This paper presents a new multi-objective jumping particle swarm optimization (MOJPSO) algorithm to solve the multi-objective multicast routing problem, which is a well-known NP-hard problem in communication networks. Each particle in the proposed MOJPSO algorithm performs four jumps, i.e. the inertial, cognitive, social and global jumps, in such a way, particles in the swarm follow a guiding particle to move to better positions in the search space. In order to rank the non-dominated solutions obtained to select the best guider of the particle, three different ranking methods, i.e. the random ranking, an entropy-based density ranking, and a fuzzy cardinal priority ranking are investigated in the paper. Experimental results show that MOJPSO is more flexible and effective for exploring the search space to find more non-dominated solutions in the Pareto Front. It has better performance compared with the conventional multi-objective evolutionary algorithm in the literature.