The multi-objective components assignment problem (MOCAP) for multi-source multi-sink flow networks when each component has an assignment cost in is never discussed. The main objective of MOCAP is to search the optimal components that maximize network reliability of multi-source multi-sink flow networks and minimized the total assignment cost. An approach based random weighted genetic algorithm (RWGA) is proposed to solve the MOCAP. The Optimal Components Assignment Problem (OCAP) has a solution that is produced by RWGA. The results demonstrated that using the suggested method, optimal component assignment yields the greatest reliability, lowest assignment cost, and shortest total lead-time. The proposed algorithm has been applied to various networks to assert its efficiency in comparison with other approaches based on single genetic algorithm. We applied it to different types of network models, including two-source two-sink networks and three-source two-sink networks, with varying numbers of available components. Also, the obtained results show that the proposed RWGA approach works well and find optimal solutions for all studied cases.