A Reconsideration Note of the Mathematical Frameworks for Fuzzy and Neutrosophic Risk Management

Uncertainty modeling underpins decision-making across diverse domains, and over the years a rich array of theoretical frameworks has emerged to capture its many facets. Notable among these are Fuzzy Sets, Rough Sets, Hyperrough Sets, Vague Sets, Intuitionistic Fuzzy Sets, Hesitant Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets, alongside ongoing advances in hybrid and higher-order uncertain frameworks. Risk management—the systematic process of identifying, quantifying, and mitigating potential losses—is indispensable in contexts ranging from project planning and system engineering to business operations. Although fuzzy-logic approaches to risk assessment have been widely studied, existing treatments often lack fully formalized, probability-theoretic foundations. In this paper, we introduce rigorously defined mathematical frameworks for fuzzy risk management and for neutrosophic risk management. Each framework extends the classical risk-optimization model by embedding fuzzy or neutrosophic membership structures into coherent risk measures, thereby enabling graded preference analysis and enhanced expressiveness. Our formulations not only generalize the crisp risk-management paradigm but also provide a unified basis for future theoretical developments and practical applications of fuzzy and neutrosophic risk models.