Some distances, similarity and entropy measures for interval-valued neutrosophic sets and their relationship

  • Jun YeEmail author
  • Shigui Du
Original Article


This paper proposes some new distance measures between interval-valued neutrosophic sets (IvNSs) and their similarity measures. Then, some entropy measures of IvNS based on the distances are proposed as the extension of the entropy measures of interval-valued intuitionistic fuzzy sets (IvIFSs). Also, we investigate the relationship between the presented entropy measures and the similarity measures for IvNSs. Finally, the comparison of the new entropy measures with existing entropy measures for IvNSs is given by the numerical and decision-making examples to demonstrate that the proposed new entropy measures for IvNSs are effective and reasonable and more intelligible in representing the degree of fuzziness of IvNSs than the existing ones.


Interval-valued neutrosophic set Distance measure Similarity measure Entropy Decision making 



This paper was supported by the National Natural Science Foundation of China (No. 71471172).

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


  1. 1.
    Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefzbMATHGoogle Scholar
  2. 2.
    Atanassov K, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aydoğdu A (2015) On entropy and similarity measure of interval valued neutrosophic sets. Neutrosophic Sets Syst 9:47–49Google Scholar
  4. 4.
    Bustince H, Burillo P (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 78:305–316MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De Luca AS, Termini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy set theory. Inf Control 20:301–312MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ji P, Zhang HY, Wang JQ (2016) A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput Appl. doi: 10.1007/s00521-016-2436-z Google Scholar
  7. 7.
    Majumder P, Samanta SK (2014) On similarity and entropy of neutrosophic sets. J Intell Fuzzy Syst 26(3):1245–1252MathSciNetzbMATHGoogle Scholar
  8. 8.
    Peng JJ, Wang JQ, Zhang HY, Chen XH (2014) An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl Soft Comput 25:336–346CrossRefGoogle Scholar
  9. 9.
    Peng JJ, Wang JQ, Wang J, Zhang HY, Chen XH (2016) Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int J Syst Sci 47(10):2342–2358CrossRefzbMATHGoogle Scholar
  10. 10.
    Peng JJ, Wang J, Wu XH (2016) An extension of the ELECTRE approach with multi-valued neutrosophic information. Neural Comput Appl. doi: 10.1007/s00521-016-2411-8 Google Scholar
  11. 11.
    Shannon CE, Weaver W (1947) The mathematical theory of communications. The University of Illinois Press, UrbanaGoogle Scholar
  12. 12.
    Shannon CE (1948) A mathematical theory of communications. Bell Syst Tech J 27:379–423MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Smarandache F (1998) Neutrosophy. Neutrosophic probability, set, and logic. American Research Press, RehobothzbMATHGoogle Scholar
  14. 14.
    Szmidt E, Kacprzyk J (2001) Entropy on intuitionistic fuzzy sets. Fuzzy Sets Syst 118:467–477MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Turksen I (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20:191–210MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tian ZP, Zhang HY, Wang J, Wang JQ, Chen XH (2016) Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets. Int J Syst Sci 47(15):3598–3608CrossRefzbMATHGoogle Scholar
  17. 17.
    Tian ZP, Wang J, Zhang HY, Wang JQ (2016) Multi-criteria decision-making based on generalized prioritized aggregation operators under simplified neutrosophic uncertain linguistic environment. Int J Mach Learn Cybern. doi: 10.1007/s13042-016-0552-9 Google Scholar
  18. 18.
    Valchos IK, Sergiadis GD (2007) Intuitionistic fuzzy information—a pattern recognition. Pattern Recognit Lett 28:197–206CrossRefGoogle Scholar
  19. 19.
    Wang H, Smarandache F, Zhang YQ, Sunderraman R (2005) Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix, USAzbMATHGoogle Scholar
  20. 20.
    Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single valued neutrosophic sets. Multisp Multistruct 4:410–413zbMATHGoogle Scholar
  21. 21.
    Wei CP, Wang P, Zhang YZ (2011) Entropy, similarity measure of interval valued intuitionistic sets and their applications. Inf Sci 181:4273–4286MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wu XH, Wang J, Peng JJ, Chen XH (2016) Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. Int J Fuzzy Syst. doi: 10.1007/s40815-016-0180-2 Google Scholar
  23. 23.
    Ye J (2010) Two effective measures of intuitionistic fuzzy entropy. Computing 87(1–2):55–62MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ye J (2010) Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Appl Math Model 34:3864–3870MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ye J (2013) Multicriteria decision-making method using the correlation coefficient under single-value neutrosophic environment. Int J Gen Syst 42(4):386–394MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ye J (2014) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26(5):2459–2466MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ye J (2014) Single valued neutrosophic cross-entropy for multicriteria decision making problems. Appl Math Model 38(3):1170–1175MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ye J (2014) Similarity measures between interval neutrosophic sets and their applications in multicriteria decision making. J Intell Fuzzy Syst 26(1):165–172zbMATHGoogle Scholar
  29. 29.
    Ye J (2014) Some aggregation operators of interval neutrosophic linguistic numbers for multiple attribute decision making. J Intell Fuzzy Syst 27(5):2231–2241MathSciNetzbMATHGoogle Scholar
  30. 30.
    Ye J (2017) Multiple attribute group decision making based on interval neutrosophic uncertain linguistic variables. Int J Mach Learn Cybern 8(3):837–848CrossRefGoogle Scholar
  31. 31.
    Zadeh LA (1965) Fuzzy sets and systems. In: Proceedings of the symposium on systems. Theory Polytechnic Institute of Brooklyn, New York, pp 29–37Google Scholar
  32. 32.
    Zadeh L (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang QS, Xing HY, Liu FC, Ye J, Tang P (2014) Some new entropy measures for interval-valued intuitionistic fuzzy sets based on distances and their relationships with similarity and inclusion measures. Inf Sci 283:55–69MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zhang HY, Wang JQ, Chen XH (2014) Interval neutrosophic sets and their application in multicriteria decision making problems. Sci World J 2014 (ID 645953, 15 pages) Google Scholar
  35. 35.
    Zhang HY, Ji P, Wang JQ, Chen XH (2015) Improved weighted correlation coefficient based on integrated weight for interval neutrosophic sets and its application in multi-criteria decision making problems. Int J Comput Intell Syst 8(6):1027–1043CrossRefGoogle Scholar
  36. 36.
    Zhang HY, Wang JQ, Chen XH (2016) An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput Appl 27(3):615–627CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Electrical and Information EngineeringShaoxing UniversityShaoxingPeople’s Republic of China
  2. 2.Department of Civil EngineeringShaoxing UniversityShaoxingPeople’s Republic of China

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