Null set concept for optimal solutions of fuzzy nonlinear optimization problems

In this work, we present a novel approach for resolving a fuzzy single-objective function with fuzzy constraints. The algorithm of the method is based on the null set concept and is focused on minimizing cases. With the null set concept, two partial subtraction orders for fuzzy numbers have been defined, namely simple subtraction and the Hukuhara difference. That allows us to define, respectively, optimal solutions and H-optimal solutions. First, the initial optimization problem is transformed into a deterministic, nonlinear, bi-objective optimization problem. Then, Karush Kuhn Tucker's (KKT) optimality conditions are applied to find deterministic optimal solutions. Finally, a few fuzzy algebraic operations are employed to transform deterministic optimal solutions into fuzzy optimal solutions for the initial solutions. In order to demonstrate the effectiveness of the approach, we have dealt with eleven test problems from the literature. Our method has been compared to those of other methods, and our method is at least the best in each instance.

Keywords:

Fuzzy nonlinear optimization , Null set , Hukuhara difference , Ranking function , Partial ordering