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 a-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.