Kybernetika 48 no. 5, 924-938, 2012

On an algorithm for testing T4 solvability of max-plus interval systems

Helena Myšková

Abstract:

In this paper, we shall deal with the solvability of interval systems of linear equations in max-plus algebra. Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\kr$, where $a\oplus b=\max\{a,b\}$, $a\kr b=a+b$. 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 solvability and give an algorithm for checking the T4 solvability.

Keywords:

max-plus algebra, interval system, T4 solvability, T4 vector

Classification:

15A06, 65G30

References:

  1. Š. Berežný and J. Plavka: Efficient algorithm for el-parametric eigenvalue-eigenvector problem in fuzzy algebra. In: AEI'2008 FEI TU, Košice 2008, pp. 53-57.   CrossRef
  2. K. Cechlárová: Solutions of interval systems in max-plus algebra. In: Proc. of SOR 2001 (V. Rupnik, L. Zadnik-stirn, and S. Drobne, eds.), Preddvor, pp. 321-326.   CrossRef
  3. K.Cechlárová and R. A. Cuninghame-Green: Interval systems of max-separable linear equations. Linear Algebra Appl. 340 (2002), 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 43 (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 Internacional Scientific Conference on Mathematical Methods in Economics and Industry, Herĺany 2007, pp. 153-157.   CrossRef
  10. K. Nachtigall: Powers of matrices over an extremal algebras with applications to periodic graphs. Math. Methods Oper. Res. 46 (1997), 87-102.   CrossRef
  11. G. J. Oldser et al.: Course notes: Max-algebra aproach to discrete event systems. In: Algebres Max-Plus et Applications an Informatique et Automatique. INRIA 1998, pp. 147-196.   CrossRef
  12. 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
  13. J. Rohn: Systems of Interval Linear Equations and Inequalities (Rectangular Case). Technical Report No. 875, Institute of Computer Science, Academy of Sciences of the Czech Republic 2002.   CrossRef
  14. J. Rohn: Complexity of some linear problems with interval data. Reliable Comput. 3 (1997), 315-323.   CrossRef
  15. K. Zimmermann: Extremální algebra. Ekonomicko-matematická laboratoř Ekonomického ústavu ČSAV, Praha 1976.   CrossRef