fuzzy sets, and we also compare the entropies and energies of the results of several of those operations.

Journal ArticleDOI
TL;DR: The substantiation for the form of the solution of this problem is shown, the problems concerning identification of fuzzy measures and integration on continuous sets are considered, and the algorithm for finding strategies is formulated.

01 Jan 1982
TL;DR: This work offers a presentation of fuzzy sets whereby they are conceived as representatives of a whole class of sets (that are themselves subsets of the universe of objects on which the fuzzy set is defined).
Abstract: Fuzzy sets have been studied in various forms. We now offer a presentation of fuzzy sets whereby they are conceived as representatives of a whole class of sets (that are themselves subsets of the universe of objects on which the fuzzy set is defined).

Journal ArticleDOI
TL;DR: F fuzzy systems described by means of fuzzy relational equations and Λ-fuzzy systems are discussed and some ideas of clustering techniques and probabilistic sets are discussed.

Journal ArticleDOI
TL;DR: It is proved that there exist fuzzy sets which do not have a decidable "sharpened" version and fuzzy, sets which are not " sharpened" versions of decidable fuzzy sets.
Abstract: The concept of "decidability" for fuzzy sets is defined. We prove that there exist fuzzy sets which do not have a decidable "sharpened" version and fuzzy, sets which are not "sharpened" versions of decidable fuzzy sets.


01 Jan 1982
TL;DR: This paper obtains systematic characterizations for three classes of n-ary fuzzy set operations, which under the mapping for any fuzzy subset A of X, yield homomorphic images, i.e., ordinary set operations applicable to the random sets S(A).
Abstract: : The fledgling discipline of fuzzy set theory has now grown to encompass well over two thousand papers. The thrust of this paper is to expand earlier unary and binary homomorphic relations between fuzzy and random set operations, by obtaining systematic characterizations for three classes of n-ary fuzzy set operations, which under the mapping for any fuzzy subset A of X, yield homomorphic images, i.e., ordinary set operations applicable to the random sets S(A): Binary operations are presented in Theorem 3, unary operations, in Theorem 4, and n-ary operations in Theorem 6: Theorem 5 demonstrates the general case, Corollary 1 exhibits characterizations for statistically independent image random sets, and Corollary 2 is concerned with a certain simplified subclass of continuous operations. In the case of binary fuzzy set operations leading to homomorphic random set compositions the characterizing structure for the membership function is a simple sum of at most four terms, each term being also an elementary combination of the individual component membership functions and/ or a functional form that satisfies a constraint equation.

Book ChapterDOI
01 Jan 1982
TL;DR: A self-supervised learning algorithm with the concept of guard zones around the class representative vectors has been presented for vowel sound recognition and a quantitative measure of image-quality is provided by the terms “index of fuzziness”, “entropy” and “π-ness” of a fuzzy set.
Abstract: The paper consists of three parts. In the first part of the paper, a self-supervised learning algorithm with the concept of guard zones around the class representative vectors has been presented for vowel sound recognition. In the second part, an algorithm consisting of histogram equalisation technique followed by a further enhancement using fuzzy S and π membership functions is described for detecting the small variation in grey levels and identifying the different regional contours of x-ray images. Finally, a quantitative measure of image-quality is provided by the terms “index of fuzziness”, “entropy” and “π-ness” of a fuzzy set.

Journal ArticleDOI
TL;DR: In this paper instructions involve fuzzy unions and intersects; opinions require the introduction of fuzzy relations and compositions to transform verbal statements into quantitative expressions.
Abstract: Architects receive and give instructions and are involved with opinions. Instructions and opinions are seldom treated objectively and yet clear communication is professionally essential. One application of fuzzy set theory is to provide a quantitative interpretation of verbally expressed, subjective statements which are meaningful and yet not always clearly defined. The fuzzy calculus leads to a better understanding of the statements without the loss of vagueness and imprecision which distinguish subjectivity from sharp objectivity. In this paper instructions involve fuzzy unions and intersects; opinions require the introduction of fuzzy relations and compositions to transform verbal statements into quantitative expressions.

Journal ArticleDOI
c. Dujet1
TL;DR: New notions of complementation and partition are given, different from the usual ones, and based on the idea of separation in a set by the mean of a fuzzy set by a functional called a "separating power".

Journal ArticleDOI
TL;DR: In this paper, a system of definitions and theorems for fuzzy sets is introduced, aimed at an adequate expression of the point of view of the objective property of fuzzy sets.
Abstract: Fuzzy sets establish a mapping from the interval of values of a criterial function onto a system of subsets of a basic set. In the paper, a system of definitions and theorems is introduced, which is aimed at an adequate expression of this point of view. The criterial function, with an arbitrary interval of values, serves for expressing the really existing objective property, forming the basis for defining a fuzzy set.