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On fuzzy input data and the worst scenario method. (English) Zbl 1099.90081
Summary: In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set \(\mathcal U_{\text{ad}}\) of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by \(\mathcal U_{\text{ad}}\) and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity. The method takes all the elements of \(\mathcal U_{\text{ad}}\) as equally important though this can be unrealistic and can lead to too pessimistic conclusions. Often, however, additional information expressed through membership function of \(\mathcal U_{\text{ad}}\) is available, i.e., \(\mathcal U_{\text{ad}}\) becomes a fuzzy set. In the article, infinite-dimensional \(\mathcal U_{\text{ad}}\) are considered, two ways of introducing fuzziness into \(\mathcal U_{\text{ad}}\) are suggested, and the worst scenario method operating on fuzzy admissible sets is proposed to obtain a fuzzy set of outputs.

MSC:
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
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References:
[1] Y. Ben-Haim, I. Elishakoff: Convex Models of Uncertainties in Applied Mechanics. Studies in Applied Mechanics, Vol. 25, Elsevier, Amsterdam, 1990. · Zbl 0703.73100
[2] Y. Ben-Haim: Information Gap Decision Theory. Academic Press, San Diego, 2001. · Zbl 0985.91013
[3] A. Bernardini: What are the random and fuzzy sets and how to use them for uncertainty modelling in engineering systems? In: Whys and Hows in Uncertainty Modelling, Probability, Fuzziness and Anti-Optimization. I. Elishakoff (ed.), Springer Verlag, Wien-New York, 1999, pp. 63-125.
[4] B. V. Bulgakov: Fehleranheufung bei Kreiselapparaten. Ingenieur-Archiv 11 (1940), 461-469. · JFM 66.0996.03
[5] B. V. Bulgakov: On the accumulation of disturbances in linear systems with constant coefficients. Dokl. Akad. Nauk SSSR 51 (1940), 339-342.
[6] J. Chleboun: On a reliable solution of a quasilinear elliptic equation with uncertain coefficients. Nonlinear Anal. Theory Methods Appl. 44 (2001), 375-388. · Zbl 1002.35041
[7] I. Elishakoff: An idea of the uncertainty triangle. Shock Vib. Dig. 22 (1990), 1.
[8] Whys and Hows in Uncertainty Modelling, Probability, Fuzziness and Anti-Optimization. CISM Courses and Lectures No. 338, I. Elishakoff (ed.), Springer Verlag, Wien, New York, 1999.
[9] R. G. Ghanem, P. D. Spanos: Stochastic Finite Elements: A Spectral Approach. Springer Verlag, Berlin, 1991. · Zbl 0722.73080
[10] I. Hlaváček: Reliable solutions of problems in the deformation theory of plasticity with respect to uncertain material function. Appl. Math. 41 (1996), 447-466. · Zbl 0870.65095
[11] I. Hlaváček: Reliable solutions of elliptic boundary value problems with respect to uncertain data. Nonlinear Anal. Theory Methods Appl. 30 (1997), 3879-3890, Proceedings of the WCNA-96. · Zbl 0896.35034
[12] Uncertainty: Models and Measures. Proceedings of the International Workshop (Lambrecht, Germany, July 22-24, 1996), Mathematical Research, Vol. 99, H. G. Natke, Y. Ben-Haim (ed.), Akademie Verlag, Berlin, 1997. · Zbl 0868.00034
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