In order to solve the problem of multi-source multi-sink multicast network coding, an algorithm for computing achievable information rate region and an approach for constructing linear network coding scheme were proposed. Based on the previous studies, the multi-source multi-sink multicast network coding problem was transformed into a specific single-source multicast network coding scenario with a constraint at the source node. By theoretical analyses and formula derivation, the constraint relationship among the multicast rate of source nodes was found out. Then a multi-objective optimization model was constructed to describe the boundary of achievable information rate region. Two methods were presented for solving this model. One was the enumeration method, the other was multi-objective optimization method based on genetic algorithm. The achievable information rate region could be derived from Pareto boundary of the multi-objective optimization model. After assigning the multicast rate of source nodes, the linear network coding scheme could be constructed by figuring out the single-source multicast network coding scenario with a constraint. The simulation results show that the proposed methods can find out the boundary of achievable information rate region including integral points and construct linear network coding scheme.