In this paper we introduce a novel methodology to face the problem of finding, for every fuzzy set of the real line, a fuzzy number which can be considered as an approximation of the first one in some reasonable sense. This methodology depends on a wide variety of initial parameters that each researcher may set depending on his/her own interests. The main objective of this new methodology is to ensure that many of the techniques that are currently available for fuzzy numbers can also be extended to the setting of fuzzy sets of the real line which are, in many ways, much more enriching. To do this, we carry out a study of the families of nested sets that can determine fuzzy numbers through their level sets. Next, we describe some of the main properties that this approximation methodology verifies and we show some examples to illustrate how the initial parameters influence the result of the approximation.

This manuscript has been partially supported by Junta de Andalucía by Project FQM-365 of the Andalusian CICYE, and also by Projects PID2020-119478GB-I00 and PID2019-108392GB-I00 (AEI/10.13039/501100011033 , Ministerio de Ciencia e Innovación).