Uncertainty theory examines the behaviors of uncertain parameters on networks. We consider uncertain and predetermined capacities, respectively, on arcs and intermediate vertices of an uncertain network. An objective is to find the non-conservative flow that is maximum at the destination and also at intermediate nodes with minimum cost. This goal is achieved by determining minimum cost paths which send the maximum flow to the sink. In this paper, we solve this problem by incorporating the idea of uncertainty theory. We define this problem, give its mathematical model and present efficient algorithms to solve our newly formulated problem. The illustrated example verifies more efficiency of our approach since it increases the flow values by 19% to 44% for different confidence levels than that of the classical solutions. The relation of confidence level with the maximum flow value and the minimum cost are also observed.