Fuzzy integral of vector valued functions and its mathematical model

In this paper, a fuzzy integral of vector valued functions is developed by extending the mapping Φ:R×R→R of utility function with mutual utility independence to the mapping Φ^*:V×V→R. The extended mapping Φ ^* 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 A²(V) respectively. If Φ is a monotone increasing function, because any measure is constituted by a fuzzy measure, then Φ^* 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