Skip to main content

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

Abstract

Several local and global properties of (extended) aggregation functions are discussed and their relationships are examined. Some special classes of averaging, conjunctive and disjunctive aggregation functions are reviewed. A special attention is paid to the weighted aggregation functions, including some construction methods

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

Access this chapter

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. Alsina C, Frank MJ, Schweizer B (2003) Problems on associative functions. Aequationes Math 66:128–140.

    Article  MATH  MathSciNet  Google Scholar 

  2. Alsina C, Nelsen RB, Schweizer B (1993) On the characterization of a class of binary operations on distributions functions. Stat Probab Lett 17:85–89.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bajraktarevič M (1963) Sur une généralisation des moyennes quasilin\’eaire. Publ Inst Math Beograd 17:69–76.

    Google Scholar 

  4. Bassan B, Spizzichino F (2005) Relations among univariate aging, bivariate aging and dependence for excheangable lifetimes. J Multivariate Anal 93:313–339.

    Article  MATH  MathSciNet  Google Scholar 

  5. Calvo T, Mesiar R (2001) Criteria importances in median-like aggregation. IEEE Trans on Fuzzy Systems 9:662–666.

    Article  Google Scholar 

  6. Calvo T, Mesiar R (2001) Weighted means based on triangular conorms. Int J Uncertainty, Fuzziness and Knowledge–Based Systems 9:183–196.

    MATH  MathSciNet  Google Scholar 

  7. Calvo T, Kolesárová A, Komorníková M, Mesiar R (2002) Aggregation Operators: Properties, Classes and Construction Methods. In: Calvo T, Mayor G, Mesiar R (eds) Aggregation Operators. Physica-Verlag, Heidelberg, pp. 3–107.

    Google Scholar 

  8. Calvo T, Mesiar R (2003) Fusion with quantitative weights. In: Proc. of Eusflat’03, Zittau, Germany, pp. 312–317.

    Google Scholar 

  9. Calvo T, Mesiar R, Yager RR (2004) Quantitative weights and aggregation. IEEE Transactions on Fuzzy Systems 12:62–69.

    Article  Google Scholar 

  10. Calvo T, Mesiar R (2004) How to incorporate weights in the aggregation process? In: Atanassov KT, Hryniewicz O, Kacprzyk J (eds) Soft Computing Foundations and Theoretical Aspects. Akademicka Oficyna Wydawnicza, Warszawa.

    Google Scholar 

  11. Calvo T, Mesiarová A, Valášková Ľ (2003) Construction of aggregation operators: new composition method. Kybernetika 39:643–650.

    MathSciNet  Google Scholar 

  12. Choquet G (1953–54) Theory of capacities. Ann Inst Fourier 5:131–295.

    MathSciNet  Google Scholar 

  13. Darsow WF, Nguyen B, Olsen ET (1992) Copulas and Markov processes. Illinois J Math 36:600–642.

    MATH  MathSciNet  Google Scholar 

  14. De Baets B (1995) Model implicators and their characterization. PhD Thesis, University Gent.

    Google Scholar 

  15. De Baets B (1999) Idempotent uninorms. Europ J Oper Research 180:631–642.

    Article  Google Scholar 

  16. De Baets B, Fodor JC (eds) (2003) Principles of Fuzzy Preference Modelling and Decision Making. Academia Press, Ghent.

    Google Scholar 

  17. De Baets B, Mesiar R (1997) Pseudo-metrics and T-equivalences. J Fuzzy Mathematics 5:471–481.

    MATH  Google Scholar 

  18. Del Amo A, Montero J, Molina E (2001) Representation of recursive rules. Europ J Operat Research 130:29–53.

    Article  MATH  MathSciNet  Google Scholar 

  19. Denneberg D (1994) Non–additive Measure and Integral. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  20. Driankov D, Hellendoorn H, Reinfrank M (1996) An Introduciton to Fuzzy Control. Second Edition, Springer Verlag, Berlin.

    Google Scholar 

  21. Dubois E, Prade H (1985) A review of fuzzy set aggregation connectives. Inform Sci 36:85–121.

    Article  MATH  MathSciNet  Google Scholar 

  22. Dubois D, Prade H (2004) On the use of aggregation operations in information fusion processes. Fuzzy Sets and Systems 142:143–161.

    Article  MATH  MathSciNet  Google Scholar 

  23. Fodor JC, Yager RR, Rybalov A (1997) Structure of uninorms. Int J Uncertainty, Fuzziness and Knowledge–Based Systems 5:411–427.

    Article  MathSciNet  Google Scholar 

  24. Fodor JC, Roubens M (1994) Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  25. Frank MJ (1979) On the simultaneous associativity of F(x,y) and x+y-F(x,y). Aequationes Math 19:194–226.

    Article  MATH  MathSciNet  Google Scholar 

  26. Grabisch M, Marichal J-L, Mesiar R, Pap E (200x) Aggregation Functions for Decision Making. Monograph in preparation.

    Google Scholar 

  27. Genest C, Quesada Molina JJ, Rodrí guez Lallena JA, Sempi C (1999) A characterization of quasi–copulas. J Multivariate Anal 69:193–205.

    Article  MATH  MathSciNet  Google Scholar 

  28. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  29. Klement EP, Mesiar R, Pap E (2004) Measure–based aggregation operators. Fuzzy Sets and Systems 142:3–14.

    Article  MATH  MathSciNet  Google Scholar 

  30. Klement EP, Mesiar R, Pap E (2002) Invariant copulas. Kybernetika 38:275–285.

    MathSciNet  Google Scholar 

  31. Kolesárová A, Mesiar R, Sempi C (2006) Compatibility of copulas. In: Proccedings IPMU’06, Paris, 1:658–663.

    Google Scholar 

  32. Kolesárová A (2003) 1–Lipschitz aggregation operators and quasi–copulas. Kybernetika 39:615–629.

    MathSciNet  Google Scholar 

  33. Marques Pereira RA, Ribeiro RA (2003) Aggregation with generalized mixture operators using weighting functions. Fuzzy Sets and Systems 137:43–58.

    Article  MATH  MathSciNet  Google Scholar 

  34. Mesiar R (200x) Fuzzy set approach to the utility, preference relations, and aggregation operators. Europ J Operational Research.

    Google Scholar 

  35. Mesiarová A (200x) A note on two open problems of Alsina, Frank and Schweizer. Aequationes Math.

    Google Scholar 

  36. Montero J (2004) Some new results on recursive aggregation rules. In: Proc. IPMU’2004, Perugia, pp. 1235–1242.

    Google Scholar 

  37. Nelsen RB (1999) An Introduction to Copulas. Lecture Notes in Statistics 139, Springer Verlag, New York.

    Google Scholar 

  38. Pap E (1995) Null–Additive Set Functions. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  39. Pradera A, Trillas E (2002) A note on pseudo-metric aggregation. Int J General Systems 31:41–51.

    Article  MATH  MathSciNet  Google Scholar 

  40. Saminger S, Mesiar R, Bodenhofer U (2002) Domination of aggregation operators and preservation of transitivity. Int J Uncertainty, Fuzziness Knowledge-Based Systems 10:11–35.

    Article  MATH  MathSciNet  Google Scholar 

  41. Schweizer B, Sklar A (1983) Probabilistic Metric Spaces. North–Holland, New York.

    MATH  Google Scholar 

  42. Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Statist Univ Paris 8:229–231.

    MathSciNet  Google Scholar 

  43. Sugeno M (1974) Theory of Fuzzy Integrals and Applications. PhD Thesis, Tokyo Inst of Technology, Tokyo.

    Google Scholar 

  44. Šabo M, Kolesárová A, Varga Š (2001) Ret operators generated by triangular norms and copulas. Int J Uncertainty, Fuzziess and Knowlwdge–Based Systems 9:169–181.

    Google Scholar 

  45. Šabo M, Strežo P (2005) On reverses of some binary operations. Kybernetika 41:437–452.

    Google Scholar 

  46. Špirková J (2005) Weighting functions for aggregation operators. In: Proc. AGOP’ 2005, Lugano, pp. 127–130.

    Google Scholar 

  47. Struk P (2006) Extremal fuzzy integrals. Soft Computing 10:502–505.

    Article  MATH  Google Scholar 

  48. Narukawa Y, Torra V (2003) Twofold integral and multi–step Choquet integral. In: Proc. AGOP’ 2003, Alcal\’a de Henares, pp. 135–140.

    Google Scholar 

  49. Yager RR (1988) On ordered weighted averaging aggregation operators in multi–criteria decision making. IEEE Transactions on Systems, Man and Cybernetics 18:183–190.

    Article  MATH  MathSciNet  Google Scholar 

  50. Yager RR, Filev D (1994) Essentials of Fuzzy Modelling and Control. J. Wiley&Sons, New York.

    Google Scholar 

  51. Yager RR (2001) Uninorms in fuzzy system modeling. Fuzzy Sets and Systems 122:167–175.

    Article  MATH  MathSciNet  Google Scholar 

  52. Yager RR, Rybalov A (1997) Understanding the median as a fusion operator. Int J General Systems 26:239–263.

    Article  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Mesiar, R., Kolesárová, A., Calvo, T., Komorníková, M. (2008). A Review of Aggregation Functions. 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_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73723-0_7

  • 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