Hesitant fuzzy sets

TLDR

Fuzzy set theory has been extended by concepts such as intuitionistic fuzzy sets, type‑2 fuzzy sets, and fuzzy multisets. The paper introduces hesitant fuzzy sets, a new type of fuzzy set whose interpretation differs from existing fuzzy multisets. Hesitant fuzzy sets are defined with basic operations and their relationship to intuitionistic fuzzy sets is examined. The envelope of hesitant fuzzy sets is shown to be an intuitionistic fuzzy set, and the proposed operations are consistent with those of intuitionistic fuzzy sets when applied to this envelope. © 2010 Wiley Periodicals, Inc.

Abstract

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.