This article examines the concept of Ordered Fuzzy Numbers (OFNs), which extend classical fuzzy numbers by introducing a direction parameter to capture trends in data variability and enhance uncertainty modeling. Despite their conceptual advantages, OFNs have faced significant criticism, particularly regarding the necessity of directionality, the lack of a standard comparison method, the emergence of improper non-convex membership functions, and the development of edge waviness in membership functions after repeated arithmetic operations. In response to these challenges, the authors propose several improvements aimed at enhancing the robustness and usability of OFN arithmetic. Vectorial Ordered Fuzzy Numbers (vOFNs) are introduced as a vector-based alternative to traditional polynomial representations, effectively eliminating edge waviness. The Shape Normalization Operator (SNO) is presented as a solution to convert improper fuzzy numbers into convex forms, improving interpretability. Additionally, the JC Metric is proposed as a novel method for comparing OFNs that considers both value and direction, while reducing distortions caused by near-zero values. These enhancements contribute to a more consistent and practical framework for OFN applications, supporting their broader adoption in fields such as fuzzy control systems, time series forecasting, and intelligent data analysis.

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Selected Aspects of the Imperfections of Ordered Fuzzy Numbers Calculus

  • Jacek M. Czerniak,
  • Leonid Rusanov,
  • Jacek Zalewski,
  • Andrzej Zak

摘要

This article examines the concept of Ordered Fuzzy Numbers (OFNs), which extend classical fuzzy numbers by introducing a direction parameter to capture trends in data variability and enhance uncertainty modeling. Despite their conceptual advantages, OFNs have faced significant criticism, particularly regarding the necessity of directionality, the lack of a standard comparison method, the emergence of improper non-convex membership functions, and the development of edge waviness in membership functions after repeated arithmetic operations. In response to these challenges, the authors propose several improvements aimed at enhancing the robustness and usability of OFN arithmetic. Vectorial Ordered Fuzzy Numbers (vOFNs) are introduced as a vector-based alternative to traditional polynomial representations, effectively eliminating edge waviness. The Shape Normalization Operator (SNO) is presented as a solution to convert improper fuzzy numbers into convex forms, improving interpretability. Additionally, the JC Metric is proposed as a novel method for comparing OFNs that considers both value and direction, while reducing distortions caused by near-zero values. These enhancements contribute to a more consistent and practical framework for OFN applications, supporting their broader adoption in fields such as fuzzy control systems, time series forecasting, and intelligent data analysis.