This thesis considers a flow problem of finding a quickest multicast routing for two commodities on a virtual circuit network where two sources should inform distinctly different messages to all other nodes. The objective is to find a routing that minimizes the maximum transmission time. It is assumed in the problem that the data can not be partitioned into smaller parts(i.e., nonbifurcated routing). Two heuristic algorithms are suggested and tested for their effectivenesses with several numerical examples.