A MATHEMATICAL MODEL FOR DECISION-MAKING UNDER UNCERTAINTY HYBRID BASED ON FUZZY LOGIC AND GENERATIVE ARTIFICIAL INTELLIGENCE

Abstract

. The article proposes a hybrid mathematical model for decision-making under uncertainty that integrates fuzzy logic, Mamdani-type rule-based inference, Explainable Artificial Intelligence (XAI), and a Large Language Model (LLM). The relevance of the study is determined by the need to improve the adaptability, interpretability, and reliability of intelligent systems operating with incomplete, ambiguous, and qualitatively described information. Unlike traditional discrete assessment algorithms, the developed model simultaneously considers user performance and decision-making time, transforms numerical indicators into linguistic variables, and produces a deterministic index of the recommended difficulty level. Fuzzy inference is implemented using IF–THEN production rules, which makes it possible to formalize expert knowledge and account for intermediate states between strictly defined classes. The generative LLM module is not used to replace the mathematical decision-making mechanism, but rather to provide semantic interpretation of the obtained result, generate adaptive content, and form personalized feedback based on the calculated fuzzy inference. A software architecture was developed using Python/Flask, scikit-fuzzy, SQLite, REST API, and a visual XAI module. Experimental validation demonstrated a 95.4% agreement between the decisions generated by the hybrid model and expert assessments; the agreement reached 94% in the HR scenario and 91.3% in the pilot adaptive testing scenario. The complete decision-generation and adaptive-response cycle requires 1.5–2.0 s. The obtained results confirm the potential of the proposed approach for adaptive learning, HR analytics, and intelligent decision support systems.