The Improper Fuzzy Riemann Integral and its Numerical Integration

The improper fuzzy Riemann integral and its numerical integration are proposed in this paper. Under the setting of the improper fuzzy Riemann integral, we can ®nd the expectation of fuzzy random variable. The a-level set of the improper fuzzy Riemann integral is a closed interval whose end points are the improper Riemann integrals. Thus we provide a numerical method to approximate the improper fuzzy Riemann integral by invoking the Simpson's rule. We also ®t the end points (closed interval) of the a-level set of the improper fuzzy Riemann integral as polynomials with variable a in a least-squares sense. Finally, the membership function of the improper fuzzy Riemann integral can be transformed into nonlinear programming problem and be solved by any presented optimizer. Ó 1998 Elsevier Science Inc. All rights reserved. Kewwords: Closed fuzzy number; Improper fuzzy Riemann integral; Least-squares; Nonlinear programming; Simpson's rule

• Type: Journal
• Year: 1998
• Where: ISCI
• Authors: Hsien-Chung Wu
• Keywords: Improper Fuzzy Riemann, Improper Riemann Integrals, ISCI 1998, Riemann Integral