Geodesic
Interval fuzzy matrix equations
Kybernetika, Tome 53 (2017) no. 1, pp. 99-112
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This paper deals with the solvability of interval matrix equations in fuzzy algebra. Fuzzy algebra is the algebraic structure in which the classical addition and multiplication are replaced by maximum and minimum, respectively. The notation $\mathbf{A} \otimes X\otimes \mathbf{C}=\mathbf{B}$, where $\mathbf{A}, \mathbf{B}, \mathbf{C}$ are given interval matrices and $X$ is an unknown matrix, represents an interval system of matrix equations. We can define several types of solvability of interval fuzzy matrix equations. In this paper, we shall deal with four of them. We define the tolerance, weak tolerance, left-weak tolerance, and right-weak tolerance solvability and provide polynomial algorithms for checking them.
This paper deals with the solvability of interval matrix equations in fuzzy algebra. Fuzzy algebra is the algebraic structure in which the classical addition and multiplication are replaced by maximum and minimum, respectively. The notation $\mathbf{A} \otimes X\otimes \mathbf{C}=\mathbf{B}$, where $\mathbf{A}, \mathbf{B}, \mathbf{C}$ are given interval matrices and $X$ is an unknown matrix, represents an interval system of matrix equations. We can define several types of solvability of interval fuzzy matrix equations. In this paper, we shall deal with four of them. We define the tolerance, weak tolerance, left-weak tolerance, and right-weak tolerance solvability and provide polynomial algorithms for checking them.
DOI : 10.14736/kyb-2017-1-0099
Classification : 15A06, 65G30
Keywords: fuzzy algebra; interval matrix equation; tolerance solvability; weak tolerance solvability 
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Draženská, Emília; Myšková, Helena. Interval fuzzy matrix equations. Kybernetika, Tome 53 (2017) no. 1, pp. 99-112. doi: 10.14736/kyb-2017-1-0099

[1] Asse, A., Mangin, P., Witlaeys, D.: Assisted diagnosis using fuzzy information. NAFIPS 2 Congress, Schenectudy 1983. | DOI

[2] Butkovič, P., Fiedler, M.: Tropical tensor product and beyond. http://web.mat.bham.ac.uk/P.Butkovic/Pubs.html

[3] Cechlárová, K.: Solutions of interval systems in fuzzy algebra. In: Proc. SOR 2001 (V. Rupnik, L.Zadnik-Stirn, and S. Drobne, eds.), Preddvor, pp. 321-326.

[4] K.Cechlárová, Cuninghame-Green, R. A.: Interval systems of max-separable linear equations. Lin. Algebra Appl. 340 (2002), 215-224. | DOI | MR | Zbl

[5] Cuninghame-Green, R. A.: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems 1966, Springer, Berlin 1979. | DOI | MR | Zbl

[6] Gavalec, M., Plavka, J.: Monotone interval eigenproblem in fuzzy algebra. Kybernetika 46 (2010), 3, 387-396. | MR

[7] Kreinovich, J., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity of Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht 1998. | DOI | MR

[8] Myšková, H.: Interval systems of max-separable linear equations. Lin. Algebra Appl. 403 (2005), 263-272. | DOI | MR | Zbl

[9] Myšková, H.: Control solvability of interval systems of max-separable linear equations. Lin. Algebra Appl. 416 (2006), 215-223. | DOI | MR | Zbl

[10] Myšková, H.: On an algorithm for testing T4 solvability of fuzzy interval systems. Kybernetika 48 (2012), 5, 924-938. | MR

[11] Myšková, H.: Interval max-plus matrix equations. Lin. Algebra Appl. 492 (2016), 111-127. | DOI | MR

[12] Nola, A. Di, Salvatore, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publishers, Dordrecht 1989. | DOI | MR

[13] Plavka, J.: On the $O(n^3)$ algorithm for checking the strong robustness of interval fuzzy matrices. Discrete Appl. Math. 160 (2012), 640-647. | DOI | MR

[14] Rohn, J.: Systems of Interval Linear Equations and Inequalities (Rectangular Case). Technical Report 875, Institute of Computer Science, Academy of Sciences of the Czech Republic, Praha 2002. | MR

[15] Rohn, J.: Complexity of some linear problems with interval data. Reliable Computing 3 (1997), 315-323. | DOI | MR | Zbl

[16] Sanchez, E.: Medical diagnosis and composite relations. In: Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, R. R. Yager, eds.), North-Holland, Amsterdam-New York 1979, pp. 437-444. | MR

[17] Terano, T., Tsukamoto, Y.: Failure diagnosis by using fuzzy logic. In: Proc. IEEE Conference on Decision Control, New Orleans 1977, pp. 1390-1395. | DOI

[18] Zadeh, L. A.: Toward a theory of fuzzy systems. In: Aspects of Network and Systems Theory (R. E. Kalman and N. De Claris, eds.), Hold, Rinehart and Winston, New York 1971, pp. 209-245.

[19] Zimmermann, K.: Extremální algebra. Ekonomicko-matematická laboratoř Ekonomického ústavu ČSAV, Praha 1976.

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