Membership function

About: Membership function is a research topic. Over the lifetime, 15795 publications have been published within this topic receiving 418366 citations.

Similarity in fuzzy reasoning

Abstract: Fuzzy set theory is based on a `fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL-monoids that can be understood as a generalization of MV-algebras. Residuation is a basic concept in GL-monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval.

Rough Set Based Generalized Fuzzy $C$ -Means Algorithm and Quantitative Indices

Abstract: A generalized hybrid unsupervised learning algorithm, which is termed as rough-fuzzy possibilistic C-means (RFPCM), is proposed in this paper. It comprises a judicious integration of the principles of rough and fuzzy sets. While the concept of lower and upper approximations of rough sets deals with uncertainty, vagueness, and incompleteness in class definition, the membership function of fuzzy sets enables efficient handling of overlapping partitions. It incorporates both probabilistic and possibilistic memberships simultaneously to avoid the problems of noise sensitivity of fuzzy C-means and the coincident clusters of PCM. The concept of crisp lower bound and fuzzy boundary of a class, which is introduced in the RFPCM, enables efficient selection of cluster prototypes. The algorithm is generalized in the sense that all existing variants of C-means algorithms can be derived from the proposed algorithm as a special case. Several quantitative indices are introduced based on rough sets for the evaluation of performance of the proposed C-means algorithm. The effectiveness of the algorithm, along with a comparison with other algorithms, has been demonstrated both qualitatively and quantitatively on a set of real-life data sets.

Symmetric Pythagorean Fuzzy Weighted Geometric/Averaging Operators and Their Application in Multicriteria Decision-Making Problems

Abstract: Pythagorean fuzzy sets PFSs, originally proposed by Yager, are a new tool to deal with vagueness with the square sum of the membership degree and the nonmembership degree equal to or less than 1, which have much stronger ability than Atanassov's intuitionistic fuzzy sets to model such uncertainty. In this paper, we modify the existing score function and accuracy function for Pythagorean fuzzy number to make it conform to PFSs. Associated with the given operational laws, we define some novel Pythagorean fuzzy weighted geometric/averaging operators for Pythagorean fuzzy information, which can neutrally treat the membership degree and the nonmembership degree, and investigate the relationships among these operators and those existing ones. At length, a practical example is provided to illustrate the developed operators and to make a comparative analysis.

Fuzzy Sets and Fuzzy Decision-Making

Abstract: The Basics of Fuzzy Set Theory Fuzzy Phenomena and Fuzzy Concepts Naive Thoughts of Fuzzy Sets Definition of Fuzzy Sets Basic Operations of Fuzzy Sets The Resolution Theorem A Representation Theorem Extension Principles References Factor Spaces What are "Factors"? The State Space of Factors Relations and Operations Between Factors Axiomatic Definition of Factor Spaces Describing Concepts in a Factor Space References The Basics of Fuzzy Decision-Making Feedback Extension and Its Applications Feedback Ranks and Degrees of Coincidence Equivalence Between Sufficient Factors and Coincident Factors How to Improve the Precision of a Feedback Extension Representation of the Intention of a Concept Basic Forms of Fuzzy Decision-Making Limitations of the Weighted Average Formula References Determination of Membership Functions A General Method for Determining Membership Functions The Three-Phase Method The Incremental Method The Multiphase Fuzzy Statistical Method The Method of Comparisons The Absolute Comparison Method The Set-Valued Statistical Iteration Method Ordering by Precedence Relations The Relative Comparison Method and the Mean Pair-Wise Comparison Method References Multifactorial Analysis Background of the Problem Multifactorial Functions Axiomatic Definition of Additive Standard Multifactorial Functions Properties of ASMm-funcs Generations of ASMm-funcs Applications of ASMm-funcs in Fuzzy Decision-Making A General Model of Multifactorial Decision-Making References Variable Weights Analysis Defining the Problem An Empirical Variable Weight Formula Principles of Variable Weights References Multifactorial Decision-Making with Multiple Objectives Background and Models Multifactorial Evaluation The Multifactorial Evaluation Approach to the Classification of Quality Incomplete Multifactorial Evaluation Multi-Level Multifactorial Evaluation An Application of Multifactorial Evaluation in Textile Engineering References Set-Valued Statistics and Degree Analysis Fuzzy Statistics and Random Sets The Falling Shadow of Random Sets Set-Valued Statistics Degree Analysis Random and Set-Valued Experiments A Mathematical Model for Employee Evaluation References Refinements of Fuzzy Operators The Axiomatic Structure of Zadeh's Operators Common Fuzzy Operators Generalized Fuzzy Operators The Strength of Fuzzy Operators "AND" and "OR" Fuzzy Operators Based on the Falling Shadow Theory References Multifactorial Decision Based on Theory of Evidence A Brief Introduction to Theory of Evidence Composition of Belief Measures Multifactorial Evaluation Based on the Theory of Evidence Two Special Types of Composition Functions The Maximum Principle for Multiple Object Evaluations References