Hesitant fuzzy entropy and cross‐entropy and their use in multiattribute decision‐making

By Zeshui Xu, Meimei Xia. International Journal of Intelligent Systems, 2012

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A VIKOR-based method for hesitant fuzzy multi-criteria decision making (2013, background): “Subsequently, Xu and Xia (2011a) investigated the distance, similarity, and correlation measures for HFSs. Xu and Xia (2011b, 2012) focused on the distance and correlation, entropy measures of hesitant fuzzy information.”
Application of decision-making techniques in supplier selection: A systematic review of literature (2013, background): “A new generalization of fuzzy sets called hesitant fuzzy sets has recently been proposed by Torra (2010), it’s the theoretical basis of which has received considerable attention (e.g. Rodriguez, Martinez, & Herrera, 2012; Wei, 2012; Xu & Xia, 2011; Xu & Xia, 2012).”
Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems (2014, background): “Farhadinia [11], Xu and Xia [68], Peng et al.”
Hesitant Fuzzy Sets: State of the Art and Future Directions (2014, methodology): “[52] Let h1 and h2 be two HFEs and h σ(i) 1 (i = 1, .”
Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets (2013, methodology): “The significant term in the definition of the entropy EA (respectively, EB) is the cross-entropy measure of HFEs, denoted by CA (respectively, CB), which has to satisfy the following conditions (see [36]): ð1Þ C ðhMðxÞ;hNðxÞÞP 0; ð2Þ C ðhMðxÞ;hNðxÞÞ 1⁄4 0 iff h M ðxÞ 1⁄4 h rðjÞ N ðxÞ; j 1⁄4 1;2; .”