Synchronization and Linearity: An Algebra for Discrete Event Systems.

The analytic hierarchy process, max algebra and multi-objective optimisation

“…The relations between the sets C A,r , the error function (1) and μ(A) were clarified by Elsner and van den Driessche [11] and are recalled in the following propositions, which we easily extend (based on [5,6,28]) to the general reducible case.…”

“…The max algebra consists of the set of nonnegative numbers together with the two basic operations a ⊕ b = max(a, b) and a ⊗ b = ab. This is isomorphic to the max-plus algebra [1,5] via the natural isomorphism x → log(x). These operations extend to nonnegative matrices and vectors in the obvious manner [1,2,7].…”

“…This is isomorphic to the max-plus algebra [1,5] via the natural isomorphism x → log(x). These operations extend to nonnegative matrices and vectors in the obvious manner [1,2,7]. For A in R n×n + , the eigenequation in the max algebra is given by A ⊗ x = λx, x 0, λ 0. μ(A) is the largest max eigenvalue of A [5].…”

“…For A in R n×n + , the eigenequation in the max algebra is given by A ⊗ x = λx, x 0, λ 0. μ(A) is the largest max eigenvalue of A [5]. If A is irreducible, then it is the unique max eigenvalue of A and there is a positive max eigenvector x ∈ int(R n + ) corresponding to it [1,2]. The eigenvector v is unique up to a scalar multiple if and only if A C is irreducible.…”

A Note on the Solution of the Characteristic Equation Over the Symmetrized Max-Plus Algebra

“…Because the matrix over the symmetrized max-plus algebra can be defined as a determinant [7], [8]. The equation characteristics can be formulated as a system of equations and polynomial inequality multivariate algebra max.…”