YinYang Bipolar Fuzzy Sets and Fuzzy Equilibrium Relations: For Clustering, Optimization, and Global Regulation

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International Journal of Information Technology and Decision Making




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 BF = [-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 ⇒ BF. 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.