Linear optimization over the approximate solutions of a system of max-min equations

Abstract

When a system of max-min equations is inconsistent, which is frequently encountered in modelling with fuzzy relations, the approximate solutions may be considered instead within an admissible error bound. This paper tackles the linear optimization problem over the approximate solutions with respect to the L∞ and the L1 residual error, respectively. It demonstrates that such an optimization problem can be reformulated as either a generalized set covering problem for the L∞ scenario or a mixed integer linear programming problem for the L1 scenario. As a result, it may be solved to optimality with the aid of a state-of-the-art optimization solver and hence does not demand a tailored solving method except for very large instances.