Bipolar Fuzzy Sets and Equilibrium Relations

2011 ◽  
pp. 129-158
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Clustering ◽  
Theoretical Basis ◽  
Cognitive Map ◽  
Global Regulation ◽  
Quantum Lattices ◽  
Bipolar Fuzzy Sets

Based on bipolar sets and quantum lattices, the concepts of bipolar fuzzy sets and equilibrium relations are presented in this chapter for bipolar fuzzy clustering, coordination, and global regulation. Related theorems are proved. Simulated application examples in multiagent macroeconomics are illustrated. Bipolar fuzzy sets and equilibrium relations provide a theoretical basis for cognitive-map-based bipolar decision, coordination, and global regulation.

YINYANG BIPOLAR FUZZY SETS AND FUZZY EQUILIBRIUM RELATIONS: FOR CLUSTERING, OPTIMIZATION, AND GLOBAL REGULATION

2006 ◽  
Vol 05 (01) ◽  
pp. 19-46 ◽  
Author(s):  
WEN-RAN ZHANG
Keyword(s):  
Fuzzy Sets ◽  
Square Lattice ◽  
Equilibrium Relation ◽  
Global Regulation ◽  
Similarity Relations ◽  
Equilibrium Function ◽  
Clustering Optimization ◽  
Holistic Theory ◽  
Α Level ◽  
Bipolar Fuzzy Sets

Based on the notions of bipolar lattices and L-sets, YinYang bipolar fuzzy sets and fuzzy equilibrium relations are presented for bipolar clustering, optimization, and global regulation. While a bipolar L-set is defined as a bipolar equilibrium function L that maps a bipolar object set X over an arbitrary bipolar lattice B as L:X ⇒ B, this work focuses on the unit square lattice B F = [-1, 0] × [0, 1]. A strong or weak bipolar fuzzy equilibrium relation in a bipolar set X is then defined as a reflexive, symmetric, and bipolar interactive (or transitive) fuzzy relation μR: X ⇒ B F . Three types of bipolar α-level sets are presented for bipolar defuzzification and depolarization. It is shown that a fuzzy equilibrium relation is a non-linear bipolar generalization and/or fusion of multiple similarity relations, which induces disjoint or joint bipolar fuzzy subsets including quasi-coalition, conflict, and harmony sets. Equilibrium energy and stability analysis can then be utilized on different clusters for optimization and global regulation purposes. Thus, this work provides a unified approach to truth, fuzziness, and polarity and leads to a holistic theory for cognitive-map-based visualization, optimization, decision, global regulation, and coordination. Basic concepts are illustrated with a simulation in macroeconomics.

scholarly journals Modeling Imprecise and Bipolar Algebraic and Topological Relations using Morphological Dilations

2021 ◽  
Vol 5 (1) ◽  
pp. 1-20
Author(s):  
Isabelle Bloch
Keyword(s):  
Information Processing ◽  
Fuzzy Sets ◽  
Mathematical Morphology ◽  
Complete Lattice ◽  
Spatial Reasoning ◽  
Topological Relations ◽  
Preference Modeling ◽  
Logical Sense ◽  
Bipolar Fuzzy Sets ◽  
Core Features

Abstract In many domains of information processing, such as knowledge representation, preference modeling, argumentation, multi-criteria decision analysis, spatial reasoning, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools, such as fusion and aggregation, similarity and distances, mathematical morphology. Here we propose to extend these tools by defining algebraic and topological relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators. These definitions are shown to have the desired properties and to be consistent with existing definitions on sets and fuzzy sets, while providing an additional bipolar feature. The proposed relations can be used for instance for preference modeling or spatial reasoning. They apply more generally to any type of functions taking values in a poset or a complete lattice, such as L-fuzzy sets.

Granular Models

2010 ◽  
pp. 243-263
Author(s):  
Witold Pedrycz ◽  
Athanasios Vasilakos
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Clustering ◽  
Feasible Region ◽  
Processing Unit ◽  
Information Granules ◽  
The Given ◽  
Output Space ◽  
Linguistic Models ◽  
Existing Data

In contrast to numeric models, granular models produce results coming in a form of some information granules. Owing to the granularity of information these constructs dwell upon, such models become highly transparent and interpretable as well as operationally effective. Given also the fact that information granules come with a clearly defined semantics, granular models are often referred to as linguistic models. The crux of the design of the linguistic models studied in this paper exhibits two important features. First, the model is constructed on a basis of information granules which are assembled in the form of a web of associations between the granules formed in the output and input spaces. Given the semantics of information granules, we envision that a blueprint of the granular model can be formed effortlessly and with a very limited computing overhead. Second, the interpretability of the model is retained as the entire construct dwells on the conceptual entities of a well-defined semantics. The granulation of available data is accomplished by a carefully designed mechanism of fuzzy clustering which takes into consideration specific problem-driven requirements expressed by the designer at the time of the conceptualization of the model. We elaborate on a so-called context – based (conditional) Fuzzy C-Means (cond-FCM, for brief) to demonstrate how the fuzzy clustering is engaged in the design process. The clusters formed in the input space become induced (implied) by the context fuzzy sets predefined in the output space. The context fuzzy sets are defined in advance by the designer of the model so this design facet provides an active way of forming the model and in this manner becomes instrumental in the determination of a perspective at which a certain phenomenon is to be captured and modeled. This stands in a sharp contrast with most modeling approaches where the development is somewhat passive by being predominantly based on the existing data. The linkages between the fuzzy clusters induced by the given context fuzzy set in the output space are combined by forming a blueprint of the overall granular model. The membership functions of the context fuzzy sets are used as granular weights (connections) of the output processing unit (linear neuron) which subsequently lead to the granular output of the model thus identifying a feasible region of possible output values for the given input. While the above design is quite generic addressing a way in which information granules are assembled in the form of the model, we discuss further refinements which include (a) optimization of the context fuzzy sets, (b) inclusion of bias in the linear neuron at the output layer.

scholarly journals Foldness of Bipolar Fuzzy Sets and Its Application in BCK/BCI-Algebras

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1036
Author(s):  
Young Bae Jun ◽  
Seok-Zun Song
Keyword(s):  
Information Processing ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Negative Information ◽  
Positive Information ◽  
Fuzzy Ideal ◽  
Recent Trends ◽  
Modern Information ◽  
Bipolar Fuzzy Sets

Recent trends in modern information processing have focused on polarizing information, and and bipolar fuzzy sets can be useful. Bipolar fuzzy sets are one of the important tools that can be used to distinguish between positive information and negative information. Positive information, for example, already observed or experienced, indicates what is guaranteed to be possible, and negative information indicates that it is impossible, prohibited, or certainly false. The purpose of this paper is to apply the bipolar fuzzy set to BCK/BCI-algebras. The notion of (translated) k-fold bipolar fuzzy sets is introduced, and its application in BCK/BCI-algebras is discussed. The concepts of k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are introduced, and related properties are investigated. Characterizations of k-fold bipolar fuzzy subalgebra/ideal are considered, and relations between k-fold bipolar fuzzy subalgebra and k-fold bipolar fuzzy ideal are displayed. Extension of k-fold bipolar fuzzy subalgebra is discussed.

Application of Bipolar Fuzzy Sets in Planar Graphs

2016 ◽  
Vol 3 (2) ◽  
pp. 773-785 ◽  
Author(s):  
Muhammad Akram ◽  
Sovan Samanta ◽  
Madhumangal Pal
Keyword(s):  
Fuzzy Sets ◽  
Planar Graphs ◽  
Bipolar Fuzzy Sets
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