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 ⊕ and \kr, where a⊕b=max{a,b},a\krb=min{a,b}. The notation \mbfA\krx=\mbfb represents an interval system of linear equations, where \mbfA=[\pA,\nA] and \mbfb=[\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 is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and \kr, where a⊕b=max{a,b},a\krb=min{a,b}. The notation \mbfA\kr\mbfx=\mbfb represents an interval system of linear equations, where \mbfA=[\pA,\nA], \mbfb=[\pb,\nb] are given interval matrix and interval vector, respectively, and a solution is from a given interval vector \mbfx=[\px,\nx]. We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.
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 ⊕ and \kr, where a⊕b=max{a,b}, a\krb=a+b. The notation \mbfA\krx=\mbfb represents an interval system of linear equations, where \mbfA=[\pA,\nA] and \mbfb=[\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.
In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix A is called strongly robust if the orbit x,A⊗x,A2⊗x,… reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong \textit{\textbf{X}}-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong \textit{\textbf{X}}-robustness is introduced and efficient algorithms for verifying the strong \textit{\textbf{X}}-robustness is described. The strong \textit{\textbf{X}}-robustness of a max-min matrix is extended to interval vectors \textit{\textbf{X}} and interval matrices \textit{\textbf{A}} using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong \textit{\textbf{X}}-robustness of interval circulant matrices is presented.
Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted ⊕ and ⊗ , respectively). The eigenproblem is the search for a vector x (an eigenvector) and a constant λ (an eigenvalue) such that A⊗x=λ⊗x , where A is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation A⊗x=λ⊗B⊗x with given matrices A,B and unknown constant λ and vector x . Generalized eigenvectors have interesting and useful properties in the various computational tasks with inexact (interval) matrix and vector inputs. This paper studies the properties of generalized interval eigenvectors of interval matrices. Three types of generalized interval eigenvectors: strongly tolerable generalized eigenvectors, tolerable generalized eigenvectors and weakly tolerable generalized eigenvectors are proposed and polynomial procedures for testing the obtained equivalent conditions are presented.