Networks and Basic Solutions

A labeling procedure which could be used to carry out the steps of the simplex algorithm for a network flow problem is given. A modification of the simplex algorithm for the maximum flow problem is suggested which is equivalent to the Ford-Fulkerson algorithm if the solutions are kept basic in the latter. A procedure to keep the solutions basic in the Ford-Fulkerson algorithm is given. An example of a flow with gains problem with integer capacities and gains is given for which the usual labeling procedure may not lead to an optimum solution if the solutions are not kept basic. Extension of the network flow techniques to flows with gains is discussed.