Abstract

In recent decades, several types of sets, such as fuzzy sets, interval‐valued fuzzy sets, intuitionistic fuzzy sets, interval‐valued intuitionistic fuzzy sets, type 2 fuzzy sets, type n fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, use a notion of nested interval to reflect their common ground, then propose an extension principle of DHFSs. Additionally, we give an example to illustrate the application of DHFSs in group forecasting.

Highlights

  • Since Zadeh 1 introduced fuzzy sets FSs and gave intensive research 2–5, several famous extensions have been developed, such as intuitionistic fuzzy sets IFSs 6, type 2 fuzzy sets T2FSs 3, 7, type n fuzzy sets TnFSs, fuzzy multisets FMSs 8–14, interval-valued fuzzy sets IVFSs 3, interval-valued intuitionistic fuzzy sets IVIFSs, and hesitant fuzzy sets HFSs 17–20

  • We introduce dual hesitant fuzzy set DHFS, which is a new extension of FS

  • The existing sets, including FSs, IFSs, HFSs, and FMSs, can be regarded as special cases of DHFSs; we do not confront an interval of possibilities as in IVFSs or IVIFSs, or some possibility distributions as in T2FSs on the possible values, or multiple occurrences of an element as in FMSs, but several different possible values indicate the epistemic degrees whether certainty or uncertainty

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Summary

Introduction

Since Zadeh 1 introduced fuzzy sets FSs and gave intensive research 2–5 , several famous extensions have been developed, such as intuitionistic fuzzy sets IFSs 6 , type 2 fuzzy sets T2FSs 3, 7 , type n fuzzy sets TnFSs , fuzzy multisets FMSs 8–14 , interval-valued fuzzy sets IVFSs 3, , interval-valued intuitionistic fuzzy sets IVIFSs , and hesitant fuzzy sets HFSs 17–20 These sets have given various ways to assign the membership degree or the nonmembership degree of an element to a given set characterized by different properties. The existing sets, including FSs, IFSs, HFSs, and FMSs, can be regarded as special cases of DHFSs; we do not confront an interval of possibilities as in IVFSs or IVIFSs , or some possibility distributions as in T2FSs on the possible values, or multiple occurrences of an element as in FMSs , but several different possible values indicate the epistemic degrees whether certainty or uncertainty.

Preliminaries
The Notion of DHFS
Basic Operations and Properties of DHFSs
Extension Principle
Concluding Remarks
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