The main purpose of the paper is to develop a general analytical framework of uncertainty measures, which provides a fresh new look into the uncertain information of hesitant fuzzy sets (HFSs). We first refine the axiomatic principles of hesitant fuzzy entropy based on the fuzzy factor and hesitant factor, and propose some distance-based entropy formulas of HFSs. Then, a novel hesitant fuzzy cross-entropy is defined to measure the discrimination of uncertain information of different HFSs. Meanwhile, the relationship between cross-entropy and entropy of HFSs is also discussed, and the hesitant fuzzy entropy may be divided into arithmetic average of fuzzy entropy and hesitant entropy. Moreover, some parameterized uncertainty measures are investigated, and two comparative examples are presented to demonstrate their reasonability. Finally, we apply these proposed measures in multiple attribute decision-making problem to illustrate their efficiency and applicability.
BedregalB., et al., Typical hesitant fuzzy negations, International Journal of Intelligent Systems26 (2014), 525–543.
3.
ChenN., XuZ.S. and XiaM.M., Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modelling37 (2013), 2197–2211.
4.
FarhadiniaB., Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets, Information Sciences240 (2013), 129–144.
5.
FarhadiniaB., A series of score functions for hesitant fuzzy sets, Information Sciences277 (2014), 102–110.
6.
HungW.L. and YangM.S., Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent Systems114 (2006), 443–451.
7.
LiaoH.C. and XuZ.S., Subtraction and division operations over hesitant fuzzy sets, Journal of Intelligent & Fuzzy Systems27 (2014), 65–72.
8.
LiaoH.C. and XuZ.S., Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment, Journal of Intelligent & Fuzzy Systems26 (2014), 1601–1617.
9.
LiaoH.C. and XuZ.S., Extended hesitant fuzzy hybrid weighted aggregation operators and their application in decision making, Soft Computing19 (2015), 2551–2564.
10.
LiaoH.C. and XuZ.S., A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optimization and Decision Making12 (2014), 373–392.
11.
LiaoH.C. and XuZ.S., Satisfaction degree based interactive decision making under hesitant fuzzy environment with incomplete weights, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems22 (2014), 553–572.
12.
LiaoH.C., XuZ.S. and XiaM.M., Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, International Journal of Information Technology & Decision Making13 (2014), 47–76.
13.
LiaoH.C., XuZ.S. and XuJ.P., An approach to hesitant fuzzy multi-stage multi-criterion decision making, Kybernetes43 (2014), 1447–1468.
14.
LiaoH.C., XuZ.S. and ZengX.J., Novel correlation coefficients between hesitant fuzzy sets and their application in decision making, Knowledge-Based Systems82 (2015), 115–127.
15.
LiuH.F., XuZ.S. and LiaoH.C., The multiplicative consistency index of hesitant fuzzy preference relation, IEEE Transactions on Fuzzy Systems (2015). doi: 10.1109/TFUZZ.2015.2426315
16.
LiD.Q., ZengW.Y. and LiJ.H., New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making, Engineering Applications of Artificial Intelligence40 (2015), 11–16.
17.
MaoJ.J., YaoD.B. and WangC.C., A novel cross-entropy and entropy measures of IFSs and their applications, Knowledge-Based Systems48 (2013), 37–45.
18.
PalN.R., et al., Uncertainties with Atanassov’s intuitionistic fuzzy sets: Fuzziness and lack of knowledge, Information Sciences228 (2013), 61–74.
19.
QuirósP., et al., An entropy measure definition for finite interval-valued hesitant fuzzy sets, Knowledge-Based Systems84 (2015), 121–133.
20.
RodriguezR.M., et al., Hesitant fuzzy sets: State of the art and future directions, International Journal of Intelligent Systems29 (2014), 495–524.
21.
SevastjanovP. and DymovaL., Generalised operations on hesitant fuzzy values in the framework of Dempster-Shafer theory, Information Sciences311 (2015), 39–58.
22.
ShangX.G. and JiangW.S., A note on fuzzy information measures, Pattern Recognition Letter18 (1997), 425–432.
23.
TorraV., Hesitant fuzzy sets, International Journal of Intelligent Systems25 (2010), 529–539.
24.
VlachosI.K. and SergiadisG.D., Intuitionistic fuzzy information-applications to pattern recognition, Pattern Recognition Letters28 (2007), 197–206.
25.
XiaM.M. and XuZ.S., Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52 (2011), 395–407.
26.
XiaM.M. and XuZ.S., Entropy/cross entropy-based group decision making intuitionistic fuzzy environment, Information Fusion13 (2012), 31–47.
27.
XuZ.S., Hesitant fuzzy sets theory, Studies in Fuzziness and Soft Computing 314, New York: Springer International Publishing Switzerland, 2014.
28.
XuZ.S. and XiaM.M., Distance and similarity measures for hesitant fuzzy sets, Information Sciences181 (2011), 2128–2138.
29.
XuZ.S. and XiaM.M., On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligent Systems26 (2011), 410–425.
30.
XuZ.S. and XiaM.M., Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making, International Journal of Intelligent Systems27 (2012), 799–822.
31.
XuZ.S. and ZhangX.L., Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information, Knowledge-Based Systems52 (2013), 53–64.
32.
YuD.J., ZhangW.Y. and XuY.J., Group decision making under hesitant fuzzy environment with application to personnel evaluation, Knowledge-Based Systems52 (2013), 1–10.
33.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
34.
ZadehL.A., Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications23 (1968), 421–427.
35.
ZadehL.A., The concept of a linguistic variable and its application to approximate reasoning, Information Sciences8 (1975), 199–249.
36.
ZhangZ.M., Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making, Information Sciences234 (2013), 150–181.
37.
ZhuB., XuZ.S. and XiaM.M., Dual hesitant fuzzy sets, Journal of Applied Mathematics2012 (2012), 13. Article ID 879629.
38.
ZhaoN., XuZ.S. and LiuF.J., Uncertainty measures for Hesitant fuzzy information, International Journal of Intelligent Systems (2015), 1–19.
39.
ZhangY.J., WangY.Z. and WangJ.P., Objective attributes weights determining based on shannon information entropy in hesitant fuzzy multiple attribute decision making, Mathematical Problems in Engineering2014 (2014), 7. Article ID 463930.
