Fuzzy Optimization and Decision Making

, Volume 4, Issue 2, pp 119–130 | Cite as

Learning Weights in the Generalized OWA Operators

  • Gleb Beliakov


This paper discusses identification of parameters of generalized ordered weighted averaging (GOWA) operators from empirical data. Similarly to ordinary OWA operators, GOWA are characterized by a vector of weights, as well as the power to which the arguments are raised. We develop optimization techniques which allow one to fit such operators to the observed data. We also generalize these methods for functional defined GOWA and generalized Choquet integral based aggregation operators.


aggregation operators ordered weighted averaging Choquet integral fuzzy sets 


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  1. Aczel, J. 1969On Applications and Theory of Functional EquationsAcademic PressNew YorkGoogle Scholar
  2. Barrodale, I., Roberts, F. 1980“Algorithm 552: Solution of the Constrained l1 Linear Approximation Problem”ACM Transactions Mathematical Software6231235Google Scholar
  3. Beliakov, G. 2000“Shape Preserving Approximation Using Least Squares Splines”Approximation Theory and Applications168098Google Scholar
  4. Beliakov, G. 2002“Monotone Approximation of Aggregation Operators Using Least Squares Splines”International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems10659676Google Scholar
  5. Beliakov, G. 2003a“Geometry and Combinatorics of the Cutting Angle Method”Optimization52379394Google Scholar
  6. Beliakov, G. 2003b“How to Build Aggregation operators from data?”International Journal of Intelligent Systems18903923Google Scholar
  7. Beliakov, G., T. Calvo, E. García Barriocanal, and M. Sicilia. (2004a), ‘Choquet Integral-Based Aggregation of Interface Usability Parameters: Identification of Fuzzy Measure’. In: 6th International Conference on Optimization Techniques and Applications. Ballarat, Australia.Google Scholar
  8. Beliakov, G., Mesiar, R., Valášková, L. 2004b“Fitting Generated Aggregation Operators to Empirical Data”International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems12219236Google Scholar
  9. Beliakov, G., Warren, J. 2001“Appropriate Choice of Aggregation Operators in Fuzzy Decision Support Systems”IEEE Transactions On Fuzzy Systems9773784Google Scholar
  10. Benvenuti, P., Mesiar, R. 2000“Integrals with Respect to a General Fuzzy Measure”Grabisch, M.Murofushi, T.Sugeno, M eds. Fuzzy Measures and Integrals. Theory and ApplicationsPhysica-VerlagHeidelberg205232Google Scholar
  11. Bezdek, J. C. 1981Pattern Recognition with Fuzzy Objective Function Algorithms, Advanced applications in pattern recognitionPlenum PressNew YorkGoogle Scholar
  12. Calvo, T., Kolesarova, A., Komornikova, M., Mesiar, R. 2002“Aggregation Operators: Properties, Classes and Construction Methods”Calvo, T.Mayor, G.Mesiar, R. eds. Aggregation Operators. New Trends and ApplicationsPhysica-VerlagHeidelberg, New York3104Google Scholar
  13. Chiclana, F., F. Herrera, and E. Herrera-Viedma. (2000). “The Ordered Weighted Geometric Operator: Properties and Applications”, In 8th International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems. Madrid, pp. 985–991.Google Scholar
  14. Denneberg, D. 1994Non-Additive Measure and IntegralKluwerDordrechtGoogle Scholar
  15. Dyckhoff, H., Pedrycz, W. 1984“Generalized Means as Model of Compensative Connectives”Fuzzy Sets and Systems14143154Google Scholar
  16. Filev, D., Yager, R. 1998“On the Issue of Obtaining OWA Operator Weights”Fuzzy Sets and Systems94157169Google Scholar
  17. Grabisch, M. 1997“k-order Additive Discrete Fuzzy Measures and Their Representation”Fuzzy Sets and Systems92167189Google Scholar
  18. Grabisch, M. 2000“The Interaction and Moebius Representation of Fuzzy Measures on Finite Spaces, k-Additive Measures: A Survey”Grabisch, M.Murofushi, T.Sugeno, M. eds. Fuzzy Measures and Integrals. Theory and Applications.Physica-VerlagHeidelberg7093Google Scholar
  19. Grabisch, M., Nguyen, H., Walker, E. 1995Fundamentals of Uncertainty Calculi, with Applications to Fuzzy InferenceKluwerDordrechtGoogle Scholar
  20. Hanson, R., Haskell, K. 1982“Algorithm 587. Two Algorithms for the Linearly Constrained Least Squares Problem”ACM Transactions Mathematical Software8323333Google Scholar
  21. Haskell, K., Hanson, R. 1981“An Algorithm for Linear Least Squares Problems with Equality and Nonnegativity Constraints”Mathematical Programming2198118Google Scholar
  22. Horst, R.Pardalos, P.Thoai, N. eds. 2000“Introduction to Global Optimization, Vol. 48 of Nonconvex Optimization and its Applications”Kluwer Academic PublishersDordrechtGoogle Scholar
  23. Lawson, C., Hanson, R. 1995Solving Least Squares ProblemsSIAMPhiladelphiaGoogle Scholar
  24. Pijavski, S. 1972“An Algorithm For Finding the Absolute Extremum of a Function”USSR Computational Mathematics and Mathematical Physics25767Google Scholar
  25. Rubinov, A. 2000Abstract Convexity and Global Optimization, Vol. 44 of Nonconvex optimization and its applications.Kluwer Academic PublishersDordrecht; BostonGoogle Scholar
  26. Sicilia, M., Garcia Barriocanal, E., Calvo, T. 2003“An Inquiry-Based Method for Choquet integral-based Aggregation of Interface Usability Parameters”Kybernetica39601614Google Scholar
  27. Watson, G. 2000“Approximation in normed linear spaces”Journal of Computational and Applied Mathematics121136Google Scholar
  28. Xu, Z., Da, Q. 2002“The Ordered Weighted Geometric Averaging Operator”International Journal of Intelligent Systems17709716Google Scholar
  29. Yager, R. 1988“On Ordered Weighted Averaging Aggregation Operators in Multicriteria Decision Making”IEEE Transactions on Systems, Man and Cybernetics18183190Google Scholar
  30. Yager, R. 1993“Families of OWA operators”Fuzzy Sets and Systems59125148Google Scholar
  31. Yager, R. 1996“Quantifier Guided Aggregation Using OWA Operators”International Journal of Intelligent Systems114973Google Scholar
  32. Yager, R. 2004“Generalized OWA Aggregation operators”Fuzzy Optimization and Decision Making393107Google Scholar
  33. Yager, R.Kacprzyk, J. eds. 1997The Ordered Weighted Averaging Operators. Theory and Applications.KluwerBostonGoogle Scholar
  34. Zimmermann, H.-J. 1996Fuzzy Set Theory–and its ApplicationsKluwerBostonGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.School of Information TechnologyDeakin UniversityBurwoodAustralia

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