FLOW OVER TIME PROBLEM WITH INFLOW-DEPENDENT TRANSIT TIMES

 Network flow over time is an important area for the researcher relating to the traffic assignment problem. Transmission times of the vehicles directly depend on the number of vehicles entering the road. Flow over time with fixed transit times can be solved by using classical (static) flow algorithms in a corresponding time expanded network which is not exactly applicable for flow over time with inflow dependent transit times. In this paper we discuss the time expanded graph for inflow-dependent transit times and non-existence of earliest arrival flow on it. Flow over time with inflow-dependent transit times are turned to inflow-preserving flow by pushing the flow from slower arc to the fast flow carrying arc. We gave an example to show that time horizon of quickest flow in bow graph GB was strictly smaller than time horizon of any inflow-preserving flow over time in GB satisfying the same demand. The relaxation in the modified bow graph turns the problem into the linear programming problem.