Skip to main content
SpringerLink
Log in

New operations over hesitant fuzzy sets

  • Original Article
  • Published:
Fuzzy Information and Engineering

Abstract

Hesitant fuzzy sets are considered to be the way to characterize vague phenomenon. Their study has opened a new area of research and applications. Set operations on them lead to a number of properties of these sets which are not evident in classical (crisp) sets make the area mathematically also very productive. Since these sets are defined in terms of functions and set of functions, which is not the case when the sets are crisp, it is possible to define several set operations. Such a study enriches the use of these sets. In this paper, four new operations are envisaged, defined and taken up to study a score of new identities on hesitant fuzzy sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zadeh L A (1965) Fuzzy sets. Information and Control 8(3): 338–353

    Article  MathSciNet  MATH  Google Scholar 

  2. Zadeh L A (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Information Sciences 8(3): 199–249

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Atanassov K T (1999) Intuitionistic fuzzy sets. Springer Physica-Verlag, Heidelberg

    Book  MATH  Google Scholar 

  5. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New York

    MATH  Google Scholar 

  6. Yager R R (1986) On the theory of bags. International Journal of General Systems 13(1): 23–37

    Article  MathSciNet  Google Scholar 

  7. Mizumoto M, Tanaka K (1981) Fuzzy sets of type-2 under algebraic product and algebraic sum. Fuzzy Sets and Systems 5(3): 277–290

    Article  MathSciNet  MATH  Google Scholar 

  8. Mizumoto M, Tanaka K (1981) Some properties of fuzzy sets of type-2. Information and Control 31(4): 312–340

    Article  MathSciNet  Google Scholar 

  9. Karnik N N, Mendel J M (2001) Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122(2): 327–348

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Szmidt E, Kacprzyk J (2001) Intuitionistic fuzzy sets in intelligent data analysis for medical diagnosis. Lecture Notes in Computer Science 2074: 263–271

    Article  Google Scholar 

  12. Liu HW, Wang G J (2007) Multi-criteria decision-making methods based on intuitionistic fuzzy sets. European Journal of Operation Research 179(1): 220–233

    Article  MATH  Google Scholar 

  13. Vlachos I K, Sergiadis G D (2007) Intuitionistic fuzzy information-application to pattern recognition. Pattern Recognition Letters 28(2): 197–206

    Article  Google Scholar 

  14. Wei G W (2010) GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting. Knowledge Based Systems 23(3): 243–247

    Article  Google Scholar 

  15. Wu J Z, Zhang Q (2011) Multicriteria decision making method based on intuitionistic fuzzy weighted entropy. Expert Systems with Applications 38(1): 916–922

    Article  Google Scholar 

  16. Verma R, Sharma B D (2011) Intuitionistic fuzzy sets: some new results. Notes on Intuitionistic Fuzzy Sets 17(3): 1–10

    MathSciNet  MATH  Google Scholar 

  17. Verma R, Sharma B D (2013) Some new equalities connected with intuitionistic fuzzy sets. Notes on Intuitionistic Fuzzy Sets (in press)

    Google Scholar 

  18. Verma R, Sharma B D (2013) Some new results on intuitionistic fuzzy sets. Proceedings of the Jangjeon Mathematical Society 16(1): 101–114.

    Google Scholar 

  19. Verma R, Sharma B D (2012) On generalized intuitionistic fuzzy divergence (relative information) and their properties. Journal of Uncertain Systems 6(4): 308–320

    MathSciNet  Google Scholar 

  20. Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. The 18th IEEE international Conference on Fuzzy Systems, Jeju Island, Korea: 1378–1382

    Google Scholar 

  21. Torra V (2010) Hesitant fuzzy sets. International Journal of Intelligent Systems 25(6): 529–539

    MATH  Google Scholar 

  22. Xia M, Xu Z S (2011) Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning 52(3): 395–407

    Article  MathSciNet  MATH  Google Scholar 

  23. Xia M M, Xu Z S, Chen N (2011) Induced aggregation under confidence levels. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 19(2): 201–227

    Article  MathSciNet  Google Scholar 

  24. Xu Z S, Xia M M (2011) Distance and similarity measures on hesitant fuzzy sets. Information Sciences 181(11): 2128–2138

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu Z S, Xia M M (2011) On distance and correlation measures of hesitant fuzzy information. International Journal of Intelligent Systems 26(5): 410–425

    Article  MATH  Google Scholar 

  26. Xia M M, Xu Z S, Chen N (2013) Some hesitant fuzzy aggregation operators with their application in decision making. Group Decision and Negotiation 22(2): 259–279

    Article  Google Scholar 

  27. Zhu B, Xu Z S, Xia M (2012) Hesitant fuzzy geometric Bonferroni means. Information Sciences 205: 72–85

    Article  MathSciNet  MATH  Google Scholar 

  28. Xu Z S (2007) Intuitionistic fuzzy aggregation operators. IEEE Transaction on Fuzzy Systems 15(6): 1179–1187

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jaypee Institute of Information Technology, A-10, Sector-62, Noida, 201307, India

    Rajkumar Verma & Bhu Dev Sharma

Authors
  1. Rajkumar Verma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bhu Dev Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rajkumar Verma.

About this article

Cite this article

Verma, R., Sharma, B.D. New operations over hesitant fuzzy sets. Fuzzy Inf. Eng. 5, 129–146 (2013). https://doi.org/10.1007/s12543-013-0137-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12543-013-0137-1

Keywords

Search

Navigation