and 6.855J. Network Simplex Animations
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1 .8 and 6.8J Network Simplex Animations
2 Calculating A Spanning Tree Flow A tree with supplies and demands. (Assume that all other arcs have a flow of ) What is the flow in arc (,)?
3 Calculating A Spanning Tree Flow To calculate flows, iterate up the tree, and find an arc whose flow is uniquely determined. What is the flow in arc (,)?
4 Calculating A Spanning Tree Flow -6 7 What is the flow in arc (,)? 6 -
5 Calculating A Spanning Tree Flow What is the flow in arc (,6)?
6 Calculating A Spanning Tree Flow What is the flow in arc (7,)? 6
7 Calculating A Spanning Tree Flow What is the flow in arc (,6)? 7
8 Calculating A Spanning Tree Flow Note: there are two different ways of calculating the flow on (,), and both ways give a flow of. Is this a coincidence? 8
9 Calculating Simplex Multipliers for a Spanning Tree Here is a spanning tree with arc costs. How can one choose node potentials so that reduced costs of tree arcs is? Recall: the reduced cost of (i,j) is c ij - π i + π j 9
10 Calculating Simplex Multipliers for a Spanning Tree There is a redundant constraint in the minimum cost flow problem. One can set π arbitrarily. We will let π i =. What is the simplex multiplier for node?
11 Calculating Simplex Multipliers for a Spanning Tree The reduced cost of (,) is c - π + π =. Thus - + π =. What is the simplex multiplier for node 7?
12 Calculating Simplex Multipliers for a Spanning Tree The reduced cost of (,) is c 7 - π 7 + π =. Thus -6 - π + =. What is the simplex multiplier for node?
13 Calculating Simplex Multipliers for a Spanning Tree What is the simplex multiplier for node 6?
14 Calculating Simplex Multipliers for a Spanning Tree What is the simplex multiplier for node?
15 Calculating Simplex Multipliers for a Spanning Tree What is the simplex multiplier for node?
16 Calculating Simplex Multipliers for a Spanning Tree These are the simplex multipliers associated with this tree. They do not depend on arc flows, nor on costs of non-tree arcs. 6
17 Network Simplex Algorithm -, $, $, $, $, $, $, $ -, $ T L U The minimum Cost Flow Problem 7
18 Spanning tree flows - - T L U An Initial Spanning Tree Solution 8
19 Simplex Multipliers and Reduced Costs The initial simplex multipliers and reduced costs? c = T L U What arcs are violating? 9
20 Add a violating arc to the spanning tree, creating a cycle,,,,,, u, x, Arc (,) is added to the tree, T L U What is the cycle, and how much flow can be sent?
21 Send Flow Around the Cycle,,,,,, u, x, units of flow were sent along the cycle., T L U What is the next spanning tree?
22 After a pivot,,,,,, u, x, The Updated Spanning Tree, T L U In a pivot, an arc is added to T and an arc is dropped from T.
23 Updating the Multipliers The current multipliers and reduced costs T L U How can we make c π = and have other tree arcs have a reduced cost?
24 Deleting (,) from T splits T into two parts Adding to nodes on one side of the tree does not effect the reduced costs of any tree arc except (,). Why? T L U What value of should be chosen to make the reduced cost of (,) =?
25 The updated multipliers and reduced costs - - The updated multipliers and reduced costs T L U Is this tree solution optimal?
26 Add a violating arc to the spanning tree, creating a cycle,,,,, Add arc (,) to the spanning tree,,, T L U What is the cycle, and how much flow can be sent? 6
27 Send Flow Around the Cycle,,,,,, unit of flow was sent around the cycle.,, T L U What is the next spanning tree solution? 7
28 The next spanning tree solution,,,,,,,, T L U Here is the updated spanning tree solution 8
29 Updated the multipliers - - Here are the current multipliers T L U How should we modify the multipliers? 9
30 Updated the multipliers Here are the current multipliers T L U What value should be?
31 The updated multipliers Here are the updated multipliers. T L U Is the current spanning tree solution optimal?
32 The Optimal Solution -6 - Here is the optimal solution. - T L U No arc violates the optimality conditions.
33 Finding the Cycle
34 Use Depth and Predecessor depth() = ; depth() = ; replace node by pred()
35 Use Depth and Predecessor depth(9) = ; depth() = ; replace node 9 by pred(9)
36 Use Depth and Predecessor depth() = ; depth() = ; replace node by pred(); replace node by pred() 6
37 Use Depth and Predecessor depth(8) = ; depth(7) = ; replace node 8 by pred(8); replace node 7 by pred() 7
38 Use Depth and Predecessor The least common ancestor of nodes and has been found. 8
39 Updating the multipliers: use the thread and depth Suppose that arc (,8) will drop out of the tree. What is the subtree rooted at node 8? 9
40 Follow the thread starting with node What is thread(8)? 6 9
41 Follow the thread starting with node What is thread()? 6 9
42 Follow the thread starting with node What is thread()? 6 9
43 Follow the thread starting with node What is thread()? 6 9
44 Follow the thread starting with node What is thread(6)? 6 9
45 The stopping rule depth = Stopping rule: stop when depth(current node) depth(8) depth = 9
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