Entropy, Distance and Similarity Measures under Interval Valued Intuitionistic Fuzzy Environment

Pratiksha Tiwari, Priti Gupta


This paper presents new axiomatic definitions of entropy measure using concept of probability and distance for interval valued intuitionistic fuzzy sets (IvIFSs) by considering degree of hesitancy which is consistent with the definition of entropy given by De Luca and Termini. Thereafter, we propose some entropy measures and also derived relation between distance, entropy and similarity measures for IvIFSs. Further, we checked the performance of proposed entropy and similarity measures on the basis of intuition and compared with the existing entropy and similarity measures using numerical examples. Lastly, proposed similarity measures are used to solve problems in the field of pattern recognition and medical diagnoses. 

Full Text:



Atanassov, K. Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1) (1986) 87–96.

Atanassov, K. T. (1999). Intuitionistic Fuzzy Sets. Heidelberg, New York: Physica-Verlag.

Atanassov, K. , & Gargov, G. . Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (3) (1989) 343–349.

De Luca A.,& Termini S. (1972). A definition of a non-probabilistic entropy in setting of fuzzy sets. Information and Control, 20, 301-312.

Gau, W.L. ,& Buehrer, D.J. , Vague sets, IEEE Transactions on systems, Man and Cybernetics 23 (2) (1993) 610–614.

Grzegorzewski, P. (2004). Distances between intuitionistic fuzzy sets and/orinterval-valued fuzzy sets based on the hausdorff metric. Fuzzy Sets andSystems, 48, 319–328.

Hu, K. & Li, J. (2013), “The entropy and similarity measure of interval valued intuitionistic fuzzy sets and their relationship”, International Journal of Fuzzy Systems, Vol. 15(3),279-288.

Hung, W., & Yang, M., (2006), “Fuzzy entropy in intuitionistic fuzzy sets”, International Journal of Intelligent Systems, Vol21, 443-451.

Kacprzyk, J. (1997). Multistage Fuzzy Control. Chichester: Wiley.

Liu, X.C. , Entropy, distance measure and similarity measure of fuzzy sets andtheir relations, Fuzzy Sets and Systems 52 (1992) 305–318.

Park, J. H., Lin, K.M., Park, J.S. &Kwun, Y.C. (2007), “Distance between interval- valued intuitionistic fuzzy sets ” Journal of Physics: Conference Series 96. Doi:10.1088/1742-6596/96/1/012089.

Singh, P.(2012), “ A new method on measure of similarity between interval-valued intuitionistic fuzzy sets for pattern recognition”, Journal of Applied & Computational Mathematics, Vol.1(1), 1-5.

Sun, M. & Liu, J.,(2012), “New entropy and similarity measures for interval- valued intuitionistic fuzzy sets”, Journal of information & Computational Sciences, vol.9(18), 5799-5806.

Szmidt, E., & Kacprzyk, J. (2000). Distance between intuitionistic fuzzy sets. Fuzzysets and systems, 114(3), 505–518.

Wei, C.P. , Wang, P. , & Zhang, Y.Z., Entropy, similarity measures of interval-valued intuitionistic fuzzy sets and their applications, Inform. Sci. 181 (2011)4273–4286.

Wu, C., Luo, P., Li, Y. &Ren, X. (2014), “ A new similarity measure of interval-valued intuitionistic fuzzy sets considering its hesitancy degree and applications in expert systems”, Mathematical Problems in Engineering, doi:http://dx.doi.org./10.115/2014/359214.

Xia, M. &Xu, Z. (2010), “Some new similarity measures for intuitionistic fuzzy values and their application in group decision making”, Journal of Systems Science and System Engineering, Vol. 19(4), 430-452.

Xu, Z., On similarity measures of interval-valued intuitionistic fuzzy sets and their application to pattern recognition's, Journal of Southeast University(English Edition) 23 (1) (2007 a) 139–143.

Xu, Z. (2007 b), “Some similarity measures of intuitionistic fuzzy sets and their application to multiple attribute decision making”, Fuzzy Optimization and Decision Making, Vol. 6(2), 109-121.

Xu, Z.S. &Yager, R. R., (2009), “Intuitionistic and interval valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group ”, Fuzzy Optimization and decision making, Vol. 8(2), 123-139.

Yang, Y., & Chiclana, F. (2012), “Consistency of 2D and 3D distances of intuitionistic fuzzy sets”, Expert Systems with Applications, Vol. 39(10), 8665–8670.

Yang, Y.J. & Hinde, C., A new extension of fuzzy sets using rough sets: R-fuzzy sets, Information Sciences 180 (2010) 354–365.

Ye, J. (2012), “Multicriteria decision making method using the Dice similarity measure based on the reduct intuitionistic fuzzy sets of interval- valued intuitionistic fuzzy sets”, Applied Mathematical Modelling, vol.36(9), 4466-4472.

Zadeh,L. A., Fuzzy sets, Information and Control 8 (3) (1965) 338–356.

Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I, Information Science 8 (1975) 199–249.

Zang, Q.S., Jiang, S.Y., Jia, B. G., & Luo, S. H., (2010), “Some information measures for interval- valued intuitionistic fuzzy sets”, Information sciences, Vol.180, 5130-5145.

Zhang, Y.J., Ma, P.J., Su, H.,& Zhang, C.P.(2011), “Entropy on interval-valued intuitionistic fuzzy sets and its application in multi-attribute decision making”, 2011 proceedings of 14th International Conference on Fusion (FUSION),1-7.

Zhang, Q., Xing, H., Liu, F., Ye, J. & Tang, P. (2014), “Some entropy measures for interval –valued intuitionistic fuzzy sets based on distances and their relationships with similarity and inclusion measures”, information sciences, 283, 55-69.

Zhang, H., Zhang, W., & Mei, C. (2009), “Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure”, Knowledge-Based Systems, Vol. 22, 449-454.

DOI: https://doi.org/10.31449/inf.v42i4.1303

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.