Computing with Words Using Weighted Power Mean Aggregation Operators

  • John T. Rickard
  • Janet Aisbett
  • Ronald R. Yager
  • Greg Gibbon
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 291)


Weighted power means are a flexible and powerful family of aggregation functions. The simplest member of this family, the weighted arithmetic mean, previously has been adapted for interval type-2 fuzzy scores and weights. This operator has been termed a “linguistic weighted average,” and has been a primary instantiation of a “perceptual computer” in recent literature. We present an algorithm for computing weighted power means of arbitrary power for type-1 or interval type-2 fuzzy inputs and weights, which we call “linguistic weighted power means.” We compare the linguistic weighted power mean and the linguistic weighted average on an “investment judgment advisor” example. Our results illustrate the flexibility and range of logical inference provided by this very versatile aggregation operator for computing with words applications.


aggregation operators type-2 fuzzy logic computing with words perceptual computing 


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  1. 1.
    Dubois, D., Prade, H.: A review of fuzzy set aggregation connectives. Inf. Sci. 36, 85–121 (1985)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Salton, G., Fox, E.A., Wu, H.: Extended Boolean information retrieval. Communications of the ACM 26(12), 1022–1036 (1983)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Hong, W.S., Chen, S.J., Wang, L.H., et al.: A new approach for fuzzy information retrieval based on weighted power-mean averaging operators. Computers & Mathematics with Applications 53(12), 1800–1819 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dujmović, J.: Continuous preference logic for system evaluation. IEEE Trans. on Fuzzy Systems 15(6), 1082–1099 (2007)CrossRefGoogle Scholar
  5. 5.
    Gärdenfors, P.: Conceptual Spaces: The Geometry of Thought. MIT Press (2000)Google Scholar
  6. 6.
    Goldstone, R.: Influences of categorization on perceptual discrimination. J. Experimental Psychology: General 123, 178–200 (2000)CrossRefGoogle Scholar
  7. 7.
    Schyns, P., Goldstone, R., Thibaut, J.-P.: The development of features in object concepts. Behavioural and Brain Science 21, 1–54 (1998)Google Scholar
  8. 8.
    Gigerenzer, G., Todd, P., and the ABC Research Group: Simple Heuristics That Make Us Smart. Oxford University Press, NY (1999)Google Scholar
  9. 9.
    Liu, F., Mendel, J.M.: Aggregation using the fuzzy weighted average as computed using the Karnik-Mendel algorithms. IEEE Trans. Fuzzy Syst. 17(6) (2009)Google Scholar
  10. 10.
    Mendel, J.M., Wu, D.: Perceptual Computing. John Wiley & Sons, Inc., Hoboken (2010)CrossRefGoogle Scholar
  11. 11.
    Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Trans. Fuzzy Syst. 4, 103–111 (1996)CrossRefGoogle Scholar
  12. 12.
    Special Section on Computing with Words. IEEE Trans. Fuzzy Syst. 18 (June 2010)Google Scholar
  13. 13.
    Mendel, J.M.: Computing with words: Zadeh, Turing, Popper and Occam. IEEE Computational Intelligence Magazine 2, 10–17 (2007)CrossRefGoogle Scholar
  14. 14.
    Yager, R.R.: Approximate reasoning as a basis for computing with words. In: Zadeh, L.A., Kacprzyk, J. (eds.) Computing with Words in Information/Intelligent Systems I: Foundations, pp. 50–77. Physica-Verlag, Heidelberg (1999)Google Scholar
  15. 15.
    Yager, R.R.: On the retranslation process in Zadeh’s paradigm of computing with words. IEEE Trans. on Systems, Man, and Cybernetics—Part B: Cybernetics 34, 1184–1195 (2004)CrossRefGoogle Scholar
  16. 16.
    Zadeh, L.A.: Toward human-level machine intelligence—Is it achievable? The need for a new paradigm shift. IEEE Computational Intelligence Magazine 3, 11–22 (2008)CrossRefGoogle Scholar
  17. 17.
    Wagner, C., Hagras, H.: Toward general type-2 fuzzy logic systems based on zSlices. IEEE Trans. on Fuzzy Systems 18(4), 637–660 (2010)CrossRefGoogle Scholar
  18. 18.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man Cybern. 18, 183–190 (1988)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Larssen, H.: Efficient andness-directed importance weighted averaging operators. Int. J. Uncertainty, Fuzziness, Knowledge-Based Syst. 12, 67–82 (2003)CrossRefGoogle Scholar
  20. 20.
    Dujmović, J., Larsen, H.L.: Generalized conjunction/disjunction. Int. J. Approx. Reas. 46, 423–446 (2007)MATHCrossRefGoogle Scholar
  21. 21.
    Dujmović, J.: Partial absorption functions. J. Univ. Belgrade, EE Dept. Ser. Math & Physics (659), 156–163 (1979)Google Scholar
  22. 22.
    Rickard, J.T., Aisbett, J., Yager, R., Gibbon, G.: Fuzzy weighted power means in evaluation decisions. In: Proc. World Symposium on Soft Computing, Paper #100, San Francisco, CA (2011)Google Scholar
  23. 23.
    Wu, D., Mendel, J.M.: Aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Trans. on Fuzzy Systems 15(6), 1145–1161 (2007)CrossRefGoogle Scholar
  24. 24.
    Wu, D., Mendel, J.M.: Corrections to ‘Aggregation using the linguistic weighted average and interval type-2 fuzzy sets’. IEEE Trans. on Fuzzy Systems 16(6), 1664–1666 (2008)CrossRefGoogle Scholar
  25. 25.
    Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice-Hall, Upper Saddle River (1995)MATHGoogle Scholar
  26. 26.
    Tong, R.M., Bonissone, P.P.: A linguistic approach to decision making with fuzzy sets. IEEE Trans. Syst., Man, Cybern.  SMC-10, 716–723 (1980)Google Scholar
  27. 27.
    Liu, F., Mendel, J.M.: Aggregation using the fuzzy weighted average as computed using the Karnick-Mendel algorithms. IEEE Trans. Fuzzy Syst. 17 (December 2009)Google Scholar
  28. 28.
    Rickard, J.T., Aisbett, J., Yager, R., Gibbon, G.: Linguistic weighted power means: comparison with the linguistic weighted average. In: Proc. FUZZ-IEEE 2011, 2011 World Congress on Computational Intelligence, Taipei, Taiwan, pp. 2185–2192 (June 2011)Google Scholar
  29. 29.
    Trillas, E., Moraga, C., Guadarrama, S., Cubillo, S., Castiñeira, E.: Computing with Antonyms. In: Nikravesh, M., Kacprzyk, J., Zadeh, L.A. (eds.) Forging New Frontiers: Fuzzy Pioneers I. STUDFUZZ, vol. 217, pp. 133–153. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • John T. Rickard
    • 1
  • Janet Aisbett
    • 2
  • Ronald R. Yager
    • 3
  • Greg Gibbon
    • 2
  1. 1.Distributed Infinity, Inc.LarkspurUnited States
  2. 2.Faculty of Science and ITThe University of NewcastleCallaghanAustralia
  3. 3.Machine Intelligence InstituteIona CollegeNew RochelleUnited States

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