@article{Myšková2024, 
author = {Helena Myšková and Ján Plavka},
title = {Optimizing the max-min function with a constraint on a two-sided linear system},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {4},
pages = {7791-7809},
keywords = {max-min algebra, interval value, max-min optimization, two-sided system, reachability},
url = {https://www.sciopen.com/article/10.3934/math.2024378},
doi = {10.3934/math.2024378},
abstract = {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.}
}