Showing papers on "Fuzzy number published in 1980"
995 citations
TL;DR: The theory of fuzzy power sets is shown very naturally to require the use of a fuzzy implication operator and emphasis is placed on the dependence of the choice of operators upon the purposes the user has in hand.
462 citations
TL;DR: This paper illustrates the application of “fuzzy subsets” concepts to goal programming in a fuzzy environment by considering a fuzzy goal-programming problem with multiple goals having equal weights associated with them and developing a solution approach based on linear programming.
Abstract: This paper illustrates the application of “fuzzy subsets” concepts to goal programming in a fuzzy environment. In contrast to a typical goal-programming problem, the goals are stated imprecisely when the decision environment is fuzzy. The paper first considers a fuzzy goal-programming problem with multiple goals having equal weights associated with them. A solution approach based on linear programming is developed. Next, the solution approach is extended to the case where unequal fuzzy weights are associated with multiple goals. Numerical examples are provided for both cases to illustrate the solution procedure.
460 citations
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TL;DR: This paper defines the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure, and shows that the monotone convergence theorem and Fatou's lemma are still true in this new setting.
389 citations
TL;DR: It is demonstrated that two points of view can be considered to extend classical linear constraints: either tolerance constraints, or approximate (in)equality constraints can be obtained.
326 citations
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TL;DR: The decision trees method is extended to the case when the involved data appear as words belonging to the common language whose semantic representations are fuzzy sets, and a reformalization of the basic concepts of probability and utility theory is carried out.
293 citations
Book•
01 Jan 1980
TL;DR: Fuzzy Logic in Data Modeling Semantics, Constraints and Database Design, Kluwer Academic Publ.
Abstract: R. Babuska, Fuzzy Modeling for Control, Kluwer Academic Publ., Dordrecht, 1998, 288 p. B. Bouchon-Meunier, L. Foulloy, M. Ramdani, Logique Floue Exercices Corriges et Exemples d' Applications, Cepadues Editions. Toulouse, 199B, 200 p. G.Q. Chen, Fuzzy Logic in Data Modeling Semantics, Constraints and Database Design, Kluwer Academic Publ., Boston. 1998, 240 p. K.J. Cios. W. Pedrycz, R.S. Swiniarski. Data Mining Methods for Knowledge Discovery, Kluwer Academic Publ., Dordrecht. 199B. J. Godjevac, Idees Nettes sur la Logique Floue, Presses Polytechniques et Universitaires Romandes. Lausanne, 1997,200 p. E. Hisdal, Logical structures for representation of knowledge and uncertainty, Physica-Verlag, Heidelberg, Germany, 199B. M. Jamshidi, A. Titli, L.A. Zadeh, S. Boverie, Applications of Fuzzy LogiC Towards High Machine Intelligence Quotient Systems, Prentice-Hall, Englewood Cliffs, NJ, 1997,423 p. F. Lootsma, Fuzzy Logic for Planning and Decision Making Applied Optimization 8. Kluwer Academic Publ., Dordrecht, 1998. W. Mielczarski, Fuzzy Logic Techniques in Power Systems, Physica-Verlag, Berlin, 1998, 456 p. J.N. Mordeson, P.S. Nair, Fuzzy Mathematics An Introduction for Engineers and Scientists. Physica-Verlag, Berlin, 1998,270 p. W. Pedrycz, Computational Intelligence An Introduction, CRC Press, Boca Raton, FL, 1997,304 p. M. Reghis, E. Roventa, Classical and Fuzzy Concepts in Mathematical Logic and Applications, CRC Press, Boca Raton, FL, 1998. L. Reznik, Fuzzy Controllers, Newness Press, UK, 1997. Schindler, Fuzzy-Datenanalyse durch kontextbasierte Datenbankanfragen, Deutscher Universitiits Verlag, Leverkusen, Germany, 1998. O. Wolkenhauer, Possibility Theory with Applications to Data Analysis. Research Studies Press, Hertfordshire, UK, 1998, 290 p. Edited volumes
197 citations
01 Jan 1980
TL;DR: Fuzzy sets theory and fuzzy logic constitute the basis for the linguistic approach and models, based on this approach, can be constructed to simulate approximate reasoning.
Abstract: Fuzzy sets theory and fuzzy logic constitute the basis for the linguistic approach. Under this approach, variables can assume linguistic values. Each linguistic value is characterized by a label and a meaning. The label is a sentence of a language. The meaning is a fuzzy subset of a universe of discourse. Models, based on this approach, can be constructed to simulate approximate reasoning. The implementation of these models presents two major problems, namely how to associate a label to an unlabelled fuzzy set on the basis of semantic similarity (linguistic approximation) and how to perform arithmetic operations with fuzzy numbers. For each problem a solution is proposed. Two illustrative applications are discussed.
195 citations
TL;DR: The problem of how to permit a patron to represent the relative importance of various index terms in a Boolean request while retaining the desirable properties of a Boolean system is concerned.
Abstract: This article concerns the problem of how to permit a patron to represent the relative importance of various index terms in a Boolean request while retaining the desirable properties of a Boolean system. The character of classical Boolean systems is reviewed and related to the notion of fuzzy sets. The fuzzy set concept then forms the basis of the concept of a fuzzy request in which weights are assigned to index terms. The properties of such a system are discussed, and it is shown that such systems retain the manipulability of traditional Boolean requests.
191 citations
TL;DR: The construction of an ideal decision function from its approximation in terms of other decision functions is interested, and fuzzy subsets of type II, those with linguistic grades of membership, are investigated.
Abstract: We are interested in the construction of an ideal decision function from its approximation in terms of other decision functions. We investigate fuzzy subsets of type II, those with linguistic grades of membership. We also discuss fuzzy subsets whose elements are fuzzy subsets of another set.
184 citations
TL;DR: Criteria for a fuzzy matrix to be regular is given and it is proved that the row and column rank of any regular fuzzy matrix are equal, and the cardinality of any two bases are equal.
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TL;DR: Using a representation theorem for fuzzy subspaces it gives theorems for convex fuzzy sets in the proper setting and completes the theory, introducing the necessary concepts.
01 Jan 1980
TL;DR: This communication is a first step towards a general approach to fuzzy set-theoretic operators, i.e. algebraic operators which coincide with set-operators when membership values are crisp, which can be crucial in decision analysis, information retrieval, pattern recognition and more.
Abstract: This communication is a first step towards a general approach to fuzzy set-theoretic operators, i.e. algebraic operators which coincide with set-operators when membership values are crisp. Some properties of subclasses of such fuzzy set-theoretic operators are investigated. Specific examples are given. An attempt to discuss a possible interpretation of these operators is proposed. The choice of a good operator in a given practical situation can be crucial in decision analysis, information retrieval, pattern recognition for the purpose of aggregating several pieces of information.
TL;DR: The functions of minπ and maxπ are introduced as the analogues of nearest neighbour “propagation” signals of binary images as well as extending some already well known binary processes into grey level algorithms.
TL;DR: The problem of selecting the highest grade of membership of fuzzy subsets of type 2 and of choosing the most truthful of a group of fuzzy propositions involve making a choice among fuzzy subset on the unit interval is proposed.
Abstract: The problem of selecting the highest grade of membership of fuzzy subsets of type 2 and of choosing the most truthful of a group of fuzzy propositions involve making a choice among fuzzy subsets on the unit interval. A procedure is proposed for the selection of fuzzy subsets on the unit interval. This procedure involves selecting the subset closest to a linear membership function on the unit interval.
TL;DR: This paper aims to give greater understanding than hitherto, to the choice of implications rules when modelling a given situation using approximate reasoning with fuzzy logic, and asserts properties of implication which satisfy an intuitive understanding of the nature of fuzzy deduction.
01 Jul 1980
TL;DR: In this paper, the applicability of fuzzy set theory to decision analysis (DA) is examined, and it is suggested that fuzzy decision analysis should be viewed as an automatic sensitivity analysis, but that fuzzy sets may be useful with another interpretation for group decisionmaking.
Abstract: The applicability of fuzzy set theory to decision analysis (DA) is examined. It extends the ideas of an earlier paper "Fuzzy decision analysis," by Watson et al. [33]. Particular emphasis is placed on justifying the use of Zadeh's fuzzy calculus to model impression, and an axiomatic system is suggested towards this end. This is seen as an attempt at extending Savage's axioms of subjective probability to produce "approximate probabilities." It is argued that the method proposed by Watson et al. for comparing decision options is unsatisfactory, and several alternative methods are developed. Some computational anomalies are pointed out which severely limit the potential of this methodology. It is suggested that, for individual decisionmaking, fuzzy decision analysis should be viewed as an automatic sensitivity analysis, but that fuzzy sets may be useful with another interpretation for group decisionmaking. The conclusions are that the methodology has too many limitations to be of use for isolated decisions but that it may have a value for often repeated generic decisions.
TL;DR: The product of fuzzy σ-algebras is introduced which is a generalisation of the family of fuzzy events considered by L.A. Zadeh and is a comparison between classical measurability of functions and fuzzy measurable defined in this paper.
TL;DR: An interactive computer program is described which implements the procedure proposed in “A Formal System for Fuzzy Reasoning”, with each piece of evidence entered as a sentence, with an associated ‘degree of belief’ and ‘weight’; followed by a tentative conclusion.
TL;DR: Two implication operators and resulting relationships between fuzzy sets are studied and the results compared with previous ones obtained with other implication operators are compared.
01 Mar 1980
TL;DR: One of the aims of the theory of fuzzy sets is the development of a methodology for the formulation and solution of problems which are too complex or ill-defined to be susceptible to analysis by conventional techniques as discussed by the authors.
Abstract: One of the aims of the theory of fuzzy sets is the development of a methodology for the formulation and solution of problems which are too complex or ill-defined to be susceptible to analysis by conventional techniques. Because of its unorthodoxy, it has been and will continue to be controversial for some time. Eventually, though, the theory of fuzzy sets is likely to be recognized as a natural development in the evolution of scientific thinking. In retrospect, the skepticism about its usefulness will be viewed as a manifestation of the human attachment to tradition andre sistance to innovation.
TL;DR: In this paper, a theory of α -Hausdorff fuzzy topological spaces which is compatible with α -compactness and fuzzy continuity is developed. But the theory is not compatible with the theory of one-point α-compactifications.
TL;DR: It is shown that under some assumptions alternatives exist which are in fact unfuzzily dominated thus serving as unfuzzy solutions to a fuzzily formulated problem.
TL;DR: This paper briefly reviews the method from an operational viewpoint, isolating the individual processes that are used in the method, and presenting a feasible algorithm for computing each process.
01 Jan 1980
TL;DR: This short note is an exploratory paper where the difference of nature between probabilistic knowledge and possibilistic knowledge is emphasized in the framework of queuing problems.
Abstract: This short note is an exploratory paper where the difference of nature between probabilistic knowledge and possibilistic knowledge is emphasized in the framework of queuing problems. While the arrival have usually to be modelled by a stochastic process (provided that its correct identification is possible), a queuing system may also exhibit another kind of uncertainty regarding the service time or the service rule which are not always precisely known or defined. Fuzziness, rather than randomness corresponds to the deep nature of this uncertainty. Thus, it is dealt with fuzzy service time and fuzzy service rule in a queuing problem. Hints for solutions are given in various situations. Moreover, an entirely possibilistic model is discussed.
TL;DR: It is shown that these fuzzy measures can be characterized in a unique way by a finite (classical) measure and a Markoff-kernel.
TL;DR: A procedure based upon calculating the consistency of two fuzzy propositions is presented as a means of imprecisely solving fuzzy equations.
Journal Article•
TL;DR: In last years, a variety of papers on fuzzy sets and other fuzzy topics was concerned with set-algebraic operations for and properties of fuzzy sets, but only few remarks are devoted to fuzzy cardinals.
Abstract: In last years, a variety of papers on fuzzy sets and other fuzzy topics was concerned with set-algebraic operations for and properties of fuzzy sets. However, only few remarks are devoted to fuzzy cardinals. In classical set theory the cardinality of a set is a measure of its size or "power". In the fuzzy case one has to differentiate: there are measures of fuzziness and measures of power. Here measures of fuzziness are not our main concern. The interested reader may consult e.g. [1], [2], [5], [9]. Fuzzy cardinals as measures of power of fuzzy sets are considered e.g. in [2], [3], and [6]. To describe and compare these definitions needs some notation. A fuzzy set A over some universe of discourse X is a function A : X -> [0, 1]. Instead of A(x) for x e X we write also x e A for this membership value of x in A. The universe of discourse X shall be fixed throughout the paper. By ^(X) we denote the class of all fuzzy sets over X; for every A e ^(X), the support |A | of A is the classical set
01 Jan 1980
TL;DR: Using results about the algebraic manipulation of fuzzy numbers, computationally attractive algorithms for fuzzy data are provided.
Abstract: Often in real-case problems, all the numerical data are not precisely known and the nature of the uncertainty is possibilistic14 rather than probabilistic. Then, the data are said fuzzy. The adaptation of an ordinary algorithm (appropriate to precise data) to fuzzy data is not always straightforward. Theoretically, the direct application of the extension principle of fuzzy set theory solves this problem, but not generally in a computationally attractive manner. Practically, the case of forecasting algorithms, where the result may be fuzzy is different from this of decision algorithms where the result must be precise. For an illustrative purpose, we successively deal with the PERT, assignment, travelling salesman and transportation problems. Using results about the algebraic manipulation of fuzzy numbers, computationally attractive algorithms for fuzzy data are provided.
TL;DR: For the case of initial data in the problem of group choice represented as fuzzy partial orderings, two problems are solved: (1) design of a set group decisions which satisfy the Pareto unanimity principle and stay "halfway" between initial relations and (2) designing a unique group decision as discussed by the authors.