15.053
Network SimplexAnimations
Network Simplex Animations
2
Calculating A Spanning Tree Flow
Network Simplex Animations
A tree with suppliesand demands.(Assume that allother arcs have aflow of 0)
What is the flow inarc (4,3)?
3
Calculating A Spanning Tree Flow
Network Simplex Animations
To calculate flows,iterate up the tree,and find an arcwhose flow isuniquelydetermined.
What is the flow inarc (5,3)?
4
Calculating A Spanning Tree Flow
Network Simplex Animations
What is the flow inarc (3,2)?
5
Calculating A Spanning Tree Flow
Network Simplex Animations
What is the flow inarc (2,6)?
6
Calculating A Spanning Tree Flow
Network Simplex Animations
What is the flow inarc (7,1)?
7
Calculating A Spanning Tree Flow
Network Simplex Animations
What is the flow inarc (1,2)?
8
Calculating A Spanning Tree Flow
Network Simplex Animations
Note: there are twodifferent ways ofcalculating the flow on(1,2), and both waysgive a flow of 4. Is thisa coincidence?
9
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
Here is a spanningtree with arc costs.How can one choosenode potentials sothat reduced costs oftree arcs is 0?
Recall: the reducedcost of (i,j) iscij - πi + πj
10
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
Here is a spanningtree with arc costs.How can one choosenode potentials sothat reduced costs oftree arcs is 0?
Recall: the reducedcost of (i,j) iscij - πi + πj
11
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
Here is a spanningtree with arc costs.How can one choosenode potentials sothat reduced costs oftree arcs is 0?
Recall: the reducedcost of (i,j) iscij - πi + πj
12
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
The reduced cost of(1,2) isc12 - π1 + π2 = 0.
Thus 5 - 0 + π2 = 0.
What is the simplexmultiplier for node7?
13
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
The reduced cost of(1,2) isc71 – π7 + π1 = 0.
Thus -6 -π2 + 0 = 0.
What is the simplexmultiplier for node3?
14
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
What is the simplexmultiplier for node6?
15
Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
What is the simplexmultiplier for node5?
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Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
What is the simplexmultiplier for node4?
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Calculating Simplex Multipliers for a Spanning Tree
Network Simplex Animations
These are thesimplex multipliersassociated with thistree. They do notdepend on arc flows,nor on costs of non-tree arcs.
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Network Simplex Algorithm
Network Simplex Animations
The minimum Cost Flow Problem
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Network Simplex Algorithm
Network Simplex Animations
An Initial Spanning Tree Solution
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Simplex Multipliers and Reduced Costs
Network Simplex Animations
The initial simplex multipliers andreduced costs
What arcs areviolating?
21
Add a violating arc to the spanning tree, creating a cycle
Network Simplex Animations
Arc (2,1) is added to the tree
What is thecycle, andhow muchflow can besent?
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Send Flow Around the Cycle
Network Simplex Animations
2 units of flow were sent along thecycle.
What is thenext spanningtree?
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After a pivot
Network Simplex Animations
The Updated Spanning Tree
In a pivot, an arcis added to T andan arc is droppedfrom T.
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Updating the Multipliers
Network Simplex Animations
The current multipliers andreduced costs
How can wemake cπ
21 = 0and have othertree arcs have a0 reduced cost?
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Deleting (2,1) from T splits T into two parts
Network Simplex Animations
What value of∆ should bechosen to makethe reduced costof (2,1) = 0?
Adding ∆ to nodes on one side ofthe tree does not effect the reducedcosts of any tree arc except (2,1).Why?
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The updated multipliers and reduced costs
Network Simplex Animations
The current multipliers andreduced costs
Is this treesolutionoptimal?
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Send Flow Around the Cycle
Network Simplex Animations
1 unit of flow was sent around thecycle.
What is thenext spanningtree solution?
28
The updated multipliers and reduced costs
Network Simplex Animations
The current multipliers andreduced costs
What is thenext spanningtree solution?
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The next spanning tree solution
Network Simplex Animations
Here is the updated spanning treesolution
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Updated the multipliers
Network Simplex Animations
Here are the current multipliers
How shouldwe modify themultipliers?
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Updated the multipliers
Network Simplex Animations
Here are the current multipliers
What valueshould ∆ be?
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The updated multipliers
Network Simplex Animations
Here are the updated multipliers.
Is the currentspanning treesolutionoptimal?
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The Optimal Solution
Network Simplex Animations
Here is the optimal solution.
No arcviolates theoptimalityconditions.
