| Title:
|
On fuzzy input data and the worst scenario method (English) |
| Author:
|
Chleboun, Jan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
48 |
| Issue:
|
6 |
| Year:
|
2003 |
| Pages:
|
487-496 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set $\mathcal U_{\mathrm ad}$ 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. (English) |
| Keyword:
|
fuzzy sets |
| Keyword:
|
uncertainty |
| Keyword:
|
worst scenario method |
| MSC:
|
03E72 |
| MSC:
|
49N99 |
| MSC:
|
65K99 |
| MSC:
|
90C70 |
| MSC:
|
90C90 |
| idZBL:
|
Zbl 1099.90081 |
| idMR:
|
MR2025958 |
| DOI:
|
10.1023/B:APOM.0000024488.86492.fb |
| . |
| Date available:
|
2009-09-22T18:15:24Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134545 |
| . |
| Reference:
|
[1] Y. Ben-Haim, I. Elishakoff: Convex Models of Uncertainties in Applied Mechanics.Studies in Applied Mechanics, Vol. 25, Elsevier, Amsterdam, 1990. |
| Reference:
|
[2] Y. Ben-Haim: Information Gap Decision Theory.Academic Press, San Diego, 2001. Zbl 0985.91013, MR 1856675 |
| Reference:
|
[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 |
| Reference:
|
[4] B. V. Bulgakov: Fehleranheufung bei Kreiselapparaten.Ingenieur-Archiv 11 (1940), 461–469. 10.1007/BF02088988 |
| Reference:
|
[5] B. V. Bulgakov: On the accumulation of disturbances in linear systems with constant coefficients.Dokl. Akad. Nauk SSSR 51 (1940), 339–342. (Russian) |
| Reference:
|
[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, MR 1817101, 10.1016/S0362-546X(99)00274-6 |
| Reference:
|
[7] I. Elishakoff: An idea of the uncertainty triangle.Shock Vib. Dig. 22 (1990), 1. 10.1177/058310249002201001 |
| Reference:
|
[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 |
| Reference:
|
[9] R. G. Ghanem, P. D. Spanos: Stochastic Finite Elements: A Spectral Approach.Springer Verlag, Berlin, 1991. MR 1083354 |
| Reference:
|
[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 |
| Reference:
|
[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. MR 1602891, 10.1016/S0362-546X(96)00236-2 |
| Reference:
|
[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, MR 1478000 |
| . |
