Fuzzy integral of vector valued functions and its mathematical model

Abstract

In this paper, a fuzzy integral of vector valued functions is developed by extending the mapping /spl Phi/:R/spl times/R/spl rarr/R of utility function with mutual utility independence to the mapping /spl Phi//sup */:V/spl times/V/spl rarr/R. The extended mapping /spl Phi//sup */ can be regarded as the sum of the Lebesgue integral on an attribute space and an interaction space. They correspond to a vector space V and a second order alternating tensor space A/sup 2/(V) respectively. If /spl Phi/ is a monotone increasing function, because any measure is constituted by a fuzzy measure, then /spl Phi//sup */ can be considered as a fuzzy integral. In addition, numerical examples by using this theory are executed in order to show the effects of the correlation between attributes on the nonadditivity of fuzzy measures.<>