Neutrosophic Fuzzy Number Optimized Path in a Network

Abstract

Fuzzy and neutrosophic numbers are mathematical tools designed to handle un- certainty and imprecision in real-world problems. Fuzzy numbers, grounded in fuzzy set theory, represent values with gradual transitions between membership and non-membership, characterized by a membership function. These numbers effectively model uncertainty when precise values are unavailable, as seen in applications like decision-making, optimization, and engineering systems. Neutrosophic numbers extend this concept by incorporating indeterminacy into the model, a key feature of neutrosophic set theory. They consist of three components: truth (T), indeterminacy (I), and falsity (F), each of which is independently assessed. This structure enables a more nuanced representation of uncertainty, capturing scenarios where incomplete, contradictory, or vague information coexists. Both approaches play crucial roles in addressing the complexity of real-world problems where classical mathematics may fall short. While fuzzy numbers are suitable for problems with well-defined boundaries of uncertainty, neutro- sophic numbers offer superior flexibility in environments with higher levels of indeterminacy. Together, these tools have broadened the scope of uncertainty modeling across disciplines such as artificial intelligence, economics, and risk management