# An Analytical Approach to Evaluating Bivariate Functions of Fuzzy Numbers with One Local Extremum

### Arthur Seibel, Josef Schlattmann

#### Abstract

This paper presents a novel analytical approach to evaluating continuous, bivariate functions of independent fuzzy numbers with one local extremum. The approach is based on a parametric a-cut representation of fuzzy numbers and allows for the inclusion of parameter uncertainties into mathematical models.

#### References

- Buckley, J. J. and Qu, Y. (1990). On using a-cuts to evaluate fuzzy equations. Fuzzy Sets and Systems, 38(3):309- 312.
- Degrauwe, D. (2007). Uncertainty propagation in structural analysis by fuzzy numbers. PhD Thesis, Katholieke Universiteit Leuven, Belgium.
- Dong, W. and Shah, H. C. (1987). Vertex method for computing functions of fuzzy variables. Fuzzy Sets and Systems, 24(1):65-78.
- Dubois, D. and Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, NY, USA.
- Fortin, J., Dubois, D., and Fargier, H. (2008). Gradual numbers and their application to fuzzy interval analysis. IEEE Transactions on Fuzzy Systems, 16(2):388-402.
- Hanss, M. (2005). Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Springer, Berlin, Germany.
- Klimke, A. (2006). Uncertainty modeling using fuzzy arithmetic and sparse grids. PhD Thesis, University of Stuttgart, Germany.
- Moens, D. and Hanss, M. (2011). Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances. Finite Elements in Analysis and Design, 47(1):4-16.
- Scheerlinck, K. (2011). Metaheuristic versus tailormade approaches to optimization problems in the biosciences. PhD Thesis, Ghent University, Belgium.
- Seibel, A. and Schlattmann, J. (2013). An analytical approach to evaluating monotonic functions of fuzzy numbers. In EUSFLAT Conference Proceedings, pages 289-293, Milano, Italy.
- Seibel, A. and Schlattmann, J. (2014). An extended analytical approach to evaluating monotonic functions of fuzzy numbers. Advances in Fuzzy Systems. Article ID 892363, 9 pages.
- Wood, K. L., Otto, K. N., and Antonsson, E. K. (1992). Engineering design calculations with fuzzy parameters. Fuzzy Sets and Systems, 52(1):1-20.
- Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8:338-353.
- Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning-I. Information Sciences, 8:199-249.

#### Paper Citation

#### in Harvard Style

Seibel A. and Schlattmann J. (2014). **An Analytical Approach to Evaluating Bivariate Functions of Fuzzy Numbers with One Local Extremum** . In *Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)* ISBN 978-989-758-053-6, pages 89-94. DOI: 10.5220/0005026500890094

#### in Bibtex Style

@conference{fcta14,

author={Arthur Seibel and Josef Schlattmann},

title={An Analytical Approach to Evaluating Bivariate Functions of Fuzzy Numbers with One Local Extremum},

booktitle={Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)},

year={2014},

pages={89-94},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005026500890094},

isbn={978-989-758-053-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)

TI - An Analytical Approach to Evaluating Bivariate Functions of Fuzzy Numbers with One Local Extremum

SN - 978-989-758-053-6

AU - Seibel A.

AU - Schlattmann J.

PY - 2014

SP - 89

EP - 94

DO - 10.5220/0005026500890094