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fuzzy sets; uncertainty; worst scenario method
In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set $\mathcal U_{\mathrm ad}$ of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by $\mathcal U_{\mathrm ad}$ 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 $\mathcal U_{\mathrm ad}$ as equally important though this can be unrealistic and can lead to too pessimistic conclusions. Often, however, additional information expressed through a membership function of $\mathcal U_{\mathrm ad}$ is available, i.e., $\mathcal U_{\mathrm ad}$ becomes a fuzzy set. In the article, infinite-dimensional $\mathcal U_{\mathrm ad}$ are considered, two ways of introducing fuzziness into $\mathcal U_{\mathrm ad}$ are suggested, and the worst scenario method operating on fuzzy admissible sets is proposed to obtain a fuzzy set of outputs.
[1] Y.  Ben-Haim, I.  Elishakoff: Convex Models of Uncertainties in Applied Mechanics. Studies in Applied Mechanics, Vol.  25, Elsevier, Amsterdam, 1990.
[2] Y.  Ben-Haim: Information Gap Decision Theory. Academic Press, San Diego, 2001. MR 1856675 | 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. MR 1763168
[4] B. V.  Bulgakov: Fehleranheufung bei Kreiselapparaten. Ingenieur-Archiv 11 (1940), 461–469. DOI 10.1007/BF02088988
[5] B. V.  Bulgakov: On the accumulation of disturbances in linear systems with constant coefficients. Dokl. Akad. Nauk SSSR 51 (1940), 339–342. (Russian)
[6] J.  Chleboun: On a reliable solution of a quasilinear elliptic equation with uncertain coefficients. Nonlinear Anal. Theory Methods Appl. 44 (2001), 375–388. DOI 10.1016/S0362-546X(99)00274-6 | MR 1817101 | Zbl 1002.35041
[7] I.  Elishakoff: An idea of the uncertainty triangle. Shock Vib. Dig. 22 (1990), 1. DOI 10.1177/058310249002201001
[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. MR 1763168
[9] R. G.  Ghanem, P. D.  Spanos: Stochastic Finite Elements: A Spectral Approach. Springer Verlag, Berlin, 1991. MR 1083354
[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. MR 1415251
[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. DOI 10.1016/S0362-546X(96)00236-2 | MR 1602891
[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. MR 1478000 | Zbl 0868.00034
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