The alpha-ordering for a wide class of fuzzy sets of the real line: the particular case of fuzzy numbers

In this paper, a novel methodology (that we call α -ordering ) for ranking two fuzzy quantities is introduced and studied. Although this methodology is applicable to all pairs of fuzzy numbers, it has been designed to be applied to a wide family 𝒞 of fuzzy sets of the real line. Concretely, this class is characterized by the following properties: normality, bounded support and closed level sets with a finite number of connected components. The proposed methodology depends, in a local way, on a finite number of membership degrees with their respective weights and an aggregation function on the real line. We demonstrate that this procedure establishes a ranking method (i.e. contains a strict order) on 𝒞 which is total (i.e., it can be applied to all pairs of fuzzy sets on 𝒞 ). Finally, we check that this method extends the linear order of real numbers and, in a sense, the pointwise order for fuzzy sets.