Kybernetika 48 no. 5, 879-889, 2012

An iterative algorithm for testing solvability of max-min interval systems

Helena Myšková


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 $\kr$, where $a\oplus b=\max\{a,b\}, a\kr b=\min\{a, b\}$. The notation $\mbf{A}\kr x=\mbf{b}$ represents an interval system of linear equations, where $\mbf{A}=[\pA,\nA]$ and $\mbf{b}=[\pb,\nb]$ 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.


max-min algebra, interval system, T4-vector, T4 solvability, T5-vector, T5 solvability


15A06, 65G30


  1. A. Asse, P. Mangin and D. Witlaeys: Assisted diagnosis using fuzzy information. In: NAFIPS 2 Congress, Schenectudy, NY 1983.   CrossRef
  2. K. Cechlárová: 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.   CrossRef
  3. K. Cechlárová and R. A. Cuninghame-Green: Interval systems of max-separable linear equations. Linear Algebra Appl. 340 (2002), 1-3, 215-224.   CrossRef
  4. R. A. Cuninghame-Green: Minimax Algebra. Lecture Notes in Econom. and Math. Systems 1966, Springer, Berlin 1979.   CrossRef
  5. M. Gavalec and J. Plavka: Monotone interval eigenproblem in max-min algebra. Kybernetika 46 (2010), 3, 387-396.   CrossRef
  6. J. Kreinovich, A. Lakeyev, J. Rohn and P. Kahl: Computational Complexity of Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht 1998.   CrossRef
  7. H. Myšková: Interval systems of max-separable linear equations. Linear Algebra Appl. 403 (2005), 263-272.   CrossRef
  8. H. Myšková: Control solvability of interval systems of max-separable linear equations. Linear Algebra Appl. 416 (2006), 215-223.   CrossRef
  9. H. Myšková: 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.   CrossRef
  10. A. Di Nola, S. Salvatore, W. Pedrycz and E. Sanchez: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publishers, Dordrecht 1989.   CrossRef
  11. J. Plavka: On the $O(n^3)$ algorithm for checking the strong robustness of interval fuzzy matrices. Discrete Appl. Math. 160 (2012), 640-647.   CrossRef
  12. J. Rohn: 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.   CrossRef
  13. J. Rohn: Complexity of some linear problems with interval data. Reliable Comput. 3 (1997), 315-323.   CrossRef
  14. E. Sanchez: 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.   CrossRef
  15. T. Terano and Y. Tsukamoto: Failure diagnosis by using fuzzy logic. In: Proc. IEEE Conference on Decision Control, New Orleans, LA 1977, pp. 1390-1395.   CrossRef
  16. L. A. Zadeh: 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.   CrossRef
  17. K. Zimmermann: Extremální algebra. Ekonomicko-matematická laboratoř Ekonomického ústavu ČSAV, Praha 1976.   CrossRef