.8 and 6.8J Network Simplex Animations
Calculating A Spanning Tree Flow -6 7 6 - A tree with supplies and demands. (Assume that all other arcs have a flow of ) What is the flow in arc (,)?
Calculating A Spanning Tree Flow -6 7 6 - To calculate flows, iterate up the tree, and find an arc whose flow is uniquely determined. What is the flow in arc (,)?
Calculating A Spanning Tree Flow -6 7 What is the flow in arc (,)? 6 -
Calculating A Spanning Tree Flow -6 7 6 6 - What is the flow in arc (,6)?
Calculating A Spanning Tree Flow -6 7 6 6 - What is the flow in arc (7,)? 6
Calculating A Spanning Tree Flow -6 7 6 6 - What is the flow in arc (,6)? 7
Calculating A Spanning Tree Flow -6 7 6 6 - Note: there are two different ways of calculating the flow on (,), and both ways give a flow of. Is this a coincidence? 8
Calculating Simplex Multipliers for a Spanning Tree -6 7-6 - 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
Calculating Simplex Multipliers for a Spanning Tree -6 7-6 - 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?
Calculating Simplex Multipliers for a Spanning Tree -6-7 - 6 - The reduced cost of (,) is c - π + π =. Thus - + π =. What is the simplex multiplier for node 7?
Calculating Simplex Multipliers for a Spanning Tree -6-7 - -6 6 - The reduced cost of (,) is c 7 - π 7 + π =. Thus -6 - π + =. What is the simplex multiplier for node?
Calculating Simplex Multipliers for a Spanning Tree -6-7 - -6-6 - What is the simplex multiplier for node 6?
Calculating Simplex Multipliers for a Spanning Tree -6-7 - -6-6 - - What is the simplex multiplier for node?
Calculating Simplex Multipliers for a Spanning Tree - -6-7 - -6-6 - - What is the simplex multiplier for node?
Calculating Simplex Multipliers for a Spanning Tree -6-7 - -6-6 - - - - These are the simplex multipliers associated with this tree. They do not depend on arc flows, nor on costs of non-tree arcs. 6
Network Simplex Algorithm -, $, $, $, $, $, $, $ -, $ T L U The minimum Cost Flow Problem 7
Spanning tree flows - - T L U An Initial Spanning Tree Solution 8
Simplex Multipliers and Reduced Costs - - - The initial simplex multipliers and reduced costs? c = T L U What arcs are violating? 9
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?
Send Flow Around the Cycle,,,,,, u, x, units of flow were sent along the cycle., T L U What is the next spanning tree?
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.
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?
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 (,) =?
The updated multipliers and reduced costs - - The updated multipliers and reduced costs T L U Is this tree solution optimal?
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
Send Flow Around the Cycle,,,,,, unit of flow was sent around the cycle.,, T L U What is the next spanning tree solution? 7
The next spanning tree solution,,,,,,,, T L U Here is the updated spanning tree solution 8
Updated the multipliers - - Here are the current multipliers T L U How should we modify the multipliers? 9
Updated the multipliers - + - Here are the current multipliers T L U What value should be?
The updated multipliers -6 - - Here are the updated multipliers. T L U Is the current spanning tree solution optimal?
The Optimal Solution -6 - Here is the optimal solution. - T L U No arc violates the optimality conditions.
Finding the Cycle 8 7 6 9
Use Depth and Predecessor 8 7 6 9 depth() = ; depth() = ; replace node by pred()
Use Depth and Predecessor 8 7 6 9 depth(9) = ; depth() = ; replace node 9 by pred(9)
Use Depth and Predecessor 8 7 6 9 depth() = ; depth() = ; replace node by pred(); replace node by pred() 6
Use Depth and Predecessor 8 7 6 9 depth(8) = ; depth(7) = ; replace node 8 by pred(8); replace node 7 by pred() 7
Use Depth and Predecessor 8 7 6 9 The least common ancestor of nodes and has been found. 8
Updating the multipliers: use the thread and depth 8 7 6 9 Suppose that arc (,8) will drop out of the tree. What is the subtree rooted at node 8? 9
Follow the thread starting with node 8 8 7 What is thread(8)? 6 9
Follow the thread starting with node 8 8 7 What is thread()? 6 9
Follow the thread starting with node 8 8 7 What is thread()? 6 9
Follow the thread starting with node 8 8 7 What is thread()? 6 9
Follow the thread starting with node 8 8 7 What is thread(6)? 6 9
The stopping rule depth = 6 8 7 Stopping rule: stop when depth(current node) depth(8) depth = 9