The maximum flow problem of uncertain random network

The maximum flow problem is one of the important problems of network optimization and covers a wide range of engineering and management applications. Different from the existing works, this paper investigates the maximum flow problem of an uncertain random network under the framework of chance theory. An model of uncertain random α-maximum flow problem is exhibited. And then, it is proved that there exists an equivalence relationship between uncertain random α-maximum flow model and the classic deterministic maximum flow model, which builds a bridge between uncertain random maximum flow problem and deterministic maximum flow problem. Furthermore, some important properties of the model are analyzed, based on which a polynomial exact algorithm is proposed. Finally, a numerical example is presented to illustrate the model and the algorithm. Network flow problem covers a wide range of engineering and management applications. The maximum flow problem is to find the maximum amount of flow from the source to the sink within a network. Different from the existing works, this paper investigates maximum flow problem in which all arc capacities of the network are uncertain variables. Some models of maximum flow problem with uncertain arc capacities are exhibited. And then, the algorithm for solving maximum flow problem is introduced. As an illustration, an example is provided to show the effectiveness of the algorithm.