This paper proposed a new average non-dominated sorting genetic algorithm (NAVNSGA). This idea is inspired from the combination of non-elitist multi-objective evolutionary algorithms, elitist multiobjective evolutionary algorithms, and statistical calculations. The proposed NAVNSGA is improved the disadvantages of the Elitist multi-objective algorithms and Non-elitist multi-objective algorithms as possible. The NAVNSGA is compared with useful algorithms such as the non-elitist sorting genetic algorithm (NSGAI) and non-elitist sorting genetic algorithm (NSGAII) and the results obtained are showed the superiority of the proposed algorithm. Additionally, the NAVNSGA algorithm is combined with the concepts of the Game theory to propose a hybrid algorithm for determining Nash equilibrium in the game theory. The combination of the NAVNSGA algorithm with the game theory previously is used for improving engineering systems, such as I-beam designing. The results obtained are showed the advantage of the proposed algorithm with those reported in the literature.