The fuzzy Riemann integral and its numerical integration

The fuzzy Riemann integral and its numerical integration are proposed in this paper. The alpha-level set of this fuzzy Riemann integral is a closed interval whose end points are the classical Riemann integrals, thus we provide a numerical method to approximate the fuzzy Riemann integral by invoking the Simpson's rule. We fit the end points (closed interval) of the alpha-level set of the fuzzy Riemann integral as polynomials with variable alpha in a least-squares sense. Finally, the membership function of the fuzzy Riemann integral can be transformed into nonlinear programming problem and can be solved by any current optimizer. (C) 2000 Elsevier Science B.V. All rights reserved.