A Linear System of Both Equations and Inequalities in Max-Algebra

The aim of this chapter is to present a system of linear equation and inequalities in max-al‐ gebra. Max-algebra is an analogue of linear algebra developed on the pair of operations (⊕ , ⊗ ) extended to matrices and vectors, where a⊕ b = max(a, b) and a⊗ b = a + b for a, b ∈ R. The system of equations A⊗ x = c and inequalities B⊗ x ≤ d have each been stud‐ ied in the literature. We will present necessary and sufficient conditions for the solvability of a system consisting of these two systems and also develop a polynomial algorithm for solv‐ ing max-linear program whose constraints are max-linear equations and inequalities. More‐ over, some solvability concepts of an inteval system of linear equations and inequalities will also be presented.