In a capacitated network, an optimum solution of the maximum flow problem is to send as much flow as possible from the source node to the sink node as efficiently as possible by satisfying the capacity and conservation constraints. But, because of the limited capacity on the arcs, total amount of flow out going from the source may not reach to the sink. If the excess amount of flow can be stored at the intermediate nodes, total amount of flow outgoing from the source can be increased significantly. Similarly, different destinations have their own importance with respect to some circumstances. Motivated with these scenarios, we introduce the lexicographic maximum flow problems with intermediate storage in static and dynamic networks by assigning the priority order to the nodes. We extend this notion to arc reversals approach, a flow maximization technique, which is widely accepted in evacuation planning as it increases the outbound arc capacities by using the arc capacities on the opposite direction as well. Travel times along the anti-parallel arcs is considered to be unequal and we take into account the travel time of the reversed arcs to be equal to the travel time of the non-reversed arc towards which the arc is reversed. We present polynomial time algorithms for the solution of these problems.

The aim of the maximum network flow problem is to push as much flow as possible between two special vertices, the source and the sink satisfying the capacity constraints. For the solution of the maximum flow problem, there exists a number of algorithms. The existing algorithms can be divided into two families. First, augmenting path algorithms that satisfy the conservation constraints at intermediate vertices and the second preflow push relabel algorithms that violates the conservation constraints at the intermediate vertices resulting incoming flow more than outgoing flow.In this paper, we study different algorithms that determine the maximum flow in the static and dynamic networks.