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Traditional multicast routing methods have some problems in constructing a multicast tree, such as limited access to network state information, poor adaptability to dynamic and complex changes in the network, and inflexible data forwarding. To address these defects, the optimal multicast routing problem in software-defined networking (SDN) is tailored as a multi-objective optimization problem, and an intelligent multicast routing algorithm DRL-M4MR based on the deep Q network (DQN) deep reinforcement learning (DRL) method is designed to construct a multicast tree in SDN. First, the multicast tree state matrix, link bandwidth matrix, link delay matrix, and link packet loss rate matrix are designed as the state space of the DRL agent by combining the global view and control of the SDN. Second, the action space of the agent is all the links in the network, and the action selection strategy is designed to add the links to the current multicast tree under four cases. Third, single-step and final reward function forms are designed to guide the intelligence to make decisions to construct the optimal multicast tree. The experimental results show that, compared with existing algorithms, the multicast tree construct by DRL-M4MR can obtain better bandwidth, delay, and packet loss rate performance after training, and it can make more intelligent multicast routing decisions in a dynamic network environment.