Abstract
Systems of linear equations with elements being affine linear functions of fuzzy parameters are relevant to many practical problems. A method for solving such systems is proposed. It consists of two steps. First a finite number of parametric interval linear systems is solved using the direct method. Then membership functions of fuzzy solution elements are interpolated. Parameters are modeled by arbitrary fuzzy numbers with convex membership function and compact support. Conditions for existence of the fuzzy solution are given. The performance of the proposed method is presented using an illustrative example of truss structure.
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References
Buckley, J.J.: Solving fuzzy equations. Fuzzy Sets and Systems 50, 1–14 (1992)
Buckley, J.J., Eslami, E., Feuring, T.: Fuzzy mathematics in economics and engineering. Studies in Fuzziness and Soft Computing, vol. 91. Physica-Verlag, Heidelberg (2002)
Buckley, J.J., Qu, Y.: Solving systems of linear fuzzy equations. Fuzzy Sets and Systems 43, 33–43 (1991)
Jankowski, M.: Elements of Computer Graphics. Wydawnictwo Naukowo-Techniczne, Warsaw (1990)
Jansson, C.: Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side. Computing 46(3), 265–274 (1991)
Kolev, L.V.: Interval methods for circuit analysis. World Scientific, Singapore (1991)
Kolev, L.V.: Outer solution of linear systems whose elements are affine functions of interval parameters. Reliable Computing 6, 493–501 (2002)
Kolev, L.V.: Worst-case tolerance analysis of linear DC and AC electric circuits. IEEE Trans. on Circuits and Systems, I, Fund. Theory and Appl. 49, 1693–1702 (2002)
Kolev, L.V.: A method for outer interval solution of linear parametric systems. Reliable Computing 10, 227–239 (2004)
Kulpa, Z., Pownuk, A., Skalna, I.: Analysis of linear mechanical structures with uncertainties by means of interval methods. Computer Assisted Mechanics and Engineering Sciences 5(4), 443–477 (1998)
Moor, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
Muzzioli, S., Reynaerts, H.: Fuzzy linear systems of the form A1x+b1 = A2x + b2. In: 4th International Symposium on Imprecise Probabilities and Their Application. Pitsburg, Pensylwania (2005)
Neumaier, A.: Interval methods for solving systems of equations. Cambridge University Press, Cambridge (1990)
Popova, E.D.: On the solution of parametrised linear systems. Scientific Computing, Validated Numerics, Interval Methods 127-138 (2001)
Rao, S.S., Chen, L.: Numerical solution of fuzzy linear equations in engineering analysis. International Journal for Numerical Methods in Engineering 43, 391–408 (1998)
Rao, S.S., Sawyer, J.P.: Fuzzy finite element method for analysis of imprecisely defined systems. AIAA Journal 33(12), 2364–2370 (1995)
Rohn, J.: Systems of linear interval equations. Linear Algebra and its Applications 126, 39–78 (1989)
Rump, S.M.: Verification methods for dense and sparse systems of equations, The Netherlands. Studies in Computational Mathematics, vol. 5, pp. 63–135. Elsevier, Amsterdam (1994)
Skalna, I.: Evolutionary optimization method for approximating the solution set hull of parametric linear systems. In: Boyanov, T., et al. (eds.) NMA 2006. LNCS, vol. 4310, Springer, Heidelberg (2007)
Skalna, I.: A method for outer interval solution of parametrized systems of linear interval equations. Reliable Computing 12(2), 107–120 (2006)
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Skalna, I. (2007). Parametric Fuzzy Linear Systems. In: Castillo, O., Melin, P., Ross, O.M., Sepúlveda Cruz, R., Pedrycz, W., Kacprzyk, J. (eds) Theoretical Advances and Applications of Fuzzy Logic and Soft Computing. Advances in Soft Computing, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72434-6_56
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DOI: https://doi.org/10.1007/978-3-540-72434-6_56
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