A Novel Asymmetric Fuzzy Number with Damped Resonance: The AFRN Framework

Abstract

Classical fuzzy numbers are widely used for representing uncertainty and imprecision, but their conventional convex structure may be insufficient for uncertainty profiles exhibiting local oscillations or resonance-like behaviour. This paper introduces the Asymmetric Fuzzy Resonance Number (AFRN), a six-parameter fuzzy model constructed from an asymmetric Laplacian kernel modulated by a damped cosine term. The proposed membership function preserves normality at its core while allowing damped oscillatory behaviour away from the centre. We establish its basic analytical properties, including normality, boundedness, continuity, non-differentiability at the core, and recovery of the classical asymmetric Laplacian in the appropriate limiting cases. Furthermore, a closed-form expression for the Centre of Gravity (COG) defuzzification is derived, avoiding direct numerical integration. Alpha-cuts are also characterised in terms of their possible connected components. Finally, a parametric sensitivity study illustrates the effects of the resonance amplitude and frequency on the membership profile and the defuzzified value. The AFRN provides a flexible parametric framework for representing asymmetric fuzzy uncertainty profiles with damped local oscillations.