Fuzzy integrals—what are they?

We bring an overview of fuzzy integrals, including historical remarks. These integrals can be viewed as an average membership value of fuzzy sets, and they are related to fuzzy measures. The Choquet integral can be traced back to 1925. The Sugeno integral has a predecessor in the Shilkret integral from 1971. Some other fuzzy integrals and the corresponding discrete integrals are introduced too. A closer look to the geometric interpretation of fuzzy integrals is also given, resulting among others the weakest and the strongest regular fuzzy integral. An application of the Choquet integral to additive impreciseness measuring of fuzzy quantities with interesting consequences for fuzzy measures is presented. Finally, recent development and streaming of the fuzzy integral theory is discussed. © 2008 Wiley Periodicals, Inc.

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