# Minimizing and maximizing a linear objective function under a fuzzy $\max-\ast$ relational equation and an inequality constraint

Zofia Matusiewicz

# Abstract:

This paper provides an extension of results connected with the problem of the optimization of a linear objective function subject to $\max-\ast$ fuzzy relational equations and an inequality constraint, where $\ast$ is an operation. This research is important because the knowledge and the algorithms presented in the paper can be used in various optimization processes. Previous articles describe an important problem of minimizing a linear objective function under a fuzzy $\max-\ast$ relational equation and an inequality constraint, where $\ast$ is the $t$-norm or mean. The authors present results that generalize this outcome, so the linear optimization problem can be used with any continuous increasing operation with a zero element where $\ast$ includes in particular the previously studied operations. Moreover, operation $\ast$ does not need to be a t-norm nor a pseudo-$t$-norm. Due to the fact that optimal solutions are constructed from the greatest and minimal solutions of a $\max-\ast$ relational equation or inequalities, this article presents a method to compute them. We note that the linear optimization problem is valid for both minimization and maximization problems. Therefore, for the optimization problem, we present results to find the largest and the smallest value of the objective function. To illustrate this problem a numerical example is provided.

# Keywords:

fuzzy relational equations, fuzzy optimization, minimizing a linear objective function, maximizing a linear objective function, system of equations, fuzzy relational inequalities, system of inequalities, $\max -\ast$ composition, solution family, minimal solutions

# Classification:

90C05, 03E72, 15A06, 15A39, 46N10

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