1 Grey relational analysis for hesitant fuzzy sets and 2 interval - valued hesitant fuzzy sets with applications 3 to MADM problems 4

Quantitative and qualitative fuzzy measures have been proposed to hesitant fuzzy sets 9 (HFSs) from different points. However, few of the existing HFSs fuzzy measures refer to the grey 10 relational analysis (GRA) theory. Actually, the GRA theory is very useful in the fuzzy measure 11 domain, which has been developed for such the intuitionistic fuzzy sets. Therefore, in this paper, 12 we apply the GRA theory to the HFSs and interval-valued hesitant fuzzy sets (IVHFS) domain. We 13 propose the HFSs grey relational degree, HFSs slope grey relational degree, HFSs synthetic grey 14 relational degree and IVHFSs grey relational degree, which describe the closeness, the variation 15 tendency and both the closeness and variation tendency of HFSs and closeness of IVHFSs, 16 respectively, greatly enriching the fuzzy measures of HFSs. Furthermore, with the help of the 17 TOPSIS method, we develop the grey relational based MADM methodology to solve the HFSs and 18 IVHFSs MADM problems. Finally, combined with two practical MADM examples about energy 19 policy selection with HFSs information and emergency management evaluation with IVHFSs 20 information, we obtain the most desirable decision results, and compared with the previous 21 methods, the validity, effectiveness and accuracy of the proposed grey relational degree for HFSs 22 and IVHFSs are demonstrated in detail. 23