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Graph partitioning is a NP-hard problem with multiple conflicting objectives. The graph partitioning should minimize the inter-partition relationship while maximizing the intra-partition relationship. Furthermore, the partition load should be evenly distributed over the respective partitions. Therefore this is a multi-objective optimization problem. There are two approaches to multi-objective optimization using genetic algorithms: weighted cost functions and finding the Pareto front. We have used the Pareto front method to find the suitable curve of non-dominated solutions, composed of a high number of solutions. The proposed methods of this paper used to improve the performance are injecting best solutions of previous runs into the first generation of next runs and also storing the non-dominated set of previous generations to combine with later generation's non-dominated set. These improvements prevent the GA from getting stuck in the local optima and make the search more efficient and increase the probability of finding more optimal solutions. Finally, a simulation research is carried out to investigate the effectiveness of the proposed algorithm. The simulation results confirm the effectiveness of the proposed multi-objective GA method.