MIE376 Lec 11 - NP Details

MIE376 Lec 11 - NP Details - Basics of the Network Simplex...

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Basics of the Network Simplex 1

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Very simple network 0 3 1 3 4 2 0/ /3 0/ /1 0/ /1 0/ /3 0/ /2 0 -3 2 Note: There are no bounds in this problem , to first develop an initial algorithm . Bounds can easily be incorporated at the end.
Equivalent LP Min 3X 12 + X 13 + 3X 23 + X 24 + 2X 34 subject to Node 1 balance: X 12 + X 13 = 3 Node 2 balance - X 12 + X 24 + X 23 = 0 Node 3 balance -X 13 - X 23 + X 34 = 0 Node 4 balance -X 24 - X 34 = -3 X 12 , X 13 , X 23 , X 24 , X 34 0 3

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# Basic variables = # nodes -1 Node 1 balance: X 12 + X 13 = 3 Node 2 balance - X 12 + X 24 + X 23 = 0 Node 3 balance -X 13 - X 23 + X 34 = 0 Node 4 balance -X 24 - X 34 = -3 Any of the 4 constraints can be derived from the other 3 only need 3 constraints there will be 3 variables in the basis in general a problem with m nodes will have (m-1) basic variables. Is any selection of (m-1) variables a basic feasible solution? 4
According to Graph Theory Any basic feasible solution of the LP corresponds to a Spanning Tree of the network graph. A graph with m nodes forms a Spanning Tree, if: ignore the arrows on the arcs include (m-1) arcs, in this case 3 arcs connect all the nodes Such that there are no loops 5

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Sample Spanning Trees 1 3 4 2 1 3 4 2 1 3 4 2 6
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This note was uploaded on 04/20/2011 for the course MIE 376 taught by Professor Daniel during the Spring '11 term at University of Toronto- Toronto.

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MIE376 Lec 11 - NP Details - Basics of the Network Simplex...

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