On a Fuzzy Integral as the Product-Sum Calculation Between a Set Function and a Fuzzy Measure

  • Eiichiro TakahagiEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


We propose the Choquet integral with respect set to a function defined as the product-sum calculation between a set function and a fuzzy measure. The fuzzy integral is an extension of the Choquet integral. The Choquet integral assumes that the interactions among input values are interact fully but the extension assumes the values partially interaction. In this paper, we define another integral expression and analyze its properties. For an input vector the optimal set function is calculated through linear programming. Lastly, we analyze coalitions among set functions that are a cooperative game using the proposed integral.


Set function Choquet integral Fuzzy measure Möbius transformation co-Möbius transformation Linear programming Supermodular Cooperative game 


  1. 1.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton Univ Press, Princeton (1944)zbMATHGoogle Scholar
  2. 2.
    Shapley, L.: A value for n-person games. Contribution Theory of Games, II. Ann. Math. Stud. 28, 307–317 (1953)MathSciNetGoogle Scholar
  3. 3.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Rota, G.-L.C.: On the foundations of combinatorial theory: I.Theory of Möbius functions. Z Wahrscheinlichkeitstheorie und Verwandte Gebiete 2, 340–368 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Murofushi, T., Sugeno, M.: A theory of fuzzy measure: Representation, the Choquet integral and null sets. J. Math. Anal. Appl. 159, 532–549 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fujimoto, K., Murofushi, T.: Some characterizations of the systems represented by choquet and multi-linear functionals through the use of möbius inversion. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 5, 547–561 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grabisch, M., Marichal, J., Roubens, M.: Equivalent representation of set functions. Math. Oper. Res. 25(2), 157–178 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wang, Z., Leung, K.S., Wong, M., Fang, J.: A new type of nonlinear integrals and the computational algorithm. Fuzzy Sets Syst. 112, 223–231 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of CommerceSenshu UniversityKawasakiJapan

Personalised recommendations