Abstract: The shortest paths tree problem when considering cost and bandwidth constraints is addressed in this paper as multiobjective shortest paths tree problem. A multi-objective genetic algorithm is suitable to solve the presented problem. Therefore, this paper presents a multi-objective genetic algorithm based on Random Weighted Genetic Algorithm (RWGA) to solve the shortest paths tree problem subject to cost and bandwidth constraints. The objective of the proposed algorithm is to search the optimal set of edges connecting all nodes such that the sum of costs is minimized and the value bandwidth is maximized. The presented algorithm has been applied on two sample networks to illustrate their efficiency.