In this paper, we propose an algorithm called Directional Exploration Genetic Algorithm (DEGA) to resolve a function \(Phi\) over the efficient set of a multi-objective integer linear programming problem (MOILP). DEGA algorithm belongs to evolutionary algorithms, which operate on the decision space by choosing the fastest improving directions that improve the objectives functions and \(Phi\) function. Two variants of this algorithm and a basic version of the genetic algorithm (BVGA) are performed and implemented in Python. Several benchmarks are carried out to evaluate the algorithm's performances and interesting results are obtained and discussed.