=9-41 South Side Oil and Gas, a new venture in Texas, has developed an oil pipeline network to transport oil from exploration fields to the refinery and other locations. There are 10 pipelines (branches) in the network. The oil flow in hundreds of gallons and the network of pipelines is given in the following table.

(a) What is the maximum that can flow through the network?

Branch Start Node End Node Capacity Reverse Capacity Flow Branch 1 1 2 10 4 10 Branch 2 1 3 8 2 5 Branch 3 2 4 12 1 10 Branch 4 2 5 6 6 0 Branch 5 3 5 8 1 5 Branch 6 4 6 10 2 10 Branch 7 5 6 10 10 0 Branch 8 5 7 5 5 5 Branch 9 6 8 10 1 10 Branch 10 7 8 10 1 5

(b) South Side Oil and Gas needs to modify its pipeline network flow patterns. The new data is in the following table. What impact does this have on the maximum flow through the network?

Branch Start Node End Node Capacity Reverse Capacity Flow Branch 1 1 2 10 4 10 Branch 2 1 3 8 2 5 Branch 3 2 4 12 1 10 Start Node End Node Capacity Reverse Capacity Flow Branch 4 2 5 0 0 0 Branch 5 3 5 8 1 5 Branch 6 4 6 10 2 10 Branch 7 5 6 10 10 0 Branch 8 5 7 5 5 5 Branch 9 6 8 10 1 10 Branch 10 7 8 10 1 5