1 / 32

15.082 and 6.855J

15.082 and 6.855J. Network Simplex Animations. Calculating A Spanning Tree Flow. A tree with supplies and demands. (Assume that all other arcs have a flow of 0). 1. 1. -6. 2. 7. 3. 1. 3. 6. -4. What is the flow in arc (4,3)?. 2. 4. 5. 3. Calculating A Spanning Tree Flow.

alastair-scott
Download Presentation

15.082 and 6.855J

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 15.082 and 6.855J Network Simplex Animations

  2. Calculating A Spanning Tree Flow A tree with supplies and demands. (Assume that all other arcs have a flow of 0) 1 1 -6 2 7 3 1 3 6 -4 What is the flow in arc (4,3)? 2 4 5 3

  3. Calculating A Spanning Tree Flow To calculate flows, iterate up the tree, and find an arc whose flow is uniquely determined. 1 1 -6 2 7 3 1 3 6 -4 2 What is the flow in arc (5,3)? 2 4 5 3

  4. Calculating A Spanning Tree Flow 1 1 What is the flow in arc (3,2)? -6 2 7 3 1 3 6 -4 2 3 2 4 5 3

  5. Calculating A Spanning Tree Flow 1 1 What is the flow in arc (2,6)? -6 2 7 3 6 1 3 6 -4 2 3 2 4 5 3

  6. Calculating A Spanning Tree Flow 1 1 What is the flow in arc (7,1)? -6 2 7 3 6 4 1 3 6 -4 2 3 2 4 5 3

  7. Calculating A Spanning Tree Flow 1 1 What is the flow in arc (1,6)? 3 -6 2 7 3 6 4 1 3 6 -4 2 3 2 4 5 3

  8. Calculating A Spanning Tree Flow 1 1 Note: there are two different ways of calculating the flow on (1,2), and both ways give a flow of 4. Is this a coincidence? 4 3 -6 2 7 3 6 4 1 3 6 -4 2 3 2 4 5 3

  9. Calculating Simplex Multipliers for a Spanning Tree 1 Here is a spanning tree with arc costs. How can one choose node potentials so that reduced costs of tree arcs is 0? 5 -6 2 7 -4 3 3 6 -2 1 Recall: the reduced cost of (i,j) is cij - i + j 4 5

  10. Calculating Simplex Multipliers for a Spanning Tree There is a redundant constraint in the minimum cost flow problem. 0 1 5 -6 2 7 One can set 1 arbitrarily. We will let i = 0. -4 3 3 6 -2 1 What is the simplex multiplier for node 2? 4 5

  11. Calculating Simplex Multipliers for a Spanning Tree The reduced cost of (1,2) is c12 - 1 + 2 = 0. Thus 5 - 0 + 2 = 0. 0 1 5 -6 -5 2 7 -4 3 3 6 -2 1 What is the simplex multiplier for node 7? 4 5

  12. Calculating Simplex Multipliers for a Spanning Tree The reduced cost of (1,2) is c71 - 7 + 1 = 0. Thus -6 - 2 +0 = 0. 0 1 5 -6 -5 2 7 -4 3 -6 3 6 What is the simplex multiplier for node 3? -2 1 4 5

  13. Calculating Simplex Multipliers for a Spanning Tree 0 1 5 -6 -5 2 7 -4 3 -6 -2 3 6 What is the simplex multiplier for node 6? -2 1 4 5

  14. Calculating Simplex Multipliers for a Spanning Tree 0 1 5 -6 -5 2 7 -4 3 -6 -2 -1 3 6 What is the simplex multiplier for node 4? -2 1 4 5

  15. Calculating Simplex Multipliers for a Spanning Tree 0 1 5 -6 -5 2 7 -4 3 -6 -2 -1 3 6 What is the simplex multiplier for node 5? -2 1 -4 4 5

  16. Calculating Simplex Multipliers for a Spanning Tree 0 These are the simplex multipliers associated with this tree. They do not depend on arc flows, nor on costs of non-tree arcs. 1 5 -6 -5 2 7 -4 3 -6 -2 -1 3 6 -2 1 -4 4 -1 5

  17. T L U Network Simplex Algorithm 2 -4 2, $4 1 4 4, $2 3, $4 4, $2 3 4, $1 1, $4 3, $5 5, $5 2 5 5 -3 The minimum Cost Flow Problem

  18. Spanning tree flows T L U 2 -4 1 1 4 1 3 0 3 0 0 2 3 2 5 5 -3 An Initial Spanning Tree Solution

  19. Simplex Multipliers and Reduced Costs T L U 0 -4 0 1 4 0 2 3 c45 = 2 ? 3 -2 4 0 0 2 5 3 -2 What arcs are violating? The initial simplex multipliers and reduced costs

  20. Add a violating arc to the spanning tree, creating a cycle u14, x14 T L U 2, 1 1 4 4,1 3,3 4, 0 3 4,0 1, 0 3,2 5, 3 2 5 What is the cycle, and how much flow can be sent? Arc (2,1) is added to the tree

  21. Send Flow Around the Cycle u14, x14 T L U 2, 1 1 4 4,3 3,3 4, 0 3 4,2 1, 0 3,0 5, 3 2 5 What is the next spanning tree? 2 units of flow were sent along the cycle.

  22. After a pivot u14, x14 T L U 2, 1 1 4 4,3 3,3 4, 0 3 4,2 1, 0 3,0 5, 3 2 5 In a pivot, an arc is added to T and an arc is dropped from T. The Updated Spanning Tree

  23. Updating the Multipliers T L U 0 -4 0 1 4 0 2 3 4 3 -2 4 0 0 2 5 How can we make cp21 = 0 and have other tree arcs have a 0 reduced cost? 3 -2 The current multipliers and reduced costs

  24. Deleting (2,1) from T splits T into two parts T L U 0 -4 0 1 4 0 2 3 4 3 -2 4 0 0 2 5 What value of  should be chosen to make the reduced cost of (2,1) = 0? 3+ -2+ Adding  to nodes on one side of the tree does not effect the reduced costs of any tree arc except (2,1). Why?

  25. The updated multipliers and reduced costs T L U 0 -4 0 1 4 0 2 3 2 3 0 2 2 0 2 5 Is this tree solution optimal? 1 -4 The updated multipliers and reduced costs

  26. Add a violating arc to the spanning tree, creating a cycle T L U 2, 1 1 4 4,3 3,3 4, 0 3 4,2 1, 0 3,0 5, 3 2 5 What is the cycle, and how much flow can be sent? Add arc (3,4) to the spanning tree

  27. Send Flow Around the Cycle T L U 2, 2 1 4 4,2 3,2 4, 0 3 4,2 1, 0 3,0 5, 3 2 5 What is the next spanning tree solution? 1 unit of flow was sent around the cycle.

  28. The next spanning tree solution T L U 2, 2 1 4 4,2 3,2 4, 0 3 4,2 1, 0 3,0 5, 3 2 5 Here is the updated spanning tree solution

  29. Updated the multipliers T L U 0 -4 0 1 4 0 2 3 2 3 0 2 2 0 2 5 How should we modify the multipliers? 1 -4 Here are the current multipliers

  30. Updated the multipliers T L U 0 -4 + 0 1 4 0 2 3 2 3 0 2 2 0 2 5 What value should be? 1 -4 Here are the current multipliers

  31. The updated multipliers T L U 0 -6 -2 1 4 0 0 3 4 3 0 2 2 0 2 5 Is the current spanning tree solution optimal? 1 -4 Here are the updated multipliers.

  32. The Optimal Solution T L U 0 -6 -2 1 4 0 0 3 4 3 0 2 2 0 2 5 No arc violates the optimality conditions. 1 -4 Here is the optimal solution.

More Related