An algorithm for solving fuzzy relation programming with the max-t composition operator

This paper studies the problem of minimizing a linear objective function subject to max-T fuzzy relation equation constraints where T is a special class of pseudot-norms. Some sufficient conditions are presented for determination of its optimal solutions. Some procedures are also suggested to simplify the original problem. Some sufficient conditions are given for uniqueness of its optimal solution. Finally, an algorithm is proposed to find its optimal solution.Publisher's Versio

A NEW ALGORITHM FOR OPTIMIZATION OF THE FUZZY RELATION EQUATION WITH MAX- ALGEBRAIC SUM COMPOSITION

ABSTRACT This paper considers an optimization problem with a linear objective function under the constraints expressed by a system of fuzzy relation equations using max-as (Algebraic Sum) composition. First, some properties of minimal solutions of the system with fuzzy relation equations and max-as composition are shown. Then, a new algorithm for solving the optimization problem is derived. The numerical examples have been provided to illustrate the theoretical results. OPSOMMING Hierdie artikel bestudeer 'n optimiseringsprobleem met 'n lineêre doelwitfunksie en wasige randvoorwaardes met 'n algebraïese somsamestelling. Aanvanklik word sommige eienskappe van die minimale oplossings van die wasige vergelykings en die algebraïese samestelling getoon. Daarna word 'n nuwe algoritme vir die oplossing van die optimiseringsprobleem afgelei. Numeriese voorbeelde word voorsien om die teoretiese resultate te ondersteun

Optimizing the max-min function with a constraint on a two-sided linear system

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