Maximal concurrent minimal cost flow problems on extended multi-cost multi-commodity networks

The graph is a great mathematical tool, which has been effectively applied to many fields such as economy, informatics, communication, transportation, etc. It can be seen that in an ordinary graph the weights of edges and vertexes are taken into account independently where the length of a path is the sum of weights of the edges and the vertexes on this path. Nevertheless, in many practical problems, weights at vertexes are not equal for all paths going through these vertexes, but are depending on coming and leaving edges. Moreover, on a network, the capacities of edges and vertexes are shared by many goods with different costs. Therefore, it is necessary to study networks with multiple weights. Models of extended multi-cost multi-commodity networks can be applied to modelize many practical problems more exactly and effectively. The presented article studies the maximal concurrent minimal cost flow problems on multi-cost and multi-commodity networks, which are modelized as optimization problems. On the base of the algorithm to find maximal concurrent flow and the algorithm to find maximal concurrent limited cost flow, an effective polynomial approximate procedure is developed to find a good solution.