Applications of Intuitionistic Fuzzy Sets to Decision-Making Using AG-Groupoids and Klein Four-Group

Abstract

This paper proposes a novel decision-making framework that integrates algebraic structures within intuitionistic fuzzy logic to address complex real-world decision-making scenarios involving uncertainty. Two models are developed, including the Abel-Grassmann Intuitionistic Fuzzy Decision Matrix (AG-IFDM) for ranking non-associative neural connectivity patterns and the Group-theoretic Intuitionistic Fuzzy Ranking (GIFR) for evaluating agricultural treatment formulations. These models provide a reliable ranking of alternatives in uncertain environments where the presence or absence of associativity influences the outcomes. The integration of empirical intuition with algebraic reasoning enables a more realistic modelling of uncertainty, where theoretical precision is complemented by practical applications. Integrating both theoretical foundation and practical insight, the applications from medical neuroscience and agricultural optimization underscore the versatility of the proposed models in addressing order-sensitive real-world systems, with promising potential for large-scale validation. Furthermore, the practical illustrations of these models confirm their effectiveness, supported by comprehensive robustness, sensitivity and complexity evaluations. This work establishes a flexible and mathematically precise foundation for algebraic decision-making to effectively address order-sensitive problems and enhance the transparency of multi-criteria decisions under uncertain environments.