The development of information measures associated with fuzzy and intuitionistic fuzzy sets is an important research area from the past few decades. Divergence and entropy are two significant information measures in the intuitionistic fuzzy set (IFS) theory, which have gained wider attention from researchers due to their extensive applications in different areas. In the literature, the existing information measures for IFSs have some drawbacks, which make them irrelevant to use in application areas. In order to obtain more robust and flexible information measures for IFSs, the present work develops and studies some parametric information measures under the intuitionistic fuzzy environment. First, the paper reviews the existing intuitionistic fuzzy divergence measures in detail with their shortcomings and then proposes four new order-α divergence measures between two IFSs. It is worth mentioning that the developed divergence measures satisfy several elegant mathematical properties. Second, we define four new entropy measures called order-α intuitionistic fuzzy entropy measures in order to quantify the fuzziness associated with an IFS. We prove basic and advanced properties of the order-α intuitionistic fuzzy entropy measures for justifying their validity. The paper shows that the introduced measures include various existing fuzzy and intuitionistic fuzzy information measures as their special cases. Further, utilizing the conventional multi-attributive border approximation area comparison (MABAC) model, we develop an intuitionistic fuzzy MABAC method to solve real-life multiple attribute group decision-making problems. Finally, the proposed method is demonstrated by using a practical application of personnel selection.
KullbackS. and LeiblerR.A., On information and sufficiency, The Annals of Mathematical Statistics22 (1951), 79–86.
2.
ShannonC.E., A mathematical theory of communication, Bell System Technical Journal27 (1948), 623–656. https://doi.org/10.1002/j.1538-7305.1948.tb00917.x
3.
RényiA., On measures of information and entropy. In: Proceedings of the fourth Berkeley Symposiumon Mathematics, Statistics and Probability (1960), pp. 547–561.
4.
TanejaI.J., On generalized information measures and their applications. In: Advances in Electronics and Electron Physics. (1989), pp. 327–413.
5.
LinJ., Divergence measure based on Shannon entropy, IEEE Transactions on Information Theory37 (1991), 145–151.
6.
KapurJ.N., Measures of Fuzzy Information. Mathematical Sciences Trust Society, Delhi, India (1997).
7.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
8.
BhandariD. and PalN.R., Some new information measures for fuzzy sets, Information Sciences67 (1993), 209–228. https://doi.org/10.1016/0020-0255(93)90073-U
9.
ShangX.G. and JiangW.S., A note on fuzzy information measures, Pattern Recognition Letters18 (1997), 425–432. https://doi.org/10.1016/S0167-8655(97)00028-7
10.
BajajR.K. and HoodaD.S., Generalized measures of fuzzy directed-divergence, total ambiguity and information improvement, Journal of Applied Mathematics, Statistics and Informatics6 (2010), 31–44.
11.
VermaR., On generalized fuzzy divergence measure and their application to multi-criteria decision making, Journal of Combinatorics, Information & System Sciences39 (2014), 191–213.
12.
VermaR. and MaheshwariS., A new measure of divergence with its application to multi-criteria decision making under fuzzy environment, Neural Computing and Applications28 (2017), 2335–2350. https://doi.org/10.1007/s00521-016-2311-y
13.
VermaR., MerigóJ.M. and SahniM., On generalized fuzzy Jensen-exponential divergence and its application to pattern recognition. In: Proceedings of the 2018 IEEE Symposium Series on Computational Intelligence, SSCI 2018 (2019).
WeiG.W., Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting, Knowledge-Based Systems21 (2008), 833–836. https://doi.org/10.1016/j.knosys.2008.03.038
16.
VermaR. and SharmaB.D., On generalized intuitionistic fuzzy divergence (relative information) and their properties, Journal of Uncertain Systems6 (2012), 308–320.
17.
XiaM. and XuZ., Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment, Information Fusion13 (2012), 31–47. https://doi.org/10.1016/j.inffus.2010.12.001
18.
VermaR. and SharmaB.D., Intuitionistic fuzzy Jensen-Rényi divergence: applications to multiple-attribute decision making, Informatica37 (2013), 399–409.
19.
VermaR. and SharmaB.D., A new measure of inaccuracy with its application to multi-criteria decision making under intuitionistic fuzzy environment, Journal of Intelligent and Fuzzy Systems27 (2014), 1811–1824. https://doi.org/10.3233/IFS-141148
20.
MaheshwariS. and SrivastavaA., Study on divergence measures for intuitionistic fuzzy sets and its application in medical diagnosis, Journal of Applied Analysis and Computation6 (2016), 772–789.
21.
IakovidisD.K. and PapageorgiouE., Intuitionistic fuzzy cognitive maps for medical decision making, IEEE Transactions on Information Technology in Biomedicine15 (2011), 100–107. https://doi.org/10.1109/TITB.2010.2093603
22.
HungW.L. and YangM.S., On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition, Information Sciences178 (2008), 1641–1650. https://doi.org/10.1016/j.ins.2007.11.006
23.
LuoX., LiW. and ZhaoW., Intuitive distance for intuitionistic fuzzy sets with applications in pattern recognition, Applied Intelligence48 (2018), 2792–2808. https://doi.org/10.1007/s10489-017-1091-0
24.
BasE., YolcuU. and EgriogluE., Intuitionistic fuzzy time series functions approach for time series forecasting, Granular Computing (2020), 1–11. https://doi.org/10.1007/s41066-020-00220-8
25.
XiangxiaoL., HonglinO. and LijuanX., Kernel-distance based intuitionistic fuzzy c-means clustering algorithm and its application, Pattern Recognition and Image Analysis29 (2019), 592–597. https://doi.org/10.1134/S1054661819040199
26.
DoganO., OztaysiB. and Fernandez-LlatasC., Segmentation of indoor customer paths using intuitionistic fuzzy clustering: Process mining visualization, Journal of Intelligent and Fuzzy Systems38 (2020), 675–684. https://doi.org/10.3233/JIFS-179440
27.
ChairaT. and PanwarA., An Atanassov’s intuitionistic fuzzy kernel clustering for medical image segmentation, International Journal of Computational Intelligence Systems7 (2014), 360–370. https://doi.org/10.1080/18756891.2013.865830.
28.
VlachosI.K. and SergiadisG.D., Intuitionistic fuzzy information- applications to pattern recognition, Pattern Recognition Letters28 (2007), 197–206. https://doi.org/10.1016/j.patrec.2006.07.004
29.
WeiP. and YeJ., Improved intuitionistic fuzzy cross-entropy and its application to pattern recognitions. In: International Conference on Intelligent Systems and Knowledge Engineering (2010), pp. 114–116.
30.
MaoJ., YaoD. and WangC., A novel cross-entropy and entropy measures of IFSs and their applications, Knowledge-Based Systems48 (2013), 37–45. https://doi.org/10.1016/j.knosys.2013.04.011
31.
MishraA.R. and RaniP., Shapley divergence measures with VIKOR method for multi-attribute decision-making problems, Neural Computing and Applications31 (2019), 1299–1316. https://doi.org/10.1007/s00521-017-3101-x
32.
AnsariM.D., MishraA.R. and AnsariF.T., New divergence and entropy measures for intuitionistic fuzzy sets on edge detection, International Journal of Fuzzy Systems20 (2018), 474–487. https://doi.org/10.1007/s40815-017-0348-4
33.
MishraA.R., JainD. and HoodaD.S., Exponential intuitionistic fuzzy information measure with assessment of service quality, International Journal of Fuzzy Systems19 (2017), 788–798. https://doi.org/10.1007/s40815-016-0278-6
34.
SongY., FuQ., WangY.F. and WangX., Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making, Applied Soft Computing Journal84 (2019). https://doi.org/10.1016/j.asoc.2019.105703
35.
BurilloP. and BustinceH., Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems78 (1996), 305–316. https://doi.org/10.1016/0165-0114(96)84611-2
36.
SzmidtE. and KacprzykJ., Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems118 (2001), 467–477. https://doi.org/10.1016/S0165-0114(98)00402-3
37.
De LucaA. and TerminiS., A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Information and Control20 (1972), 301–312. https://doi.org/10.1016/S0019-9958(72)90199-4
38.
ZhangQ.S. and JiangS.Y., A note on information entropy measures for vague sets and its applications, Information Sciences178 (2008), 4184–4191. https://doi.org/10.1016/j.ins.2008.07.003
39.
VermaR. and SharmaB.D., Exponential entropy on intuitionistic fuzzy sets, Kybernetika49 (2013), 114–127.
40.
PalN.R. and PalS.K., Entropy: A new definition and its applications, IEEE Transactions on Systems, Man and Cybernetics21 (1991), 1260–1270. https://doi.org/10.1109/21.120079
41.
LiuM. and RenH., A new intuitionistic fuzzy entropy and application in multi-attribute decision making, Information5 (2014), 587–601. https://doi.org/10.3390/info5040587
42.
VermaR. and SharmaB.D., On intuitionistic fuzzy entropy of order-α, Advances in Fuzzy Systems (2014), 1–8. https://doi.org/10.1155/2014/789890
43.
VermaR. and SharmaB.D., R-norm entropy on intuitionistic fuzzy sets, Journal of Intelligent and Fuzzy Systems28 (2015), 327–335. https://doi.org/10.3233/IFS-141306
44.
PamučarD. and ĆirovićG., The selection of transport and handling resources in logistics centers using Multi-Attributive Border Approximation area Comparison (MABAC), Expert Systems with Applications 42 (2015), 3016–3028. https://doi.org/10.1016/j.eswa.2014.11.057
HeY., HeZ. and ChenH., Intuitionistic fuzzy interaction Bonferroni means and its application to multiple attribute decision making, IEEE Transactions on Cybernetics45 (2015), 116–128. https://doi.org/10.1109/TCYB.2014.2320910
47.
ZengW. and LiH., Relationship between similarity measure and entropy of interval valued fuzzy sets, Fuzzy Sets and Systems157 (2006), 1477–1484. https://doi.org/10.1016/j.fss.2005.11.020
48.
HungW.-L. and YangM.-S., Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent Systems21 (2006), 443–451. https://doi.org/10.1002/int.20131
49.
YeJ., Two effective measures of intuitionistic fuzzy entropy, Computing87 (2010), 55–62. https://doi.org/10.1007/s00607-009-0075-2
50.
WeiC.P., GaoZ.H. and GuoT.T., An intuitionistic fuzzy entropy measure based on the trigonometric function, Control and Decision27 (2012), 571–574.
51.
WangJ. and WangP., Intuitionistic linguistic multi-criteria decision-making method based on intuitionistic fuzzy entropy, Control and Decision27 (2012), 1694–1698.
52.
BhattacharyyaA., On a measure of divergence between two statistical populations defined by their probability distributions, Bulletin of the Calcutta Mathematical Society35 (1943), 99–109.
53.
JoshiR. and KumarS., A new intuitionistic fuzzy entropy of order-α with applications in multiple attribute decision making. In: Proceedings of Sixth International Conference on Soft Computing for Problem Solving. Springer, Patiala, India, (2017), pp. 212–219.
54.
HoodaD.S., On generalized measures of fuzzy entropy, Mathematica Slovaca54 (2004), 315–325.
55.
PengX. and YangY., Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making, International Journal of Intelligent Systems31 (2016), 989–1020. https://doi.org/10.1002/int.21814
56.
XueY.X., YouJ.X., LaiX.D. and LiuH.C., An interval-valued intuitionistic fuzzy MABAC approach for material selection with incomplete weight information, Applied Soft Computing Journal38 (2016), 703–713. https://doi.org/10.1016/j.asoc.2015.10.010
57.
YuS.M., WangJ. and qiangW.J., An interval type-2 fuzzy likelihood-based MABAC approach and its application in selecting hotels on a tourism website, International Journal of Fuzzy Systems19 (2017), 47–61. https://doi.org/10.1007/s40815-016-0217-6
58.
BoranF.E., GençS. and AkayD., Personnel selection based on intuitionistic fuzzy sets, Human Factors and Ergonomics in Manufacturing & Service Industries21 (2011), 493–503. https://doi.org/10.1002/hfm.20252
59.
YeJ., Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment, European Journal of Operational Research205 (2010), 202–204. https://doi.org/10.1016/j.ejor.2010.01.019
60.
LiD.F., WangY.C., LiuS. and ShanF., Fractional programming methodology for multi-attribute group decision-making using IFS, Applied Soft Computing Journal9 (2009), 219–225. https://doi.org/10.1016/j.asoc.2008.04.006
61.
YagerR.R., On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems, Man and Cybernetics18 (1988), 183–190. https://doi.org/10.1109/21.87068
62.
WangW. and LiuX., Intuitionistic fuzzy information aggregation using einstein operations, IEEE Transactions on Fuzzy Systems20 (2012), 923–938. https://doi.org/10.1109/TFUZZ.2012.2189405
63.
HuangJ.Y., Intuitionistic fuzzy Hamacher aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems27 (2014), 505–513. https://doi.org/10.3233/IFS-131019
64.
LiuP. and WangP., Multiple attribute group decision making method based on intuitionistic fuzzy Einstein interactive operations, International Journal of Fuzzy Systems22 (2020), 790–809. https://doi.org/10.1007/s40815-020-00809-w
