Discrete Interval–valued Choquet Integrals

For a fixed finite universe U = {u1, . . . , un}, a fuzzy subset F of U is given by its membership function F : U → [0, 1] (we will not distinguish fuzzy subsets and the corresponding membership functions notations). For several practical purposes, especially in multicriteria decision making, the expected value E(F ) of F should be introduced. The original Zadeh approach in [11] was based on a probability measure P on U, P (ui) = pi, and then E(F ) = ∑n i=1 pi F (ui). More general approach, not limited by the non–interaction of single elements of U, is based on a fuzzy measure m : 2 → [0, 1], m(∅) = 0, m(U) = 1, m(A) ≤ m(B) whenever A ⊆ B ⊆ U and the Choquet integral [3, 4],