Showing papers on "Fuzzy number published in 1978"
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TL;DR: It is shown that solutions obtained by fuzzy linear programming are always efficient solutions and the consequences of using different ways of combining individual objective functions in order to determine an “optimal” compromise solution are shown.
3,357 citations
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TL;DR: The usual algebraic operations on real numbers are extended to fuzzy numbers by the use of a fuzzification principle, and the practical use of fuzzified operations is shown to be easy, requiring no more computation than when dealing with error intervals in classic tolerance analysis.
Abstract: A fuzzy number is a fuzzy subset of the real line whose highest membership values are clustered around a given real number called the mean value ; the membership function is monotonia on both sides of this mean value. In this paper, the usual algebraic operations on real numbers are extended to fuzzy numbers by the use of a fuzzification principle. The practical use of fuzzified operations is shown to be easy, requiring no more computation than when dealing with error intervals in classic tolerance analysis. The field of applications of this approach seems to be large, since it allows many known algorithms to be fitted to fuzzy data.
2,412 citations
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01 Jan 1978
TL;DR: Experimental results are presented which indicate that more accurate clustering may be obtained by using fuzzy covariances, a natural approach to fuzzy clustering.
Abstract: A class of fuzzy ISODATA clustering algorithms has been developed previously which includes fuzzy means. This class of algorithms is generalized to include fuzzy covariances. The resulting algorithm closely resembles maximum likelihood estimation of mixture densities. It is argued that use of fuzzy covariances is a natural approach to fuzzy clustering. Experimental results are presented which indicate that more accurate clustering may be obtained by using fuzzy covariances.
1,988 citations
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TL;DR: Fuzziness is discussed in the context of multivalued logic, and a corresponding view of fuzzy sets is given.
1,161 citations
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TL;DR: It is shown that unfuzzy nondominated solutions to the decision-making problem exist, provided the original fuzzy relation satisfies some topological requirements, and a simple method of calculating these solutions is indicated.
1,141 citations
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703 citations
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TL;DR: A new definition of transitivity for fuzzy relations yields a relation-theoretic characterization of the class of all psuedo-metrics on a fixed data set into the closed unit interval.
339 citations
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01 Jan 1978TL;DR: This work investigates the question of ranking fuzzy subsets in the unit interval by presenting some initial ideas and concepts toward a solution of this problem.
Abstract: We are interested in investigating the question of ranking fuzzy subsets in the unit interval. We present some initial ideas and concepts toward a solution of this problem.
255 citations
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TL;DR: It is observed that it is possible to have degrees of compactness, which is called α-compactness (α a member of a designated lattice), and a Tychonoff Theorem is obtained for an arbitrary product of α-Compact fuzzy spaces and a 1-point compactification.
242 citations
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TL;DR: A definition of a fuzzy model is introduced, the assessment of its quality is discussed, and a systematic procedure for deriving a model from input-output data is outlined.
Abstract: Many industrial processes are examples of Zadeh's “principle of incompatibility” which states that as a system becomes more complex it becomes increasingly difficult to make mathematical statements about it which are both meaningful and precise. So that if a model of such a process is required then a fuzzy model may be the “best” that can be achieved. This paper considers the problems of building such models. It introduces a definition of a fuzzy model, discusses the assessment of its quality and outlines a systematic procedure for deriving a model from input-output data.
228 citations
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01 Dec 1978TL;DR: This paper is concerned with formulating such a framework and with introducing a bibliography of 570 items, all classified with fuzzy set theory and its applications, which will help the readers to come to grips with the literature explosion on the subjects of fuzzy sets, fuzzy algebra, fuzzy statistics, and closely related applications.
Abstract: The extension of algebraic and analytical concepts to the theory of fuzzy sets appears to play a central role in the investigation of nondeterministic techniques. Since an exact description of any real physical situation is virtually impossbile, it is necessary to develop schemes which deal analytically with decision processes in an imprecise environment. In this paper we are concerned with formulating such a framework and with introducing a bibliography of 570 items, all classified with fuzzy set theory and its applications, which will help the readers to come to grips with the literature explosion on the subjects of fuzzy sets, fuzzy algebra, fuzzy statistics, and closely related applications.
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TL;DR: A unified presentation of classical clustering algorithms is proposed both for the hard and fuzzy pattern classification problems, and two coefficients that measure the “degree of non-fuzziness” of the partition are proposed.
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TL;DR: A certain interpretation of a partially fuzzy LP-Problem is proposed and the corresponding variables of the Dual are analyzed and an economic interpretation is given.
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TL;DR: Algorithms for the determination of the greatest eigen fuzzy set associated with a given fuzzy relation, thinking of medical diagnosis applications are described.
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TL;DR: In this paper, various possible mathematical operations on fuzzy sets that are required to implement a set of control rules as a fuzzy logic control element are considered, and the influence that these operations have on the characteristics of the final control element is a factor that is used to select those operations most suitable in the control context.
Abstract: This brief paper considers the various possible mathematical operations on fuzzy sets that are required to implement a set of control rules as a fuzzy logic control element. The influence that these operations have on the characteristics of the final control element is a factor that is used to select those operations most suitable in the control context.
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TL;DR: The application of the fuzzy sets theory and the statistical decision theory to the decision problems in fuzzy events leads to a specific formulation of fuzzy decision problems and the definitions of entropy, worth of information, and quantity of information.
Abstract: For decision problems in the real world, states of nature, information, and actions should be viewed as fuzzy events. The application of the fuzzy sets theory and the statistical decision theory to the decision problems in fuzzy events leads to a specific formulation of fuzzy decision problems and the definitions of entropy, worth of information, and quantity of information. Some results which are analogous to those in the statistical decision theory are given in this paper.
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TL;DR: A general type of entropies is introduced (in the sense of De Luca and Termini) for finite fuzzy sets and its properties for the case of the so-called θ- ∗Entropies, which generalize both the logarithmic entropy, and Sugeno's fuzzy integral for finite sets.
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TL;DR: The multistage control of a deterministic and stochastic system in a fuzzy environment is considered, with fuzzy decision assumed to be the intersection of fuzzy constraints and a fuzzy goal.
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TL;DR: A category of fuzzy complemented spaces is defined and necessary and sufficient conditions for subcategories to have properties analogous to those of the category of sets and functions are given.
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TL;DR: The basic ideas discussed in the paper are, perhaps, the first steps towards a practical theory of fuzzy systems.
Abstract: This paper considers the difficulties which arise from an attempt to analyse fuzzy systems. Previous work in this area is briefly reviewed and a new way of describing closed-loop fuzzy systems is introduced. Using only finite discrete fuzzy relations, it is possible to define such concepts as state equivalence, stability and controllability, and, by restricting attention to a special class of systems, a simple control problem can be solved. The basic ideas discussed in the paper are, perhaps, the first steps towards a practical theory of fuzzy systems.
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TL;DR: The fstds system, in which 52 fuzzy-set operations are available, is implemented in fortran, and is currently running on a FACOM 230-45S computer.
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TL;DR: A survey of some aspects of many-valued logics and the theory of fuzzy sets and fuzzy reasoning, as advocated in particular by Zadeh, and shows that their definition is sound if some acceptable rationality requirements are demanded.
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TL;DR: A procedure for validating models that involve linguistic variables, based upon the obtaining of the value of a truth variable, indicating how true a model is is introduced.
Abstract: This article introduces a procedure for validating models that involve linguistic variables. First we discuss Zadeh's extension principle for fuzzy sets. Then we discuss the concept of a linguistic truth variable. Using the concepts developed in these sections we present a methodology for validation models involving fuzzy and linguistic variables. This procedure is based upon the obtaining of the value of a truth variable, indicating how true a model is. This truth variable is obtained by fitting fuzzy data to our model.
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TL;DR: The concept of a fuzzy subobject of an object in arbitrary categories, generated by the representation theorem of fuzzy sets, is defined and it is proved that C -sets can be represented by some sets of functors.
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01 Apr 1978
TL;DR: It is shown here that other interpretation of fuzzy connectives may result in entirely different results.
Abstract: Recently, Stallings compared fuzzy set theory with Bayesian statistics. This comparison is improper as it compares Bayesian statistics with a particular case of fuzzy set theory. It is shown here that other interpretation of fuzzy connectives may result in entirely different results.
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TL;DR: The aim of this paper is to raise some questions–and partly, also to answer them –in connection with two important problem groups of fuzzy mathematics: n-fuzzy objects and the sigma-properties of different interactive fuzzy structures.
Abstract: The aim of this paper is to raise some questions–and partly, also to answer them –in connection with two important problem groups of fuzzy mathematics: n-fuzzy objects and the sigma-properties of different interactive fuzzy structures. These questions are suggested by the analyzation of natural languages, the common sense thinking – which are typical fields where the most adequate mathematical model is a fuzzy one-especially by complex adjectival structures and subjective “verifying” processes, respectively. They have, however, a real practical significance also in the field of engineering, as, e.g., in learning machine problems.In the first part we try to point to the practical importance of the concept of fuzzy objects of type n (or n-fuzzy objects), from the aspect of modeling natural languages. A useful way to define n-fuzzy algebras, i.e., generalizing ordinary fuzzy algebras for n-fuzzy objects, is also given, with introducing an isomorphism mapping from the fuzzy to the n-fuzzy object space. As an ...
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01 Jan 1978TL;DR: Under some weak topological conditions unfuzzy non-dominated solutions to the decision-making problem exist and the questions of equivalence of these solutions are studied and a method of calculating these solutions is suggested.
Abstract: A general decision-making problem is considered in which a set of alternatives and a non-strict preference relation specified in this set are of fuzzy nature. The corresponding fuzzy equivalence and strict preference relations are defined which allow to introduce a fuzzy set of non-dominated solutions. It is shown that under some weak topological conditions unfuzzy non-dominated solutions to the problem exist. The questions of equivalence of these solutions are studied and a method of calculating these solutions is suggested.
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TL;DR: An adaptive algorithm for recognition of ill-defined patterns using weak representative points and single pattern training procedure is presented from the standpoint of fuzzy set theory.
Abstract: An adaptive algorithm for recognition of ill-defined patterns using weak representative points and single pattern training procedure is presented from the standpoint of fuzzy set theory. The method includes both supervised and non-supervised schemes, A non-adaptive algorithm with fixed reference and weight vectors is also described to describe the efficiency of the system's adaptiveness to a new input. This was implemented to machine recognition of vowel sounds of a number of speakers in Consonant-Vowel Nucleus-Consonant (CNC) context considering the first three vowel formants as input features. The decision of the machine is governed by the maximum value of fuzzy membership function. A recognition rate, particularly for weak initial representative vectors, was seen to be dependent on the sequence of incoming patterns. As the process of classification continued, the learned moan vectors approached their respective true values of the clusters. Again, once the optimum size of training set is obtained, the r...