1. ENGM 732 Formalization of Network Flows Network Flow Models

2. Origin and Termination Lists O = [O 1, O 2, O 3,..., O m ] T = [T 1, T 2, T 3,..., T m ]

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

4. Flow f k = flow into a node f k ’ = flow out of a node f k ’ = f k, flow in = flow out f k ’ = a k f k, flow with gains f = [f 1, f 2, f 3,..., f m ]’ (flow is a column vector)

5. Cost Cost may be associated with a flow in an arc.

6. Capacity c k < f k < c k, flow is restricted between upper and lower bounds

7. External Flows External flows enter or leave the network at nodes. For most network models, external flows represent connections to the world outside the system being modeled. f si is allowable slack flow (positive or negative) h si is cost of each clack flow (positive or negative)

8. External Flows External flows enter or leave the network at nodes. For most network models, external flows represent connections to the world outside the system being modeled. f si is allowable slack flow (positive or negative) h si is cost of each clack flow (positive or negative)

9. Conservation of Flow For each node, total arc flow leaving a node - total arc flow entering a node = fixed external flow at the node. Let b i = fixed external flow at node i. Then,

10. Slack Node [3,1,1] [-5,0,0] (1,2) (4,-1) (3,5) 2 1 3 4 (2,1) (3,3) [0,2,-1] [0,-1,1] 1 2 5 4 3 [ b i, b si, h is ] (c k, h k )

11. Slack Node [3,1,1] [-5,0,0] (1,2) (4,-1) (3,5) 2 1 3 4 (2,1) (3,3) [0,2,-1] [0,-1,1] 1 2 5 4 3 [ b i ] (c k, h k ) [3] [-5] (1,2) (4,-1) (3,5) 2 1 3 4 (2,1) (3,3) [0] 1 2 5 4 3 5 8 7 6 (2-1) (1,1)

12. Slack Node [ b i ] (c k, h k ) [3] [-5] (1,2) (4,-1) (3,5) 2 1 3 4 (2,1) (3,3) [0] 1 2 5 4 3 5 8 7 6 (2-1) (1,1)

13. Delete Nonzero Lower Bound [3] [-3] (f k,1,2) 2 1 3 4 [0] 1 2 5 4 3 [ b i ] (f k, c k, c k )

14. Delete Nonzero Lower Bound [3] [-3] (f k,1,2) 2 1 3 4 [0] 1 2 5 4 3 [ b i ] (f k, c k, c k ) [3] [-3] (f’ k,0,1) 2 1 3 4 [-1] [+1] 1 2 5 4 3

15. Algebraic Model

17. Example [3,2,1] [-5,0,0] (1,2) (2,-1) (3,5) 2 1 3 4 (3,1) (5,3) [0,1,-1] [0,0,0] 1 2 5 4 3 [ b i, b si, h is ] (c k, h k )

18. Example [3,2,1] [-5,0,0] (1,2) (2,-1) (3,5) 2 1 3 4 (3,1) (5,3) [0,1,-1] [0,0,0] 1 2 5 4 3 [ b i ] (c k, h k ) [3] [-5] (1,2) (2,-1) (3,5) 2 1 4 5 (2,1) (5,3) [0] 1 2 5 4 3 5 7 6 (1,-1) (2,1)

19. Example [ b i ] (c k, h k ) [3] [-5] (1,2) (2,-1) (3,5) 2 1 4 5 (2,1) (5,3) [0] 1 2 5 4 3 5 7 6 (1,-1) (2,1)

20. Primal / Dual Review

21. Example