A Novel Approach to the Vectorial Redefinition of Ordered Fuzzy Numbers for Improved Arithmetic and Directional Representation - Hubert Zarzycki; Andrzej Żak; Jacek M. Czerniak (Applied Sciences, 2025)
This paper presents a novel formulation of Ordered Fuzzy Numbers (OFNs), referred to as Vectorial Ordered Fuzzy Numbers (vOFNs). In contrast to the traditional definition based on a pair of functions, the vOFN framework employs a pair of vectors, offering a more concise and structurally coherent representation. This reformulation addresses the key limitations of classical OFNs, such as non-convexity and difficulties in handling curvilinear boundaries during multiplication and division. The vOFN model retains compatibility with commonly used fuzzy number types—triangular, trapezoidal, and singleton—and preserves directional properties that are essential for modeling fuzzy trends. Furthermore, it simplifies comparison operations and supports a complete algebraic structure. Due to its mathematical consistency, low computational complexity, and ease of implementation, the vOFN framework is well-suited for applications in intelligent systems, particularly in domains that require reasoning under uncertainty.