The Minimum Cost Flow Problem of Uncertain Random Network

The aim of this paper is to present a novel method for solving the minimum cost flow problem on networks with uncertain-random capacities and costs. The objective function of this problem is an uncertain random variable and the constraints of the problem do not make a deterministic feasible set. Under the framework of uncertain random programming, a corresponding alpha-minimum cost flow model with a prespecified confidence level alpha, is formulated and its main properties are analyzed. It is proven that there exists an equivalence relationship between this model and the classical deterministic minimum cost flow model. Then an algorithm is proposed to find the maximum amount of alpha such that for it, the feasible set of alpha-minimum cost flow model is nonempty. Finally, a numerical example is presented to illustrate the efficiency of our proposed method.