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The behavior of discrete-event systems, in which the individual components move from event to event rather than varying continuously through time, is often described by systems of linear equations in max-min algebra, in which classical addition and multiplication are replaced by ⊕ and ⊗, representing maximum and minimum, respectively. Max-min equations have found a broad area of applications in causal models, which emphasize relationships between input and output variables. Many practical situations can be described using max-min systems of linear equations. We shall deal with a two-sided max-min system of linear equations with unknown column vector x of the form A⊗x⊕c=B⊗x⊕d, where A, B are given square matrices, c, d are column vectors and operations ⊕ and ⊗ are extended to matrices and vectors in the same way as in the classical algebra. We give an equivalent condition for its solvability. For a given max-min objective function f, we consider optimization problem of type f⊤⊗x→max or min constraint to A⊗x⊕c=B⊗x⊕d. We solve the equation in the form f(x)=v on the set of solutions of the equation A⊗x⊕c=B⊗x⊕d and extend the problem to the case of an interval function f and an interval value v. We define several types of the reachability of the interval value v by the interval function f and provide equivalent conditions for them. [ABSTRACT FROM AUTHOR]

The behavior of discrete-event systems, in which the individual components move from event to event rather than varying continuously through time, is often described by systems of linear equations in max-min algebra, in which classical addition and multiplication are replaced by $ \oplus $ and $ {\otimes} $, representing maximum and minimum, respectively. Max-min equations have found a broad area of applications in causal models, which emphasize relationships between input and output variables. Many practical situations can be described using max-min systems of linear equations. We shall deal with a two-sided max-min system of linear equations with unknown column vector $ x $ of the form $ A{\otimes} x{\oplus} c = B{\otimes} x{\oplus} d $, where $ A $, $ B $ are given square matrices, $ c $, $ d $ are column vectors and operations $ \oplus $ and $ {\otimes} $ are extended to matrices and vectors in the same way as in the classical algebra. We give an equivalent condition for its solvability. For a given max-min objective function $ f $, we consider optimization problem of type $ f^\top{\otimes} x\rightarrow \max\text{ or } \min $ constraint to $ A{\otimes} x{\oplus} c = B{\otimes} x{\oplus} d $. We solve the equation in the form $ f(x) = v $ on the set of solutions of the equation $ A{\otimes} x{\oplus} c = B{\otimes} x{\oplus} d $ and extend the problem to the case of an interval function $ {{\boldsymbol{f}}} $ and an interval value $ {{\boldsymbol{v}}} $. We define several types of the reachability of the interval value $ {{\boldsymbol{v}}} $ by the interval function $ {{\boldsymbol{f}}} $ and provide equivalent conditions for them.