Entropy and cross-entropy are very vital for information discrimination under complicated Pythagorean fuzzy environment. Firstly, the novel score factors and indeterminacy factors of intuitionistic fuzzy sets (IFSs) are proposed, which are linear transformations of membership functions and non-membership functions. Based on them, the novel entropy measures and cross-entropy measures of an IFS are introduced using Jensen Shannon-divergence (J-divergence). They are more in line with actual fuzzy situations. Then the cases of Pythagorean fuzzy sets (PFSs) are extended. Pythagorean fuzzy entropy, parameterized Pythagorean fuzzy entropy, Pythagorean fuzzy cross-entropy, and weighted Pythagorean fuzzy cross-entropy measures are introduced consecutively based on the novel score factors, indeterminacy factors and J-divergence. Then some comparative experiments prove the rationality and effectiveness of the novel entropy measures and cross-entropy measures. Additionally, the Pythagorean fuzzy entropy and cross-entropy measures are designed to solve pattern recognition and multiple criteria decision making (MCDM) problems. The numerical examples, by comparing with the existing ones, demonstrate the applicability and efficiency of the newly proposed models.
AtanassovK.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems20(1) (1986), 87–96.
2.
BilalM.A. and ShabirM., Approximations of pythagorean fuzzy sets over dual universes by soft binary relations, Journal of Intelligent & Fuzzy Systems41 (2021), 2495–2511.
3.
BurilloP. and BustinceH., Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems78(3) (1996), 305–316.
4.
FanJ.P., YanY. and WuM.Q., TOPSIS and cross entropy method for multicriteria decision making under Pythagorean fuzzy environment, Computer Engineering and Applications54(16) (2018), 146–151
5.
HungW.L. and YangM.S., Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent Systems21(4) (2006), 443–451.
6.
HungW.L. and YangM.S., On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition, Information Sciences178(6) (2008), 1641–1650.
7.
HungK.C., LinK.P. and WengC.C., Fault diagnosis of turbine using an improved intuitionistic fuzzy cross entropy approach, IEEE International Conference on Fuzzy Systems, Taipei, Taiwan: IEEE, 2011, pp. 590–594.
8.
KullbackS., Information theory and statistics, Wiley, New York, 1997.
9.
LambertiP.W., MajteyA.P., BorrasA., CasasM. and PlastinoA., Metric character of the quantum Jensen-Shannon divergence, Physical Review A77(5) (2008), 052311.
10.
LiD.Q. and ZengW.Y., Distance measure of Pythagorean fuzzy sets, International Journal of Intelligent Systems33(2) (2018), 348–361.
11.
LinJ.H., Divergence measures based on the Shannon entropy, IEEE Transactions on Information Theory37(1) (1991), 145–151.
12.
LiuZ.M., LiuP.D., LiuW.L. and PangJ.Y., Pythagorean uncertain linguistic partitioned Bonferroni mean operators and their application in multi-attribute decision making, Journal of Intelligent & Fuzzy Systems32(3) (2017), 2779–2790.
13.
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.
14.
NaeemK., RiazM. and AfzalD., Pythagorean m-polar fuzzy sets and TOPSIS method for the selection of advertisement mode, Journal of Intelligent & Fuzzy Systems37(6) (2019), 8441–8458.
15.
NaeemK., RiazM. and KaraaslanF., Some novel features of Pythagorean m-polar fuzzy sets with applications, Complex & Intelligent Systems7 (2020), 459–475.
16.
NganR.T., CuongB.C. and AliM., H-max distance measure of intuitionistic fuzzy sets in decision making, Applied Soft Computing69 (2018), 393–425.
17.
PengX.D., YuanH.Y. and YangY., Pythagorean fuzzy information measures and their applications, International Journal of Intelligent Systems32(10) (2017), 991–1029.
18.
RenP.J., XuZ.S. and GouX.J., Pythagorean fuzzy TODIM approach to multi-criteria decision making, Applied Soft Computing42 (2016), 246–259.
19.
SamuelA.E. and RajakumarS., Intuitionistic fuzzy set with modal operators in medical diagnosis, Advances in Fuzzy Mathematics12(2) (2017), 167–175.
20.
SinghS. and GanieA.H., On some correlation coefficients in Pythagorean fuzzy environment with applications, International Journal of Intelligent Systems35(4) (2020), 682–717.
21.
SongY.F., FuQ., WangY.F. and WangX.D., Divergence-based cross entropy and uncertainty measures of Atanassovąŕs intuitionistic fuzzy sets with their application in decision making, Applied Soft Computing Journal84 (2019), 105703.
22.
SzmidtE. and KacprzykJ., Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems118(3) (2001), 467–477.
23.
TengF., LiuZ.M. and LiuP.D., Some power Maclaurin symmetric mean aggregation opertors based on Pythagorean fuzzy linguistic numbers and their application to group decision making, International Journal of Intelligent Systems33(9) (2018), 1949–1985.
24.
ThaoN.X. and SmarandacheF., A new fuzzy entropy on Pythagorean fuzzy sets, Journal of Intelligent & Fuzzy Systems37(1) (2019), 1065–1074.
25.
ThaoN.X., A newcorrelation coefficient of the Pythagorean fuzzy sets and its applications, Soft Computing (2019). DOI: 10.1007/s00500-019-04457-7.
26.
VlachosI.K. and SergiadisG.D., Intuitionistic fuzzy information Applications to pattern recognition, Pattern Recognition Letters28(2) (2007), 197–206.
27.
WeiG.W. and WeiY., Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications, International Journal of Intelligent Systems33(3) (2018), 634–652.
28.
WuX.H., WangJ.Q., PengJ.J. and ChenX.H., Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems, International Journal of Intelligent Systems18(6) (2016), 1104–1116.
29.
XiaM.M. and XuZ.S., Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment, Information Fusion13(1) (2012), 31–47.
30.
XiaoF.Y. and DingW.P., Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis, Applied Soft Computing Journal79 (2019), 254–267.
31.
XueW.T., XuZ.S., ZhangX.L. and TianX.L., Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making, International Journal of Intelligent Systems33(1) (2018), 93–125.
32.
YagerR.R., Pythagorean fuzzy subsets, Proceeding of the Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, 2013, pp. 57–61.
33.
YagerR.R., Pythagorean membership grades in multicriteria decision making, IEEE Transactions on Fuzzy Systems22(4) (2014), 958–965.
34.
YeJ., Multicriteria fuzzy decision-making method based on the intuitionistic Fuzzy cross-entropy, International Conference on Intelligent Human-Machine Systems and Cybernetics, Hangzhou, Zhejiang, 2009, pp. 59–61.
35.
YeJ., Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment, European Journal of Operational Research205(1) (2010), 202–204.
36.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
37.
ZhangQ.S. and JiangS.Y., A note on information entropy measures for vague sets and its applications, Information Sciences178 (2008), 4184–4191.
38.
ZhangH.M., Entropy for intuitionistic fuzzy sets based on distance and intuitionistic index, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems21(1) (2013), 139–155.
39.
ZhangX.L. and XuZ.S., Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, International Journal of Intelligent Systems29(12) (1061), 1061–1078.
40.
ZhangX.L., Pythagorean fuzzy clustering analysis: A hierarchical clustering algorithm with the ratio index-based ranking methods, International Journal of Intelligent Systems33(9) (2018), 1798–1822.
