Three-objective optimization in linguistic function approximation

Our task is to approximately represent a nonlinear function using a small number of simple linguistic rules such as -If x₁ is small and x₂ is large then y is large”. Linguistic rules are extracted from numerical input-output data by a multi-objective fuzzy GBML, (genetics-based machine learning) algorithm. In this paper, we first formulate our rule extraction task as a three-objective combinatorial optimization problem. Three objectives are to minimize the total squared error, the number of linguistic rules, and their total length. Then we show how a fuzzy GBML algorithm can be implemented in the framework of multi-objective optimization. This algorithm does not try to find a single rule set but a number of non-dominated rule sets with respect to the three objectives. Finally we illustrate our approach to linguistic modeling through computer simulations on numerical examples