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Evaluate fuzzy Riemann integrals using the Monte Carlo method. (English) Zbl 0993.65004

Consider a function \(\widetilde f\) defined on the real line whose values are fuzzy numbers. The fuzzy Riemann integral of \(\widetilde f\) is a fuzzy set whose membership function is \(\xi(r)= \sup_{0\leq\alpha\leq 1}\alpha 1_{A_\alpha}(r)\), where \[ A_\alpha= \Biggl[\int^b_a\widetilde f^L_\alpha(x) dx,\;\int^b_a\widetilde f^R_\alpha(x) dx\Biggr] \] and \([\widetilde f^L_\alpha(x),\widetilde f^R_\alpha(x)]\) is the \(\alpha\)-cut of \(\widetilde f(x)\). After surveying several properties of these integrals, the author shows how to compute them numerically using the Monte Carlo method. The convergence properties are derived using the strong law of large numbers for fuzzy random variables. The membership function of the integral is then evaluated by transforming it into a standard optimization problem.

MSC:

65C05 Monte Carlo methods
26E50 Fuzzy real analysis
28E10 Fuzzy measure theory
65D32 Numerical quadrature and cubature formulas
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References:

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