Enhancing the Expressive Power of Sugeno Integrals for Qualitative Data Analysis

  • Miguel CouceiroEmail author
  • Didier Dubois
  • Henri Prade
  • Agnès Rico
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)


Sugeno integrals are useful for describing families of multiple criteria aggregation functions qualitatively. It is known that Sugeno integrals, as aggregation functions, can be represented by a set of rules. Each rule refers to the same threshold in the conditions about the values of the criteria and in the conclusion pertaining to the value of the integral. However, in the general case we expect rules where several thresholds appear. Some of these rules involving different thresholds can be represented by Sugeno utility functionals where criteria values are rescaled by means of utility functions associated with each criterion. But as shown in this paper, their representation power is quite restrictive. In contrast, we provide evidence to conjecture that the use of disjunctions or conjunctions of Sugeno integrals with utility functions drastically improves the expressive power and that they can capture any aggregation function on a finite scale, understood as piecewise unary aggregation functions.


Sugeno integrals Piecewise unary functions Rule-based representation 



This work is supported by ANR-11-LABX-0040-CIMI (Centre International de Mathématiques et d’Informatique) within the program ANR-11-IDEX-0002-02, project ISIPA.


  1. 1.
    Couceiro, M., Dubois, D., Prade, H., Waldhauser, T.: Decision making with Sugeno integrals. Bridging the gap between multicriteria evaluation and decision under uncertainty. Order 33(3), 517–535 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Couceiro, M., Marichal, J.-L.: Axiomatizations of quasi-polynomial functions on bounded chains. Aequationes Math. 396(1), 195–213 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Couceiro, M., Waldhauser, T.: Pseudo-polynomial functions over finite distributive lattices. Fuzzy Sets Syst. 239, 21–34 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dubois, D., Durrieu, C., Prade, H., Rico, A., Ferro, Y.: Extracting decision rules from qualitative data using Sugeno integral: a case-study. In: Proceedings of the ECSQARU 2015, LNAI 9161, pp. 14–24. Springer, New York (2015)Google Scholar
  5. 5.
    Dubois, D., Marichal, J.-L., Prade, H., Roubens, M., Sabbadin, R.: The use of the discrete Sugeno integral in decision making: a survey. Int. J. Uncertainty, Fuzziness Knowl.-Based Syst. 9, 539–561 (2001)Google Scholar
  6. 6.
    Dubois, D., Prade, H.: Qualitative possibility functions and integrals. In: Handbook of Measure Theory, (E. Pap, edn.), vol. 2, pp. 1469–1521. Elsevier, Amsterdam (2002)Google Scholar
  7. 7.
    Dubois, D., Prade, H., Rico, A.: The logical encoding of Sugeno integrals. Fuzzy Sets Syst. 241, 61–75 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dubois, D., Prade, H., Sabbadin, R.: Decision-theoretic foundations of qualitative possibility theory. Eur. J. Oper. Res. 128, 459–478 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals. Theory and Applications. Physica-verlag, Berlin (2000)zbMATHGoogle Scholar
  10. 10.
    Greco, S., Matarazzo, B., Slowinski, R.: Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. Eur. J. Operarional Res. 158, 271–292 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Marichal, J.-L.: Aggregation operations for multicriteria decision aid. Ph.D. thesis, University of Liège, Belgium (1998)Google Scholar
  12. 12.
    Marichal, J.-L.: On Sugeno integrals as an aggregation function. Fuzzy Sets Syst. 114(3), 347–365 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Prade, H., Rico, A., Serrurier, M., Raufaste, E.: Elicitating Sugeno integrals: methodology and a case study. In: Proceedings of the ECSQARU 2009, LNAI 5590, pp. 712–723. Springer, New York (2009)Google Scholar
  14. 14.
    Rico, A.: Modélisation des préférences pour l’aide à la décision par l’intégrale de Sugeno. Ph.D. thesis, Université Paris I-Panthéon-Sorbonne (2002)Google Scholar
  15. 15.
    Rico, A., Labreuche, C., Grabisch, M., Chateauneuf, A.: Preference modeling on totally ordered sets by the Sugeno integral. Discrete Appl. Math. 147, pp. 113–124 (2005)Google Scholar
  16. 16.
    Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology, Tokyo (1974)Google Scholar
  17. 17.
    Sugeno, M.: Fuzzy measures and fuzzy integrals: a survey. In: Gupta, M.M., Saridis, G.N., Gaines, B.R. (eds.) Fuzzy Automata and Decision Processes, pp. 89–102. North-Holland, Amsterdam (1977)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Miguel Couceiro
    • 1
    Email author
  • Didier Dubois
    • 2
  • Henri Prade
    • 2
  • Agnès Rico
    • 3
  1. 1.Université de Lorraine, LORIA, INRIA Nancy Grand-Est, CNRS, UMR 7503NancyFrance
  2. 2.IRIT-CNRSToulouseFrance
  3. 3.ERICUniversité de LyonLyonFrance

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