Simultaneous solution of linear equations and inequalities in max-algebra

Let a⊕b = max(a, b) and a⊗b = a+b for a, b ∈ R. Max-algebra is an analogue of linear algebra developed on the pair of operations (⊕,⊗) extended to matrices and vectors. The system of equations A ⊗ x = b and inequalities C ⊗ x ≤ d have each been studied in the literature. We consider a problem consisting of these two systems and present necessary and sufficient conditions for its solvability. We also develop a polynomial algorithm for solving max-linear program whose constraints are max-linear equations and inequalities.