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?

16

Calculating Simplex Multipliers for a Spanning Tree

Network Simplex Animations

What is the simplexmultiplier for node4?

17

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.

18

Network Simplex Algorithm

Network Simplex Animations

The minimum Cost Flow Problem

19

Network Simplex Algorithm

Network Simplex Animations

An Initial Spanning Tree Solution

20

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?

22

Send Flow Around the Cycle

Network Simplex Animations

2 units of flow were sent along thecycle.

What is thenext spanningtree?

23

After a pivot

Network Simplex Animations

The Updated Spanning Tree

In a pivot, an arcis added to T andan arc is droppedfrom T.

24

Updating the Multipliers

Network Simplex Animations

The current multipliers andreduced costs

How can wemake cπ

21 = 0and have othertree arcs have a0 reduced cost?

25

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?

26

The updated multipliers and reduced costs

Network Simplex Animations

The current multipliers andreduced costs

Is this treesolutionoptimal?

27

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?

29

The next spanning tree solution

Network Simplex Animations

Here is the updated spanning treesolution

30

Updated the multipliers

Network Simplex Animations

Here are the current multipliers

How shouldwe modify themultipliers?

31

Updated the multipliers

Network Simplex Animations

Here are the current multipliers

What valueshould ∆ be?

32

The updated multipliers

Network Simplex Animations

Here are the updated multipliers.

Is the currentspanning treesolutionoptimal?

33

The Optimal Solution

Network Simplex Animations

Here is the optimal solution.

No arcviolates theoptimalityconditions.