zbMATH — the first resource for mathematics

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.

90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
Full Text: DOI EuDML
[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 · 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.
[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 · doi:10.1016/S0362-546X(99)00274-6
[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.
[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 · eudml:32961
[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 · doi:10.1016/S0362-546X(96)00236-2
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.