Showing papers on "Membership function published in 1981"
TL;DR: A function to help in the ordering of fuzzy subsets of the unit interval is introduced, which is the integral of the mean of the level sets associated with the fuzzy subset.
1,302 citations
TL;DR: It is demonstrated how fuzzy or imprecise aspirations of the decision maker (DM) can be quantified through the use of piecewise linear and continuous functions.
408 citations
TL;DR: In this paper, the authors derived versions of known results dealing with core, equilibria and Shapley-values of cooperative games in the case of cooperative fuzzy games, i.e., games defined on fuzzy subsets of the set of n players.
Abstract: We derive versions of known results dealing with core, equilibria and Shapley-values of cooperative games in the case of cooperative fuzzy games, i.e., games defined on fuzzy subsets of the set of n players. A fuzzy coalition is an n-vector τ = τi associating with each player i his “rate of participation” τi ∈ [0, 1] in the fuzzy coalition and the real number v· is the coalition worth, assumed to be positively homogeneous. If also v is superadditive, or equivalently concave, the fuzzy game with side payments has a nonempty core. Associated with the coalition-worth function for any ordinary game with side payments is a fuzzy extension thereof, viz., the fuzzy game whose coalition-worth function is the least positively-homogeneous superadditive function majorizing the coalition-worth function of the original game. The fuzzy extension always has a nonempty core. Moreover, if the original game has a nonempty core, it coincides with that of its fuzzy extension. Analogous results are established for games without side payments. An axiomatization of “values” of fuzzy games with side payments is also given. The results are applied to show that the set of Walras equilibria coincides with the fuzzy core of an economy.
324 citations
TL;DR: The problem of deriving the uncertainty, on a sum of variables whose values lie within fuzzy intervals, is addressed, and the problem of computing mathematical expectations with fuzzy probabilities is solved.
Abstract: This paper provides an account of an approach to modeling unknown data by means of fuzzy-set theory, and addresses the problem of deriving the uncertainty, on a sum of variables whose values lie within fuzzy intervals. The first part is an extensive presentation of the theoretical background of the approach: the extension principle is stated in terms of possibility of an event; the concept of variable interaction is investigated at length. Section II gives new results regarding the effective practical computation of additions of fuzzy numbers. Its originality lies in the introduction of interaction which enables to control the growth of uncertainty, in calculations. Moreover, the problem of computing mathematical expectations with fuzzy probabilities is solved. The results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed. Typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities. More generally, fuzzy arithmetic can be an important tool for sophisticated, computationally tractable sensitivity analysis in systems modeling, computer-aided design and operations research.
266 citations
TL;DR: Using the structural forms supplied by fuzzy set theory and approximate reasoning, a new method is presented for solving multiple-objective decision problems for which the decision maker can supply only ordinal information on his preferences and the importance of the individual objectives.
Abstract: Using the structural forms supplied by fuzzy set theory and approximate reasoning, a new method is presented for solving multiple-objective decision problems for which the decision maker can supply only ordinal information on his preferences and the importance of the individual objectives.
265 citations
TL;DR: A method for estimating a project completion time in the situation when activity duration times in the project network model are given in the form of fuzzy variables—fuzzy sets on time space is presented.
240 citations
TL;DR: This paper investigates the algebraic properties of fuzzygrades under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring.
232 citations
TL;DR: It is shown that using the fuzzy min-operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem.
231 citations
TL;DR: It is shown that a fuzzy probability measure is always an integral (if the space is generated) if the authors replace the operations ∧ and ∨ by the t-norm To and its dual S0 (see [6]).
82 citations
TL;DR: The algebraic properties of fuzzy sets under these nex operations of bounded-sum and bounded-difference are investigated and the properties in the case where these new operations are combined with the well-known operations of union, intersection, algebraic product and algebraic sum are investigated.
Abstract: Among the basic operations which can be performed on fuzzy sets are the operations of union, intersection, complement, algebraic product and algebraic sum. In addition to these operations, new operations called “bounded-sum” and “bounded-difference” were defined by L. A. Zadeh to investigate the fuzzy reasoning which provides a way of dealing with the reasoning problems which are too complex for precise solution. This paper investigates the algebraic properties of fuzzy sets under these nex operations of bounded-sum and bounded-difference and the properties of fuzzy sets in the case where these new operations are combined with the well-known operations of union, intersection, algebraic product and algebraic sum.
73 citations
TL;DR: A model simulating the medical diagnostic process is presented and the process is considered as the evaluation by the clinician of his degree of belief concerning the belonging of his patient to a fuzzy set given fuzzy and partial informations.
TL;DR: In this paper, an experimental approach is described to obtaining membership functions which describe the values of linhuistic estimates of variables and mathematical tools for description of hedges and computation of composite linguistic estimates.
TL;DR: Zadeh's fuzzy sets theory provides a systematic framework for modeling qualitative fuzziness in foreign policy decision-making and international systems analysis and implications for theory building and computer simulation are discussed.
Abstract: Ambiguity and uncertainty are inherent in many historical alliances, decisions, and perceptions of International Relations (IR). Random factors, unreliable quantitative data, and inaccurate measurement do not cause all this qualitative fuzziness. Existing methods of statistical analysis and classical mathematical structures are ill equipped for modeling and analyzing this empirical real-world fuzziness. Zadeh's fuzzy sets theory provides a systematic framework for modeling qualitative fuzziness in foreign policy decision-making and international systems analysis. Basic fuzzy operations, linguistic variables, and fuzzy logic (viz., inference and algorithms) are explained informally with IR examples. Implications for theory building and computer simulation are discussed.
TL;DR: This paper attempts to shed more light on fuzzy variables in analogy with random variables by studying the problem: if X1, X2,…,Xn are mutually unrelated fuzzy variables with common membership function μ and α1, α2,….,αn are real numbers satisfying αi ⩾ o for every i, when does does Z = Σi = 1n αiXi have the same membershipfunction μ?
TL;DR: It is proved that the sum of many fuzzy variables is a variable whose membership function is approximately equal to ϱ(x) = max {1 − 1 2 c(x − α) 2 , 0} , where a and c are some constant parameters.
TL;DR: An axiomatic generalization is built for the nonprobabilistic entropy of De Luca and Termini in the setting of fuzzy sets theory which can be used like discriminant function in Pattern Recognition when patterns are described by means of fuzzy Set Theory.
01 Aug 1981-International Journal of Human-computer Studies \/ International Journal of Man-machine Studies
TL;DR: It is argued that recognition of various different roles for fuzzy logics strengthens the pragmatic case for their development but that their formal justification remains somewhat exposed to Haack's arguments.
Abstract: Haack (1979) has questioned the need for fuzzy logic on methodological and linguistic grounds. However, three possible roles for fuzzy logic should be distinguished; as a requisite apparatus—because the world poses fuzzy problems; as a prescriptive apparatus—the only proper calculus for the manipulation of fuzzy data; as a descriptive apparatus—some existing inference system demands description in fuzzy terms. Haack does not examine these distinctions. It is argued that recognition of various different roles for fuzzy logics strengthens the pragmatic case for their development but that their formal justification remains somewhat exposed to Haack's arguments. An attempt is made to reconcile pragmatic pressures and theoretical issues by introducing the idea that fuzzy operations should be carried out on subjective statements about the world, leaving standard logic as the proper basis for objective computations.
TL;DR: The theory of fuzzy subsets is applied to the multiple objective decision problem of stock selection and some procedures for subjectively evaluating the membership functions associated with these criteria are indicated.
TL;DR: This fuzzy linear programming problem based on fuzzy functions can be regarded as a model of decision problems where human estimation is influential and obtained a reasonable solution under consideration of the ambiguity of parameters.
TL;DR: The formal system is an independent first-order axiomatization of fuzzy set theory which parallels the Zermelo-Fraenkel development of classical set theory.
TL;DR: A cohesive and widely applicable structure within which human problem solving can be represented that includes fuzzy sets of recalled, applicable, useful, simple, and choosable problem solving rules is discussed.
TL;DR: The definition of fuzzy conveXity is reviewed and some results on projections of convex and fuzzy convex sets are established, and relationships among alternative definitions are investigated.
TL;DR: The determination of eigenvalue spectrum, or the density of states, for a particle in a fuzzy crystal, is obtained by using the concepts developed in fuzzy statistics.
TL;DR: A fuzzy system is formulated as an example of complex man-machine systems by a fuzzy relational equation and the behavior of the system is analyzed by simulation method.
Journal Article•
31 Mar 1981-Bulletin of the University of Osaka Prefecture, Series A Engineering and Natural Sciences
TL;DR: In this paper, a fuzzy linear programming problem is formulated to obtain a reasonable solution under consideration of the ambiguity of parameters, where the ambiguity considered here is not randomness, but fuzziness which is associated with the lack of sharp transition from membership to nonmembership.
Abstract: Linear programming problems with fuzzy parameters have been formulated by fuzzy functions. The ambiguity considered here is not randomness, but fuzziness which is associated with the lack of sharp transition from membership to nonmembership. Parameters on constraint and objective functions are given by fuzzy sets. In this paper, our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters. This fuzzy linear programming problem based on fuzzy functions can be regarded as a model of decision problems where human estimation is influential.
TL;DR: It is shown that the special kind of energy measure the so called degree of fuzziness can be more useful in many practical situations of decision making.
TL;DR: The properties on fuzzy relationships developed by Zadeh are reviewed and a metric is developed from a fuzzy similarity relation and a lattice is developedfrom a fuzzy partial ordering.
Abstract: The properties on fuzzy relationships developed by Zadeh are reviewed. We then develop a metric from a fuzzy similarity relation. We also develop a lattice from a fuzzy partial ordering.
01 Nov 1981-International Journal of Human-computer Studies \/ International Journal of Man-machine Studies
TL;DR: Some of the ideas are used to extend image processing techniques and hence generalize certain raster graphics operations from the binary image case to grey-scale and colour images.
Abstract: The notion of fuzzy sets whose characteristic function is defined over a proper subset of the universal set is discussed. Arising out of this, operations on fuzzy sets over restricted domains of interest are defined. The implications of the use of fuzzy domains of interest is explored. Domain shift operations , which yield a new domain of interest as well as operating on the fuzzy set, are introduced and used to define sequences of operations that terminate when the domain of interest is empty. Some of the ideas are used to extend image processing techniques and hence generalize certain raster graphics operations from the binary image case to grey-scale and colour images.