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

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Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 610))

Abstract

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.

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References

  1. Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton Univ Press, Princeton (1944)

    MATH  Google Scholar 

  2. Shapley, L.: A value for n-person games. Contribution Theory of Games, II. Ann. Math. Stud. 28, 307–317 (1953)

    MathSciNet  Google Scholar 

  3. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grabisch, M., Marichal, J., Roubens, M.: Equivalent representation of set functions. Math. Oper. Res. 25(2), 157–178 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Commerce, Senshu University, 2-1-1, Higashimita, Tamaku, Kawasaki, 214-8580, Japan

    Eiichiro Takahagi

Authors
  1. Eiichiro Takahagi
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Corresponding author

Correspondence to Eiichiro Takahagi .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Machine Intelligence Institute, Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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© 2016 Springer International Publishing Switzerland

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Takahagi, E. (2016). On a Fuzzy Integral as the Product-Sum Calculation Between a Set Function and a Fuzzy Measure. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_9

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  • DOI: https://doi.org/10.1007/978-3-319-40596-4_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40595-7

  • Online ISBN: 978-3-319-40596-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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