1. ENGM 535 Optimization Networks

2. Transportation Models
(Flow, Cost)
[External Flow]
[2]
[4]
[3]
[-3]
(3,3)
(1,1)
(0,4)
(2,2)
(0,3) 1 2 3 4 5 6

3. Transportation Models
(Flow, Cost)
[External Flow]
[2]
[4]
[3]
[-3]
(3,3)
(1,1)
(0,4)
(2,2)
(0,3) 1 2 3 4 5 6
Properties
All arcs have infinite capacity.
All nodes have nonzero fixed external flows.
The sum of the external flows over all nodes is zero.

4. Assignment Models (Flow, Cost) [External Flow] [-1] [1] 1 4 [1] 2 5
(0,4)
[-1]
[1] 1 4
(1,1)
(1,3)
(0,8)
[1] 2 5
[-1]
(0,4)
(0,2)
(1,2)
[-1]
[1] 3 6

5. Assignment Models (Flow, Cost) [External Flow] [-1] Assignment [1] 1 4
(0,4)
[-1]
Assignment
All demands and supplies are unity.
Find the one-to-one pairing of the two sets that minimizes the sum of the pairing costs.
[1] 1 4
(1,1)
(1,3)
(0,8)
[1] 2 5
[-1]
(0,4)
(0,2)
(1,2)
[-1]
[1] 3 6

6. Shortest Path (Flow, Cost) [External Flow] 2 [1] 1 5 [-1] 3 4 (0,4)
(0,5)
(0,3)
(1,2)
(0,6)
[1] 1 5
[-1] 3
(1,1)
(0,5)
(1,4) 4

7. Shortest Path Shortest Path (Flow, Cost) [External Flow] 2 [1] 1 5
One node is the source.
One node is the sink.
Optimal path is the sequence of arcs such that the sum of the arc costs on the path are minimized.
(Flow, Cost)
[External Flow] 2
(0,4)
(0,5)
(0,3)
(1,2)
(0,6)
[1] 1 5
[-1] 3
(1,1)
(0,5)
(1,4) 4

8. Maximum Flow Models (Flow, Capacity) 2 4 6 1 3 5 (4,4) (6,8) (4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

9. Maximum Flow Models (Flow, Capacity) Maximal Flow 2 4 6 1 3 5
Capacity is only relevant parameter.
Find maximal flow from source to sink.
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

10. Maximum Flow Models (Flow, Capacity) Maximal Flow 2 4 6 1 3 5
Capacity is only relevant parameter.
Find maximal flow from source to sink.
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

11. Maximum Flow Models (Flow, Capacity) [External Flow] Maximal Flow 2 4
Capacity is only relevant parameter.
Find maximal flow from source to sink.
(4,4) 2 4
(6,8)
(4,10)
[-M]
[M]
(2,2)
(0,3)
(0,8) 6 S S 1
(3,6)
(5,7)
(3,3) 3 5

12. (Flow, capacity, gain, cost)
Network with Gains
(Flow, capacity, gain, cost)
[External Flow] 2
(1,2,.5,3)
(1.5,4,2,1)
[3]
[-3] 1 4
(0,2,1,1)
(1,2,1,-1)
(2,2,.5,2)
(0,4,2,5) 3

13. Relationships General Min Cost Flow Shortest Path Pure Min Cost Flow
Assignment
Transpor-
tation
General
Min Cost
Flow
Shortest
Path
Pure Min
Cost Flow
Linear
Program
Maximal
Flow
Less General
More General

14. Network Primal/Dual Relationship

15. Primal / Dual Review

16. Transportation Models
(Flow, Cost)
[External Flow]
[2]
[4]
[3]
[-3]
(3,3)
(1,1)
(0,4)
(2,2)
(0,3) 1 2 3 4 5 6

17. Example [ bi ] (ck , hk) [3] [-5] (1,2) (2,-1) (3,5) 2 1 4 5 (2,1)
(5,3)
[0] 3 7 6
(1,-1)
[ bi ]
(ck , hk)

18. Example

19. Moving to a Solution
For each basic arc
For each non-basic arc

20. Moving to a Solution
For each basic arc
For each non-basic arc