An algorithm for maximum profit flows

A maximum profit flows (MPF) problem is a network optimization problem aiming to get maximum transport profit. A feasible MPF can be decomposed into a few path flows and a few circle flows, and consequently the profit of MPF is divided into the profit of those path flows and circle flows. We present a basic algorithm to find the solution by augmenting flow on profit augmenting path until no path exists based on the theorem that a feasible flow fmax is a maximum profit flow if and only if there is no augmenting path with respect to fmax.