A unified approach to define fuzzy integrals

Abstract

Recently, the two most general and well-known fuzzy integrals (Sugeno's integral and Choquet's integral) were characterized. It was also shown how both integrals fitted the same formal model, and they only differed in the operators used. In this paper we try to analyze whether it is possible to improve those characterization theorems, and we also study the possibility of defining other fuzzy integrals following the same model but using other operators. In a recent paper, Campos and Bolafios (6) characterized two general and well-known fuzzy integrals: Sugeno's integral (11) and Choquet's integral (7), the last one also called monotone expectation (2, 4, 5, 6). The two characterization theorems are based on the same formal model of four properties, which only differ in the operators used: sum and product for Choquet's integral, maximum and minimum for Sugeno's integral. Two important questions remained open: one involved the characterization theorems of these integrals in an attempt to further refine them; the second question was to investigate the possibility of defining other fuzzy integrals using different operators but following the same model. In this paper we give answers to those problems. Section 2 is devoted to describing fuzzy integrals (defined on Sugeno's fuzzy measures), particularly Sugeno's and Choquet's integrals, and their characterizations. In Section 3, we show how the characterization of Choquet's integral can be refined, whereas the same cannot be made for Sugeno's integral. Finally, other fuzzy integrals fitting our formal model are obtained in Section 4.