This paper addresses the permutation of a flexible job shop problem that minimizes the makespan and total idleness as a bi-objective problem. This optimization problem is an NP-hard one because a large solution space allocated to it. We use a duplicate genetic algorithm (DGA) to solve the problem, which is developed a genetic algorithm procedure. Since the proposed DGA is working based on the GA, it often offers a better solution than the standard GA because it includes the rational and appropriate justification. The proposed DGA is used the useful features and concepts of elitism and local search, simultaneously. It provides local search for the best solution in every generation with the neighborhood structure in several stages and stores them in an external list for reuse as a secondary population of the GA. The performance of the proposed GA is evaluated by a number of numerical experiments. By comparing the results of the DGA other algorithms, we realize that our proposed DGA is efficient and appropriate for solving the given problem.