JACIII Vol.17 No.4 pp. 493-503
doi: 10.20965/jaciii.2013.p0493


Obtaining Admissible Preference Orders Using Hierarchical Bipolar Sugeno and Choquet Integrals

Katsushige Fujimoto* and Michio Sugeno**

*College of Symbiotic Systems Science, Fukushima University, 1 Kanayagawa, Fukushima 960-1296, Japan

**European Centre for Soft Computing, Gonzalo Gutiérrez Quirós S/N, 33600 Mieres, Spain

February 28, 2013
April 12, 2013
July 20, 2013
admissible preference order, hierarchical bipolar Sugeno and Choquet integral, ordinal preference, piecewise linear functional
In this paper, we demonstrate the modeling capabilities of certain types of fuzzy integrals such as hierarchical bipolar/cumulative-prospect-theory-type Sugeno and Choquet integrals. The notion of admissible preference structures, introduced by Nakama and Sugeno, is one of the weakest restrictions in rational preference structures. Nakama and Sugeno also proved that, under a certain condition, any admissible preference can be modeled by a hierarchical bipolar Sugeno integral. Here, we extend this result and show that if we use an extra dummy attribute, we can use the hierarchical Choquet integral to generate any admissible preference.
Cite this article as:
K. Fujimoto and M. Sugeno, “Obtaining Admissible Preference Orders Using Hierarchical Bipolar Sugeno and Choquet Integrals,” J. Adv. Comput. Intell. Intell. Inform., Vol.17 No.4, pp. 493-503, 2013.
Data files:
  1. [1] A. Chateauneuf, “Modeling attitudes towards uncertainty and risk through the use of choquet integral,” Annals of Operations Research, Vol.52, pp. 1-20, 1994.
  2. [2] Ch. Labreuche and M. Grabisch, “The Representation of Conditional Relative Importance between Criteria,” Annals of Operations Research, Vol.154, pp. 93-112, 2007.
  3. [3] M. Grabisch and C. Labreuche, “A decade of application of the Choquet and Sugeno integrals in multicriteria decision aid,” 4OR, Vol.6, pp. 1-44, 2008.
  4. [4] T. Murofushi, “Semiatoms in Choquet Integral Models of Multiattribute Decision Making,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.9, No.5, pp. 477-483, 2005.
  5. [5] M. Sugeno, “Theory of Fuzzy Integrals and Its Applications,” Ph.D. dissertation, Tokyo Institute of Technology, 1974.
  6. [6] M. Grabisch, “The symmetric Sugeno integral,” Fuzzy Sets and Systems, Vol.139, pp. 473-490, 2003.
  7. [7] T. Nakama and M. Sugeno, “Admissibility of preferences and modeling capability of fuzzy integrals,” In: Proc. of 2012 IEEE World Congress on Computational Intelligence, Brisbane, Australia, pp. 1-8, 2012.
  8. [8] T. Murofushi, M. Sugeno, and K. Fujimoto, “Separated hierarchical decomposition of the Choquet integral,” Int. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems, Vol.5, No.5, pp. 563-585, 1997.
  9. [9] M. Sugeno, T. Murofushi, and K. Fujimoto, “A Hierarchical Decomposition of Choquet Integral Model,” Int J. of Uncertainty, Fuzziness, and Knowledge-Based Systems, Vol.3, No.1, pp. 1-15, 1995.
  10. [10] P. Benvenuti and R. Mesiar, “A note on Sugeno and Choquet integrals,” In: Proc. of IPMU 2000, Madrid, Spain, pp. 582-585, 2000.
  11. [11] R. Mesiar and D. Vivona, “Two-step integral with respect to fuzzy measure,” Tetra Mt. Math. Publ., Vol.16, pp. 359-368, 1999.
  12. [12] Y. Narukawa and V. Torra, “Twofold integral and multi-step Choquet integral,” Kybernetika, Vol.40, No.1, pp. 39-50, 2004.
  13. [13] V. Torra and Y. Narukawa, “On the meta-knowledge Choquet integral and related models,” Int. J. of Intelligent Systems, Vol.20, pp. 1017-1036, 2005.
  14. [14] V. Torra, “On hierarchically S-decomposable fuzzy measures,” Int. J. of Intelligent Systems, Vol.14, pp. 923-934, 1999.
  15. [15] V. Torra and Y. Narukawa, “Modeling Decisions: Information Fusion and Aggregation Operators,” Springer-Verlag Berlin Heidelberg, 2007.
  16. [16] M. Sugeno, “Ordinal Preference models based on S-integrals and their verification,” In: S. Li et al. (Eds.), Nonlinear Mathematics for Uncertainty and its Applications, Springer-Verlag, Berlin-Heidelberg, pp. 1-18, 2011.
  17. [17] L. J. Savage, “The Foundation of Statistics,” John Wiley and Sons, New York, 1954.
  18. [18] M. Grabisch, “The Möbius function on symmetric ordered structures and its application to capacities on finite sets,” Discrete Mathematics, Vol.287, pp. 17-34, 2004.
  19. [19] M. Grabisch and C. Labreuche, “Bi-capacities. Part I: definition, Möbius transform and interaction,” Fuzzy Sets and Systems, Vol.151, pp. 211-236, 2005.
  20. [20] J.M. Bilbao, J. R. Fernández, N. Jiménez, and J. J. López, “A survey of bicooperative games,” In: Pareto Optimality, Game Theory And Equilibria, Springer, New York, pp. 187-216, 2008.
  21. [21] K. Fujimoto and T. Murofushi, “Some characterizations of k-monotonicity through the bipolar Möbius transform in bicapacities,” J. of Advanced Computational Intelligence and Intelligent informatics, Vol.9, No.5, pp. 484-495, 2005.
  22. [22] M. Grabisch and C. Labreuche, “Bi-capacities for Decision Making on Bipolar Scales,” In: Proc. of EUROFUSE Workshop on Information Systems, Varenna, Italy, pp. 185-190, 2002.
  23. [23] S. Ovchinnikov, “Max-min representation of piecewise linear functions,” Contributions to Algebra and Geometry, Vol.43, pp. 297-302, 2002.
  24. [24] T. Murofushi and Y. Narukawa, “A Characterization of Multi-level Discrete Choquet Integral over a finite set,” In: Proc. of Seventh Workshop on Heart and Mind, Kitakyushu, Japan, pp. 33-36, 2002 (in Japaneses).

*This site is desgined based on HTML5 and CSS3 for modern browsers, e.g. Chrome, Firefox, Safari, Edge, Opera.

Last updated on Apr. 05, 2024