Soft Computing

, Volume 22, Issue 15, pp 5051–5071 | Cite as

Entropy measures for Atanassov intuitionistic fuzzy sets based on divergence

  • Ignacio MontesEmail author
  • Nikhil R. Pal
  • Susana Montes


In the literature, there are two different approaches to define entropy of Atanassov intuitionistic fuzzy sets (AIFS, for short). The first approach, given by Szmidt and Kacprzyk, measures how far is an AIFS from its closest crisp set, while the second approach, given by Burrillo and Bustince, measures how far is an AIFS from its closest fuzzy set. On the other hand, divergence measures are functions that measure how different two AIFSs are. Our work generalizes both types of entropies using local measures of divergence. This results in at least two benefits: depending on the application, one may choose from a wide variety of entropy measures and the local nature provides a natural way of parallel computation of entropy, which is important for large data sets. In this context, we provide the necessary and sufficient conditions for defining entropy measures under both frameworks using divergence measures for AIFS. We show that the usual examples of entropy measures can be obtained as particular cases of our more general framework. Also, we investigate the connection between knowledge measures and divergence measures. Finally, we apply our results in a multi-attribute decision-making problem to obtain the weights of the experts.


Atanassov intuitionistic fuzzy sets Divergence measures Entropy 



A shorter version of this paper was published at EUSFLAT Conference (Montes et al. 2018). The current version has been updated with additional results, comments and proofs. We acknowledge the financial support by project TIN2014-59543-P.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefzbMATHGoogle Scholar
  2. Bhandari D, Pal N (1993) Some new information measures for fuzzy sets. Inf Sci 67(3):209–228MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bhandari D, Pal N, Majumder D (1992) Fuzzy divergence, probability measure of fuzzy events and image thresholding. Pattern Recogn Lett 13(12):857–867CrossRefGoogle Scholar
  4. Burrillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 78:305–316MathSciNetCrossRefzbMATHGoogle Scholar
  5. De Luca A, Termini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy theory. J Gen Syst 5:301–312MathSciNetzbMATHGoogle Scholar
  6. Deng G, Jiang Y, Fu J (2015) Monotonic similarity measures between intuitionistic fuzzy sets and their relationship with entropy and inclusion measure. Inf Sci 316:348–369CrossRefzbMATHGoogle Scholar
  7. Dubois D, Prade H (eds.) (2000) Fundamentals of Fuzzy Sets. SpingerGoogle Scholar
  8. Farnoosh R, Rahimi M, Kumar P (2016) Removing noise in a digital image using a new entropy method based on intuitionistic fuzzy sets. In: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)Google Scholar
  9. Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst 148:319–328MathSciNetCrossRefzbMATHGoogle Scholar
  10. Guo K (2016) Knowledge measure for Atanassov’s intuitionistic fuzzy sets. IEEE Trans Fuzzy Syst 24(5):1072–1078CrossRefGoogle Scholar
  11. Guo K, Song Q (2014) On the entropy for Atanassov’s intuitionistic fuzzy sets: an interpretation from the perspective of amount of knowledge. Appl Soft Comput 24:328–340CrossRefGoogle Scholar
  12. Hong D, Kim C (1999) A note on similarity measures between vague sets and between elements. Inf Sci 115:83–96MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hung W, Yang M (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distances. Pattern Recogn Lett 25:1603–1611CrossRefGoogle Scholar
  14. Joshi R, Kumar S (2017) Parametric (r, s)-norm entropy on intuitionistic fuzzy sets with a new approach in multi attribute decision making. Fuzzy Inf Eng 9(2):181–203MathSciNetGoogle Scholar
  15. Kacprzyk J, Pedrycz, W (eds.) (2015) Springer Handbook of Computational Intelligence. SpringerGoogle Scholar
  16. Klement E, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  17. Kosko B (1986) Fuzzy entropy and conditioning. Inf Sci 40:166–177MathSciNetCrossRefzbMATHGoogle Scholar
  18. Liang Z, Shi P (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recogn Lett 24:2687–2693CrossRefzbMATHGoogle Scholar
  19. Liu X (1992) Entropy, distance measure and similarity measure of fuzzy sets and their relation. Fuzzy Sets Syst 53:305–318MathSciNetzbMATHGoogle Scholar
  20. Melo-Pinto P, Couto P, Bustince H, Barrenechea E, Pagola M, Fernandez J (2013) Image segmentation using Atanassov’s intuitionistic fuzzy sets. Expert Syst Appl 40(1):15–26CrossRefzbMATHGoogle Scholar
  21. Meng F, Chen X (2016) Entropy and similarity measure of atanassov’s intuitionistic fuzzy sets and their application to pattern recognition based on fuzzy measures. Pattern Analysis and Applications 19(1):11–20. MathSciNetCrossRefGoogle Scholar
  22. Montes I, Janis V, Montes S (2011) An axiomatic definition of divergence for intuitionistic fuzzy sets. Proc EUSFLAT Conf 1:547–553zbMATHGoogle Scholar
  23. Montes I, Janis V, Montes S (2012) Local IF-divergences. Commun Comput Inf Sci 298:491–500zbMATHGoogle Scholar
  24. Montes I, Janis V, Pal N, Montes S (2016) Local divergences for atanassov intuitionistc fuzzy sets. IEEE Trans Fuzzy Syst 24(2):360–373CrossRefGoogle Scholar
  25. Montes I, Montes S, Pal N (2018) On the use of divergences for defining entropies for Atanassov intuitionistic fuzzy sets. Adv Intell Syst Comput 642:554–565Google Scholar
  26. Montes I, Pal N, Janis V, Montes S (2015) Divergence measures for intuitionistic fuzzy sets. IEEE Trans Fuzzy Syst 23(2):444–456CrossRefGoogle Scholar
  27. Montes S, Couso I, Bertoluzza C (1998) Some classes of fuzziness measures from local divergences. Belg J Oper Res Stat Comput Sci 38(2–3):37–49MathSciNetzbMATHGoogle Scholar
  28. Montes S, Couso I, Gil P, Bertoluzza C (2002) Divergence measure between fuzzy sets. Int J Approx Reason 30(2):91–105MathSciNetCrossRefzbMATHGoogle Scholar
  29. Nguyen H (2015) A new knowledge-based measure for intuitionistic fuzzy sets and its application in multiple attribute group decision making. Expert Syst Appl 42:8766–8774CrossRefGoogle Scholar
  30. Pal N, Bejdek J (1994) Measuring fuzzy uncertainty. IEEE Trans Fuzzy Syst 2(2):107–118CrossRefGoogle Scholar
  31. Pal N, Bustince H, Pagola M, Mukherjee U, Goswami D, Beliakov G (2013) Uncertainty with Atanassov intuitionistc fuzzy sets: fuzziness and lack of knowledge. Inf Sci 228:61–74CrossRefzbMATHGoogle Scholar
  32. Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst 114:505–518MathSciNetCrossRefzbMATHGoogle Scholar
  33. Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118:467–477MathSciNetCrossRefzbMATHGoogle Scholar
  34. Szmidt E, Kacprzyk J (2006) An application of intuitionistic fuzzy set similarity measures to a multi-criteria decision making problem. In: Artificial Intelligence and Soft Computing ICAISC. Lecture Notes in Computer Science, vol. 4029, pp. 314–323. SpringerGoogle Scholar
  35. Szmidt E, Kacprzyk J, Bujnowski P (2014) How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets. Inf Sci 257:276–285MathSciNetCrossRefzbMATHGoogle Scholar
  36. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539zbMATHGoogle Scholar
  37. Trillas E, Riera T (1978) Entropies in finite fuzzy sets. Inf Sci 15:159–168MathSciNetCrossRefzbMATHGoogle Scholar
  38. Xu Z (2007) Some similarity measures of intuitionistic fuzzy sets and their application to multiple attribute decision making. Fuzzy Optim Decis Making 6:109–121MathSciNetCrossRefzbMATHGoogle Scholar
  39. Xu Z, Cai X (2010) Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information. Int J Intell Syst 25(6):489–513zbMATHGoogle Scholar
  40. Yager R (1982) On measure of fuzziness and fuzzy complements. Int J Gen Syst 8:169–180MathSciNetCrossRefGoogle Scholar
  41. Zadeh L (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  42. Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning. Inf Sci 8:199–249MathSciNetCrossRefzbMATHGoogle Scholar
  43. Zimmermann HJ (2001) Fuzzy set theory and its applications. SpringerGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and O.R.University of OviedoOviedoSpain
  2. 2.Electronics and Communication Sciences UnitIndian Statistical InstituteCalcuttaIndia

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