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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: This paper is the first of two dealing with the analysis and design of a class of complex control systems using a combination of fuzzy logic and modern control theory, which can be represented by a fuzzy aggregation of a set of local linear models.
338 citations
TL;DR: An interval‐valued Pythagorean fuzzy ELECTRE method is proposed to solve uncertainty MAGDM problem and an illustrative example for evaluating the software developments is given to verify the developed approach and to demonstrate its practicality and effectiveness.
Abstract: In this paper, we investigate the multiple attribute group decision making MAGDM problems with interval-valued Pythagorean fuzzy sets IVPFSs. First, the concept, operational laws, score function, and accuracy function of IVPFSs are defined. Then, based on the operational laws, two interval-valued Pythagorean fuzzy aggregation operators are developed for aggregating the interval-valued Pythagorean fuzzy information, such as interval-valued Pythagorean fuzzy weighted average IVPFWA operator and interval-valued Pythagorean fuzzy weighted geometric IVPFWG operator. A series of inequalities of aggregation operators are studied. Later, we develop some interval-valued Pythagorean fuzzy point operators. Moreover, combining the interval-valued Pythagorean fuzzy point operators with IVPFWA operator, we present some interval-valued Pythagorean fuzzy point weighted averaging IVPFPWA operators, which can adjust the degree of the aggregated arguments with some parameters. Then, we propose an interval-valued Pythagorean fuzzy ELECTRE method to solve uncertainty MAGDM problem. Finally, an illustrative example for evaluating the software developments is given to verify the developed approach and to demonstrate its practicality and effectiveness.
337 citations
TL;DR: In Part I, a generalization of the Fuzzy c-Means and FuzzY c-Lines algorithms are developed and shown to be special cases of a more general class of fuzzy algorithms, the fuzzy scatter matrices.
Abstract: In Part I, a generalization of the Fuzzy c-Means (or Fuzzy ISODATA) clustering algorithms is developed. Necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a Picard iteration scheme which generates simultaneously (i) c fuzzy clusters in the data; (ii) a set of c prototypical straight lines in feature space which best fit the data in a well-defined sense; (iii) a set of c prototpyical centers of mass (on the c lines) which characterize the “core” of each linear fuzzy cluster. Theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices. A convergence theorem for each algorithm in the infinite family is given. The algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially “linear” substructure. In Part II, the Fuzzy c-Means and Fuzzy c-Lines algorithms are shown to be special cases of a more general class of fuzzy algorithms, the...
337 citations
01 Jul 2012
TL;DR: Multi-Dimensional Data Semi-Tensor Product of Matrices Multilinear Mappings Among Vector Spaces Right and General Semi- Tensor Products Rank, Pseudo-Inverse, and Positivity of STP Matrix.
Abstract: Multi-Dimensional Data Semi-Tensor Product of Matrices Multilinear Mappings Among Vector Spaces Right and General Semi-Tensor Products Rank, Pseudo-Inverse, and Positivity of STP Matrix Expression of Logic Mix-Valued Logic Logical Matrix, Fuzzy Set and Fuzzy Logic Fuzzy Relational Equation Fuzzy Control with Coupled Fuzzy Relations Boolean Function with Galois Field Structure Decomposition of Logical Functions Boolean Calculus Lattice, Graph, and Universal Algebra Boolean Network Boolean Control System Game Theory Multi-Variable Polynomials Some Applications to Differential Geometry and Algebra Morgan's Problem Linearization of Nonlinear Control Systems Stability Region of Dynamic System.
337 citations
TL;DR: This paper presents the correct centroid formulae for fuzzy numbers and justify them from the viewpoint of analytical geometry and a numerical example demonstrates that Cheng's formULae can significantly alter the result of the ranking procedure.
336 citations