The multicast routing problem lies in the composition of a multicast routing tree including a source node and multiple destinations. There is a trade-off relationship between cost and delay, and the multicast routing problem of optimizing these two conditions at the same time is a difficult problem to solve and it belongs to a multi-objective optimization problem (MOOP). A multi-objective genetic algorithm (MOGA) is efficient to solve MOOP. A micro-genetic algorithm(${mu}GA$) is a genetic algorithm with a very small population and a reinitialization process, and it is faster than a simple genetic algorithm (SGA). We propose a multi-objective micro-genetic algorithm (MO${mu}GA$) that combines a MOGA and a ${mu}GA$ to find optimal solutions (Pareto optimal solutions) of multicast routing problems. Computational results of a MO${mu}GA$ show fast convergence and give better solutions for the same amount of computation than a MOGA.