# Article

 Title: An iterative algorithm for testing solvability of max-min interval systems (English) Author: Myšková, Helena Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 48 Issue: 5 Year: 2012 Pages: 879-889 Summary lang: English . Category: math . Summary: This paper is dealing with solvability of interval systems of linear equations in max-min algebra. Max-min algebra is the algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, where $a\oplus b=\max\{a,b\}, a\otimes b=\min\{a, b\}$. The notation ${\mathbb A}\otimes x={\mathbb b}$ represents an interval system of linear equations, where ${\mathbb A}=[\underline{A},\overline{A}]$ and ${\mathbb b}=[\underline{b},\overline{b}]$ are given interval matrix and interval vector, respectively. We can define several types of solvability of interval systems. In this paper, we define the T4 and T5 solvability and give necessary and sufficient conditions for them. (English) Keyword: max-min algebra Keyword: interval system Keyword: T4-vector Keyword: T4 solvability Keyword: T5-vector Keyword: T5 solvability MSC: 15A06 MSC: 65G30 idMR: MR3086857 . Date available: 2012-12-17T13:29:09Z Last updated: 2013-09-24 Stable URL: http://hdl.handle.net/10338.dmlcz/143087 . Reference: [1] Asse, A., Mangin, P., Witlaeys, D.: Assisted diagnosis using fuzzy information..In: NAFIPS 2 Congress, Schenectudy, NY 1983. Reference: [2] Cechlárová, K.: Solutions of interval systems in max-plus algebra..In: Proc. SOR 2001 (V. Rupnik, L. Zadnik-Stirn, S. Drobne, eds.), Preddvor, Slovenia, pp. 321-326. MR 1861219 Reference: [3] Cechlárová, K., Cuninghame-Green, R. A.: Interval systems of max-separable linear equations..Linear Algebra Appl. 340 (2002), 1-3, 215-224. Zbl 1004.15009, MR 1869429, 10.1016/S0024-3795(01)00405-0 Reference: [4] Cuninghame-Green, R. A.: Minimax Algebra..Lecture Notes in Econom. and Math. Systems 1966, Springer, Berlin 1979. Zbl 0739.90073, MR 0580321 Reference: [5] Gavalec, M., Plavka, J.: Monotone interval eigenproblem in max-min algebra..Kybernetika 46 (2010), 3, 387-396. Zbl 1202.15013, MR 2676076 Reference: [6] Kreinovich, J., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity of Feasibility of Data Processing and Interval Computations..Kluwer, Dordrecht 1998. Reference: [7] Myšková, H.: Interval systems of max-separable linear equations..Linear Algebra Appl. 403 (2005), 263-272. Zbl 1129.15003, MR 2140286 Reference: [8] Myšková, H.: Control solvability of interval systems of max-separable linear equations..Linear Algebra Appl. 416 (2006), 215-223. Zbl 1129.15003, MR 2242726 Reference: [9] Myšková, H.: Solvability of interval systems in fuzzy algebra..In: Proc. 15th International Scientific Conference on Mathematical Methods in Economics and Industry, Herĺany 2007, pp. 153-157. Reference: [10] Nola, A. Di, Salvatore, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering..Kluwer Academic Publishers, Dordrecht 1989. MR 1120025 Reference: [11] Plavka, J.: On the $O(n^3)$ algorithm for checking the strong robustness of interval fuzzy matrices..Discrete Appl. Math. 160 (2012), 640-647. MR 2876347, 10.1016/j.dam.2011.11.010 Reference: [12] Rohn, J.: Systems of Interval Linear Equations and Inequalities (Rectangular Case)..Technical Peport No. 875, Institute of Computer Science, Academy of Sciences of the Czech Republic 2002. Reference: [13] Rohn, J.: Complexity of some linear problems with interval data..Reliable Comput. 3 (1997), 315-323. Zbl 0888.65052, MR 1616269, 10.1023/A:1009987227018 Reference: [14] Sanchez, E.: Medical diagnosis and composite relations..In: Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. R. Yager, eds.), North-Holland, Amsterdam - New York 1979, pp. 437-444. MR 0558737 Reference: [15] Terano, T., Tsukamoto, Y.: Failure diagnosis by using fuzzy logic..In: Proc. IEEE Conference on Decision Control, New Orleans, LA 1977, pp. 1390-1395. Reference: [16] 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. Reference: [17] Zimmermann, K.: Extremální algebra..Ekonomicko-matematická laboratoř Ekonomického ústavu ČSAV, Praha 1976. .

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