A Probabilistic Fuzzy Framework for Decision Making Under Uncertainty in Complex Systems

Decision-making problems in real-world systems are often affected by ambiguity, incomplete information, and uncertainty that cannot be adequately represented using classical deterministic or purely probabilistic models. While probabilistic approaches are effective in modeling random variability, they are limited to some situations where vagueness and subjectivity play a dominant role. This paper proposes a new probabilistic fuzzy decision-making framework to address such uncertainty in complex decision environments.

The proposed approach combines fuzzy set theory with probabilistic concepts and operations research principles, allowing the simultaneous treatment of stochastic uncertainty and linguistic imprecision. A generalized aggregation mechanism is developed to integrate fuzzy evaluations with probabilistic criterion importance, resulting in a comprehensive decision score for each alternative. Key theoretical properties of the framework, including consistency, boundedness, and stability with respect to uncertainty variations, are analytically investigated.

The applicability of the proposed framework is illustrated through representative multi-criteria decision-making scenarios involving conflicting criteria and imprecise information. The results demonstrate that the proposed approach provides more flexible and reliable decision outcomes compared to traditional crisp and purely probabilistic methods.

This study contributes to the advancement of decision theory and fuzzy systems, offering a mathematically sound and adaptable framework with potential applications in operations research, data analysis, and interdisciplinary decision-support problems.