A maximum flow problem. The diagram Figure 1 below shows a portion of a city where morning...

A maximum flow problem. The diagram Figure 1 below shows a portion of a city where morning traffic is flowing into the area across two bridges at nodes at 1 and 2. Virtually all the traffic exits at node 9 to a manufacturing plant. On the link between any pair of nodes i and j (intersections) is a capacity cij. This is the maximum flow in cars per hour on that link in the direction From i to j. An unknown amount of flow A1 enter node 1 and an unknown amount of flow A2 enter node 2. The sum of the

flows, A1+ A2, exits to the factory at node 9. The flows are unknown because we are seeking the maximum flow through the system, and motorist’s will divide themselves by experimentation into cars entering node 1 and cars entering node 2. In general, motorists will not backtrack to gain advantage, and men will not ask for directions.

In the evening, flow reverse and leave the factory to cross the area and exit via one of the two bridges. The flow capacities on each link are different now, because traffic is generally facing downhill into the evening sun. Set up a math program or programs that will find the maximum flow through the system.

Figure 1