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Qualitative Integrals and Desintegrals: How to Handle Positive and Negative Scales in Evaluation

  • Didier Dubois
  • Henri Prade
  • Agnès Rico
Part of the Communications in Computer and Information Science book series (CCIS, volume 299)

Abstract

Integrals are currently used in multiple criteria analysis for synthesizing into a global evaluation the advantages possessed by a potential choice. As such, integrals are operators that increase with the criteria evaluations. However, an item may be also evaluated in terms of its defects. Then the more and the greater the defects, the smaller the evaluation should be. An operator that can provide a synthesis of the defects of an item in this sense is called a desintegral. Desintegrals are maximal when no defects at all are present, while integrals are maximal when all advantages are sufficiently present. So, the greater the value of an integral, or a desintegral, the better the corresponding item since advantages are greater, or defects are smaller respectively. Desintegrals implicitly refer to a negative scale, since an order-reversing mapping of the scale used for evaluating each criterion transforms the degree to which the value is advantageous into a degree to which it is disadvantageous, and conversely. In this paper, we provide an organised description of counterparts to Sugeno integrals that synthesize positive or negative evaluations in the qualitative framework of a totally ordered residuated lattice equipped with an involutive negation. We exploit three kinds of criteria weighting schemes that are allowed by this algebraic structure.

Keywords

Global Evaluation Positive Scale Residuated Lattice Aggregation Operation Negative Scale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Beliakov, G., Bustince, H., Goswami, D.P., Mukherjee, U.K., Pal, N.R.: On averaging operators for Atanassov’s intuitionistic fuzzy sets. Inf. Sci. 181(6), 1116–1124 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonnefon, J.-F., Dubois, D., Fargier, H., Leblois, S.: Qualitative heuristics for balancing the pros and the cons. Theory and Decision 65, 71–95 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dubois, D., Fargier, H., Bonnefon, J.-F.: On the qualitative comparison of decisions having positive and negative features. J. of Artif. Intellig. Res. 32, 385–417 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dubois, D., Prade, H.: A theorem on implication functions defined from triangular norms. Stochastica 8, 267–279 (1984)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dubois, D., Prade, H.: Fuzzy rules in knowledge-based systems Modelling gradedness, uncertainty and preference. In: Yager, R.R., Zadeh, L.A. (eds.) An Introduction to Fuzzy Logic Applications in Intelligent Systems, pp. 45–68. Kluwer Acad. (1992)Google Scholar
  6. 6.
    Dvořák, A., Holčapek, M.: Fuzzy integrals over complete residuated lattices. In: Carvalho, J.P., Dubois, D., Kaymak, U., da Costa Sousa, J.M. (eds.) Proc. Joint 2009 Inter. Fuzzy Systems Association World Congress and 2009 Europ. Society of Fuzzy Logic and Technology Conf (ISFA-EUSFLAT), Lisbon, July 20-24, pp. 357–362 (2009)Google Scholar
  7. 7.
    Grabisch, M.: The Möbius transform on symmetric ordered structures and its application to capacities on finite sets. Discrete Mathematics 287(1-3), 17–34 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grabisch, M.: The symmetric Sugeno integral. Fuzzy Sets Syst. 139, 473–490 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann. Oper. Res. 175, 247–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Greco, S., Matarazzo, B., Slowinski, R.: Bipolar Sugeno and Choquet integrals. In: Proc. EUROFUSE Workshop on Informations Systems, Varenna, Italy, pp. 191–196 (September 2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Didier Dubois
    • 1
  • Henri Prade
    • 1
  • Agnès Rico
    • 2
  1. 1.IRITCNRS and Université de ToulouseFrance
  2. 2.ERICUniversité de LyonFrance

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