There is a panoply of fuzzy sets extensions, that include interval fuzzy set, intuitionistic fuzzy set, hesitant fuzzy set, and so on. So how we can make a choice and how to assess its performance?
I've just added a conference paper that may be interesting for you:
Conference Paper Against Artificial Complexification: Crisp vs. Fuzzy Informa...
It is not just a matter of choosing an extension but to clearly justify the need. Every extension needs more "precise" information to define a single datumm.