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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.

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15.082 and 6.855J

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  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.

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