Spontaneous Symmetry Breaking in Group Decision-Making with Complex Polytopic Fuzzy System

Beginning with a symmetrical multiple-choice individual as the foundation, I develop a sociophysics model of decision-making. By simplifying the range of choices, the framework incorporates the complex Polytopic fuzzy model to capture nuanced dynamics. This approach enables a deeper analysis of decision-making processes within social systems. Decision-making problems commonly involve uncertainty and complexity, posing considerable challenges for organizations and individuals. Due to their structure and variable parameters, the Einstein t-norm (ETN) and t-conorm (ETCN) offer more elasticity than the algebraic t-norm (ATN) and t-conorm (ATCN). This flexibility makes them commonly effective and valuable in fuzzy multi-attribute decision-making (MADM) problems, where nuanced valuations are critical. Their application enhances the ability to model and analyze vagueness and uncertain information, eventually leading to more informed decision outcomes. The complex Polytopic fuzzy set (CPFS) improves the Polytopic fuzzy set (PFS) and complex fuzzy set (CPFS), allowing for a more precise valuation of attributes in complex (MADM) problems. This study aims to propose a MADM scheme using the ETN and ETCN within the framework of a complex Polytopic fuzzy environment. It begins by presenting the Einstein product and sum operations for complex Polytopic fuzzy numbers (CPFNs) and explores their necessary properties. This method enhances the accuracy and applicability of DM processes in ambiguous environments. Subsequently, three complex Polytopic fuzzy operators with known weighted vectors are developed: the complex Polytopic fuzzy Einstein weighted averaging (CPFEWA) operator, complex Polytopic fuzzy Einstein ordered weighted averaging (CPFEOWA) operator, complex Polytopic fuzzy Einstein hybrid averaging (CPFEHA) operator. Moreover, some substantial properties of the operators are studied. Finally, a method based on novel operators is planned, and a numerical example is provided to prove the practicality and effectiveness of the new proposed methods.