Homework 2 - Nodes and Networks

Homework 2 - Nodes and Networks - Bus 140 Professor Denzler...

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Bus 140 Professor Denzler 9/28/08 Maximum Flow Problem Given the network the maximum amount of units we can ship from node 1 to node 8 is 14. How to solve a Maximum Flow Problem Given: Each unit between nodes can only be used once. The maximum amount of units that may travel along a given path is equal to the lowest value along that path. You may only go in the same direction the arrows are pointing. E.g.: you cannot travel from node 7 to node 5. You may travel as many paths as are possible as long as there are available units along the path (the smallest value along that path is >0). E.g.: if you have used all units from node 1 to 2 you may no longer travel that path. Steps to solve the given maximum flow problem: 1) First path to follow is as follows: 1 to 2 to 4 to 6 to 8 a. The lowest value along this path is 5, so therefore 5 units are shipped. b. 5 units were shipped so therefore nodes 1 to 2 have no more units available since they had 5 in the first place. The 9 units that could be shipped from node 2 to 4 is now down to 4 since we shipped 5 units along the path. The value from nodes 4 to 6 is down to 13 since it started at 18 and we shipped 5 units along the path. The value from 6 to 8 is down to 1 unit left. 2) Our next path we will choose is 1 to 3 to either 4 or 5, but in the case we will choose 4, then 4 to 6 to 8. a. The lowest value along the path is 1, so therefore we have 1 unit shipped to bring our total to 6 units shipped – 5 from the last route and 1 from this route. b. Now we must adjust the values along the path. The 12 from nodes 1 to 3 becomes 11, the 6 from nodes 3 to 4 becomes 5, the 13 we still have from 4 to 6 becomes 12, and the 1 that we had left from 6 to 8 becomes a 0. 3)
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Homework 2 - Nodes and Networks - Bus 140 Professor Denzler...

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