Abstract
The Choquet integral is a powerful tool in multi-criteria decision making and decision under uncertainty. This paper studies the use of its discrete form for the definition of norms, in the general case beyond the often considered case of Ordered Weighted Averages. It proposes a discussion of the characterisation based on Metric Inducing Fuzzy Measures (MIFM) introduced by Bolton et al., 2008, questioning its results. It then describes a characterisation for the discrete case that relates to the notion of properties holding almost everywhere derived from the null sets associated to a fuzzy measure. It discusses in particular the case of Choquet integrals induced by possibility measures.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bolton, J., Gader, P., Wilson, J.N.: Discrete Choquet integral as a distance metric. IEEE Trans. Fuzzy Syst. 16(4), 1107–1110 (2008)
Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994)
Dubois, D., Prade, H.: Possibility theory and its applications: where do we stand? In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 31–60. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-43505-2_3
Fodor, J., Roubens, M.: Characterization of weighted maximum and some related operations. Inf. Sci. 84(3), 173–180 (1995)
Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann. Oper. Res. 175(1), 247–290 (2010)
Murofushi, T., Sugeno, M.: A theory of fuzzy measures: representations, the Choquet integral, and null sets. J. Math. Anal. Appl. 159(2), 532–549 (1991)
Narukawa, Y.: Distances defined by Choquet integral. In: Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 511–516 (2007)
Yager, R.R.: Families of OWA operators. Fuzzy Syst. Syst. 59, 125–148 (1993)
Yager, R.R.: Norms induced from OWA operators. IEEE Trans. Fuzzy Syst. 18(1), 57–66 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Rico, A., Lesot, MJ., Marsala, C. (2023). Norms and Discrete Choquet Integrals Induced by Submodular Fuzzy Measures: A Discussion. In: Massanet, S., Montes, S., Ruiz-Aguilera, D., González-Hidalgo, M. (eds) Fuzzy Logic and Technology, and Aggregation Operators. EUSFLAT AGOP 2023 2023. Lecture Notes in Computer Science, vol 14069. Springer, Cham. https://doi.org/10.1007/978-3-031-39965-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-39965-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-39964-0
Online ISBN: 978-3-031-39965-7
eBook Packages: Computer ScienceComputer Science (R0)