Evaluate Fuzzy Riemann Integrals Using the Monte Carlo Method: Evaluate fuzzy Riemann integrals using the Monte Carlo method
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| Název: | Evaluate Fuzzy Riemann Integrals Using the Monte Carlo Method: Evaluate fuzzy Riemann integrals using the Monte Carlo method |
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| Autoři: | Wu, HC |
| Přispěvatelé: | 管理學院:資訊管理學系, Wu, HC |
| Zdroj: | Journal of Mathematical Analysis and Applications. 264:324-343 |
| Informace o vydavateli: | Elsevier BV, 2001. |
| Rok vydání: | 2001 |
| Témata: | (improper) fuzzy Riemann integrals, convergence, strong law of large numbers, Fuzzy measure theory, Applied Mathematics, Fuzzy real analysis, mathematical programming problems, Monte Carlo methods, fuzzy numbers, 02 engineering and technology, Numerical quadrature and cubature formulas, Monte Carlo method, 0202 electrical engineering, electronic engineering, information engineering, Analysis, fuzzy Riemann integral |
| Popis: | 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. |
| Druh dokumentu: | Article |
| Popis souboru: | application/xml; 104 bytes; text/html |
| Jazyk: | English |
| ISSN: | 0022-247X |
| DOI: | 10.1006/jmaa.2001.7659 |
| Přístupová URL adresa: | https://zbmath.org/1716535 https://doi.org/10.1006/jmaa.2001.7659 https://www.sciencedirect.com/science/article/pii/S0022247X01976590 http://www.sciencedirect.com/science/article/pii/S0022247X01976590 https://core.ac.uk/display/82110980 |
| Rights: | Elsevier Non-Commercial |
| Přístupové číslo: | edsair.doi.dedup.....0085f11e96d9628f73ed237a46a3ae6f |
| Databáze: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | 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. |
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| ISSN: | 0022247X |
| DOI: | 10.1006/jmaa.2001.7659 |