Abstract
Hesitant fuzzy sets (HFSs) are a useful tool to manage situations in which the decision makers (DMs) hesitate about several possible values for the membership to assess a variable, alternative, etc. However, HFSs have the information loss problem and cannot identify different DMs, which interferes with the application of HFSs in decision making. To overcome these limitations, we develop the extended hesitant fuzzy sets (EHFSs) in this paper. As an extension of HFSs, EHFSs have close relationships with existing fuzzy sets including intuitionistic fuzzy sets (IFSs), fuzzy multisets (FMSs), type-2 fuzzy sets (T2FSs), dual hesitant fuzzy sets (DHFSs), and especially HFSs. We propose a concept of extended hesitant fuzzy elements (EHFEs), then study the basic operations and the desirable properties of EHFEs in detail. Some extended hesitant distance measures are developed to illustrate their advantages comparing with the existing hesitant distance measures. To extend EHFSs to decision making, we combine the proposed distance measures with the Dempster-Shafer belief structure.
Highlights
Zadeh (1965) introduced fuzzy sets (FSs) as a powerful tool to address fuzziness, which have wide applications in practice (Baležentis et al 2012; Stankevičienė, Mencaitė 2012)
We have developed the extended hesitant fuzzy sets (EHFSs) to resolve the information loss problem of hesitant fuzzy sets (HFSs) in this paper, and have shown that intuitionistic fuzzy sets (IFSs), HFSs and dual hesitant fuzzy sets (DHFSs) are particular cases of EHFSs with certain conditions
EHFSs can be represented as fuzzy multisets (FMSs) or type-2 fuzzy sets (T2FSs)
Summary
Researchers developed some extensions of FSs, such as intuitionistic fuzzy sets (IFSs) (Atanassov 1986), type-2 fuzzy sets (T2FSs) (Zadeh 1975; Mizumoto, Tanaka 1976; Dubois 1980), fuzzy multisets (FMSs) (Yager 1986), interval-valued fuzzy sets (IVFSs) (Zadeh 1975), interval-valued intuitionistic fuzzy sets (IVIFSs) (Atanassov, Gargov 1989), hesitant fuzzy sets (HFSs) (Torra 2010) and dual hesitant fuzzy sets (DHFSs) (Zhu et al 2012b). EHFSs have close relationships with IFSs, T2FSs, FMSs, DHFSs and especially HFSs. EHFSs are an extension of HFSs, on the contrary, HFSs can be considered as a particular case of EHFSs. EHFSs increase the richness of numerical representation based on the value-groups, enhance the modeling abilities of HFSs, and can identify different DMs in decision making, which expand the applications of HFSs in practice. An energy policy example is given to illustrate our results
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