Skip to main content
SpringerLink
Log in

Aggregation Operators in Interval-valued Fuzzy and Atanassov’s Intuitionistic Fuzzy Set Theory

  • Chapter
Fuzzy Sets and Their Extensions: Representation, Aggregation and Models

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 220))

Abstract

In this chapter we give an overview of some recent advances on aggregation operators on L I, where L I is the underlying lattice of interval-valued fuzzy set theory (which is equivalent to Atanassov’s intuitionistic fuzzy set theory). We discuss some special classes of t-norms on L I and their properties. We show that the t-representable t-norms, which are constructed as a pair of t-norms on [0,1], are not the t-norms with the most interesting properties. We study additive generators of t-norms on L I, uninorms on L I and generators of uninorms on L I. We give the general definition and some special classes of aggregation operators on L I. Finally we discuss the generalization of Yager’s OWA operators to interval-valued fuzzy set theory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. T. Atanassov. Intuitionistic fuzzy sets, 1983. VII ITKR’s Session, Sofia (deposed in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. K. T. Atanassov. Intuitionistic fuzzy sets. Physica-Verlag, Heidelberg/New York, 1999.

    MATH  Google Scholar 

  4. G. Birkhoff. Lattice Theory, Vol. 25. AMS Colloquium Publications, Providence, Rhode Island, 1973.

    Google Scholar 

  5. H. Bustince and P. Burillo. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets and Systems, 79(3):403–405, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  6. T. Calvo, A. Kolesárová, M. Komorníková, and R. Mesiar. Aggregation operators: properties, classes and construction methods. In T. Calvo, G. Mayor, and R. Mesiar, editors, Aggregation Operators: New Trends and Applications, pp. 3–104. Physica-Verlag, Heidelberg/New York, 2002.

    Google Scholar 

  7. T. Calvo and R. Mesiar. Continuous generated associative aggregation operators. Fuzzy Sets and Systems, 126(2):191–197, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. De Baets and J. Fodor. On the structure of uninorms and their residual implicators. In S. Gottwald and E. P. Klement, editors, Proceedings of the 18th Linz Seminar on Fuzzy Set Theory: Enriched Lattice Structures for Many-Valued and Fuzzy Logics, pp. 81–87, 1997.

    Google Scholar 

  9. G. Deschrijver. Additive and multiplicative generators in interval-valued fuzzy set theory. IEEE Transactions on Fuzzy Systems. in press.

    Google Scholar 

  10. G. Deschrijver. Algebraic operators in interval-valued fuzzy set theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. submitted.

    Google Scholar 

  11. G. Deschrijver. The Archimedean property for t-norms in interval-valued fuzzy set theory. Fuzzy Sets and Systems. in press.

    Google Scholar 

  12. G. Deschrijver. Generators of t-norms in interval-valued fuzzy set theory. In Proceedings of the 4th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2005), pp. 253–258, 2005.

    Google Scholar 

  13. G. Deschrijver, C. Cornelis, and E. E. Kerre. On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems, 12(1):45–61, 2004.

    Article  MathSciNet  Google Scholar 

  14. G. Deschrijver and E. E. Kerre. Classes of intuitionistic fuzzy t-norms satisfying the residuation principle. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 11(6):691–709, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Deschrijver and E. E. Kerre. On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems, 133(2):227–235, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Deschrijver and E. E. Kerre. Uninorms in L*-fuzzy set theory. Fuzzy Sets and Systems, 148(2):243–262, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Deschrijver and E. E. Kerre. Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems, 153(2):229–248, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Eisenberg. Axiomatic theory of sets and classes. Holt, Rinehart and Winston, Inc., New York, 1971.

    Google Scholar 

  19. J. Fodor and B. De Baets. On uninorms and their residual implicators. In P. Sinčák, J. Vaščák, V. Kvasnička, and J. Pospíchal, editors, Intelligent Technologies - Theory and Applications, pp. 91–97. IOS Press, 2002.

    Google Scholar 

  20. J. Fodor, R. Yager, and A. Rybalov. Structure of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge Based Systems, 5:411–427, 1997.

    Article  MathSciNet  Google Scholar 

  21. J. C. Fodor and S. Ovchinnikov. On aggregation of T-transitive fuzzy binary relations. Fuzzy Sets and Systems, 72(2):135–145, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. C. Fodor and M. Roubens. Fuzzy preference modelling and multicriteria decision support. Kluwer Academic Publishers, Dordrecht, 1994.

    MATH  Google Scholar 

  23. W. L. Gau and D. J. Buehrer. Vague sets. IEEE Trans. Systems Man. Cybernet., 23(2): 610–614, 1993.

    Article  MATH  Google Scholar 

  24. J. Goguen. L-fuzzy sets. Journal of Mathematical Analysis and Applications, 18(1):145–174, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. B. Gorzalçany. A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems, 21(1):1–17, 1987.

    Google Scholar 

  26. M. Grabisch. Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems, 69(3):279–298, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. P. Klement, R. Mesiar, and E. Pap. On the relationship of associative compensatory operators to triangular norms and conorms. Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 4:129–144, 1996.

    Article  MathSciNet  Google Scholar 

  28. E. P. Klement, R. Mesiar, and E. Pap. Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets and Systems, 104(1):3–13, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. Kluwer Academic Publishers, Dordrecht, 2002.

    Google Scholar 

  30. E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy Sets and Systems, 143(1):5–26, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  31. E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. Position paper II: general constructions and parametrized families. Fuzzy Sets and Systems, 145(3):411–438, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  32. A. Kolesárová and M. Komorníková. Triangular norm-based iterative compensatory operators. Fuzzy Sets and Systems, 104(1):109–120, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  33. H. L. Larsen. Importance weighted OWA aggregation of multicriteria queries. In Proceedings of the North American Fuzzy Information Processing Society Conference (NAFIPS’99), pp. 740–744, New York, 1999.

    Google Scholar 

  34. C.-H. Ling. Representation of associative functions. Publ. Math. Debrecen, 12:189–212, 1965.

    MathSciNet  Google Scholar 

  35. H. B. Mitchell. An intuitionistic OWA operator. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12(6):843–860, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  36. H. B. Mitchell and D. D. Estrakh. An OWA operator with fuzzy ranks. International Journal of Intelligent Systems, 13:69–82, 1998.

    Article  MATH  Google Scholar 

  37. H. B. Mitchell and P. A. Schaefer. Multiple priorities in an induced ordered weighted averaging operator. International Journal of Intelligent Systems, 15:317–328, 2000.

    Article  MATH  Google Scholar 

  38. H. B. Mitchell and P. A. Schaefer. On ordering fuzzy numbers. International Journal of Intelligent Systems, 15:981–994, 2000.

    Article  MATH  Google Scholar 

  39. R. Sambuc. Fonctions Φhi-floues. Application à l’aide au diagnostic en pathologie thyroidienne. Ph. D. Thesis, Université de Marseille, France, 1975.

    Google Scholar 

  40. P. A. Schaefer and H. B. Mitchell. A generalized OWA operator. International Journal of Intelligent Systems, 14:123–144, 1999.

    Article  MATH  Google Scholar 

  41. J. Wang, S. Y. Liu, J. Zhang, and S. Y. Wang. On the parametrized OWA operators for fuzzy MCDM based on vague set theory. Fuzzy Optimization and Decision Making, 5:5–20, 2006.

    Article  MathSciNet  Google Scholar 

  42. R. Yager. On ordered weighted averaging operators in multicriteria decision making. IEEE Transactions on Systems, Man and Cybernetics, 18(1):183–190, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Yager and D. Filev. Induced ordered weighted averaging operators. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 29:141–150, 1999.

    Article  Google Scholar 

  44. R. Yager and A. Rybalov. Uninorm aggregation operators. Fuzzy Sets and Systems, 80(1): 111–120, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  45. R. Yager and A. Rybalov. Full reinforcement operators in aggregation techniques. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 28(6):757–769, 1998.

    Article  Google Scholar 

  46. H.-J. Zimmermann and P. Zysno. Latent connectives in human decision making. Fuzzy Sets and Systems, 4(1):37–51, 1980.

    Article  MATH  Google Scholar 

Download references

Authors
  1. Glad Deschrijver
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Etienne Kerre
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Automatics and Computation, Universidad Pública de Navarra, Campus Arrosadia s/n, 31006 Pamplona, Spain

    Humberto Bustince

  2. Department of Computer Science and Artificial Intelligence (DECSAI), University of Granada, Periodista Daniel Saucedo Aranda s/n, 18071 Granada, Spain

    Francisco Herrera

  3. Faculty of Mathematics, Universidad Complutense, Plaza de las Ciencias 3, 28040 Madrid, Spain

    Javier Montero

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Deschrijver, G., Kerre, E. (2008). Aggregation Operators in Interval-valued Fuzzy and Atanassov’s Intuitionistic Fuzzy Set Theory. In: Bustince, H., Herrera, F., Montero, J. (eds) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Studies in Fuzziness and Soft Computing, vol 220. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73723-0_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73723-0_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73722-3

  • Online ISBN: 978-3-540-73723-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation