Evaluate Fuzzy Riemann Integrals Using the Monte Carlo Method

Techniques for using the Monte Carlo method to evaluate fuzzy Riemann integrals and improper fuzzy Riemann integrals are proposed in this paper. Owing to the α-level set of the (improper) fuzzy Riemann integral being the closed interval whose end points are the classical (improper) Riemann integrals, it is possible to invoke the Monte Carlo method to approximate the end points of the α-level closed intervals. We develop the strong law of large numbers for fuzzy random variables in order to give the techniques proposed for evaluating the (improper) fuzzy Riemann integrals using the Monte Carlo approach more theoretical support. The membership function of the (improper) fuzzy Riemann integral can be transformed into mathematical programming problems. Therefore, we can obtain the membership value by solving the mathematical programming problems using the commercial optimizer.