This article proposes a new definition, called the generalized fuzzy integral (gFI), and efficient algorithm for calculation for fuzzy set (FS)-valued integrands; and it compares the gFI, numerically and theoretically, with the authors' non-extension-principle (EP)-based FI extension called the nondirect FI (NDFI).
The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multi-criteria decisionmaking, skeletal age-at-death estimation, and multisource (e.g., feature, algorithm, sensor, and confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno's original real-valued integrand and fuzzy measure (FM) for the case of higher order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh's extension principle (EP). Namely, existing extensions focus on fuzzy number (FN), i.e., convex and normal fuzzy set (FS)-valued integrands. In addition to proposing the gFI and efficient algorithm for calculation for FS-value integrands and comparison with the NDFI, the current project investigated examples in skeletal age-at-death estimation in forensic anthropology and outsource fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, it shows there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications. (Publisher abstract modified)