Generalisation of the Signed Distance

Rédina Berkachy () and Laurent Donzé
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Rédina Berkachy: School of Engineering and Architecture of Fribourg, HES-SO University of Applied Sciences and Arts Western Switzerland, 1700 Fribourg, Switzerland

Mathematics, 2024, vol. 12, issue 24, 1-23

Abstract: This paper presents a comprehensive study of the signed distance metric for fuzzy numbers. Due to the property of directionality, this measure has been widely used. However, it has a main drawback in handling asymmetry and irregular shapes in fuzzy numbers. To overcome this rather bad feature, we introduce two new distances, the balanced signed distance (BSGD) and the generalised signed distance (GSGD), seen as generalisations of the classical signed distance. The developed distances successfully and effectively take into account the shape, the asymmetry and the overlap of fuzzy numbers. The GSGD is additionally directional, while the BSGD satisfies the requirements for being a metric of fuzzy quantities. Analytical simplifications of both distances in the case of often-used particular types of fuzzy numbers are provided to simplify the computation process, making them as simple as the classical signed distance but more realistic and precise. We empirically analyse the sensitivity of these distances. Considering several scenarios of fuzzy numbers, we also numerically compare these distances against established metrics, highlighting the advantages of the BSGD and the GSGD in capturing the shape properties of fuzzy numbers. One main finding of this research is that the defended distances capture with great precision the distance between fuzzy numbers; additionally, they are theoretically appealing and are computationally easy for traditional fuzzy numbers such as triangular, trapezoidal, Gaussian, etc., making these metrics promising.

Keywords: asymmetrical fuzzy numbers; balanced signed distance; fuzzy metrics; fuzzy numbers; fuzzy statistics; generalised signed distance; L 2 metric; signed distance (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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