Network Simplex Animations Network Simplex Animations

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

3 Calculating A Spanning Tree Flow Network Simplex Animations To calculate flows, iterate up the tree, and find an arc whose flow is uniquely determined. What is the flow in arc (5,3)?

4 Calculating A Spanning Tree Flow Network Simplex Animations What is the flow in arc (3,2)?

5 Calculating A Spanning Tree Flow Network Simplex Animations What is the flow in arc (2,6)?

6 Calculating A Spanning Tree Flow Network Simplex Animations What is the flow in arc (7,1)?

7 Calculating A Spanning Tree Flow Network Simplex Animations What is the flow in arc (1,2)?

8 Calculating A Spanning Tree Flow Network Simplex Animations 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?

9 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations Here is a spanning tree with arc costs. How can one choose node potentials so that reduced costs of tree arcs is 0? Recall: the reduced cost of (i,j) is c ij - π i + π j

10 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations Here is a spanning tree with arc costs. How can one choose node potentials so that reduced costs of tree arcs is 0? Recall: the reduced cost of (i,j) is c ij - π i + π j

11 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations Here is a spanning tree with arc costs. How can one choose node potentials so that reduced costs of tree arcs is 0? Recall: the reduced cost of (i,j) is c ij - π i + π j

12 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations The reduced cost of (1,2) is c 12 - π 1 + π 2 = 0. Thus π 2 = 0. What is the simplex multiplier for node 7?

13 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations The reduced cost of (1,2) is c 71 – π 7 + π 1 = 0. Thus -6 -π = 0. What is the simplex multiplier for node 3?

14 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations What is the simplex multiplier for node 6?

15 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations What is the simplex multiplier for node 5?

16 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations What is the simplex multiplier for node 4?

17 Calculating Simplex Multipliers for a Spanning Tree Network Simplex Animations These are the simplex multipliers associated with this tree. They do not depend 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 and reduced costs What arcs are violating?

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 the cycle, and how much flow can be sent?

22 Send Flow Around the Cycle Network Simplex Animations 2 units of flow were sent along the cycle. What is the next spanning tree?

23 After a pivot Network Simplex Animations The Updated Spanning Tree In a pivot, an arc is added to T and an arc is dropped from T.

24 Updating the Multipliers Network Simplex Animations The current multipliers and reduced costs How can we make c π 21 = 0 and have other tree arcs have a 0 reduced cost?

25 Deleting (2,1) from T splits T into two parts Network Simplex Animations What value of ∆ should be chosen to make the reduced cost of (2,1) = 0? Adding ∆ to nodes on one side of the tree does not effect the reduced costs of any tree arc except (2,1). Why?

26 The updated multipliers and reduced costs Network Simplex Animations The current multipliers and reduced costs Is this tree solution optimal?

27 Send Flow Around the Cycle Network Simplex Animations 1 unit of flow was sent around the cycle. What is the next spanning tree solution?

28 The updated multipliers and reduced costs Network Simplex Animations The current multipliers and reduced costs What is the next spanning tree solution?

29 The next spanning tree solution Network Simplex Animations Here is the updated spanning tree solution

30 Updated the multipliers Network Simplex Animations Here are the current multipliers How should we modify the multipliers?

31 Updated the multipliers Network Simplex Animations Here are the current multipliers What value should ∆ be?

32 The updated multipliers Network Simplex Animations Here are the updated multipliers. Is the current spanning tree solution optimal?

33 The Optimal Solution Network Simplex Animations Here is the optimal solution. No arc violates the optimality conditions.