In this paper, we present a new approach to solve the minimum cost flow problem using the out-of-kilter idea. Our algorithm uses Minty's Lemma to transform all the out-of-kilter arcs into in-kilter arcs. This algorithm gives a geometrical explanation to the optimality concept. For a network with n nodes and m arcs, the algorithm performs O(log(nU)) phases and runs in O(m(m+n log n)log (nU)) time (where U is the largest absolute arc bound ), which is O(m(m+n log n) log n) under the similarity assumption[1]. This time is the running time of the algorithm in [2] which is the best strongly polynomial-time algorithms to solve this problem.
