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ENGM 732 Formalization of Network Flows

ENGM 732 Formalization of Network Flows. Network Flow Models. Origin and Termination Lists . O = [O 1 , O 2 , O 3 , . . . , O m ] T = [T 1 , T 2 , T 3 , . . . , T m ]. Shortest Path . (Flow, Cost) [External Flow]. O = [1,1,1,2,3,3,3,4] T = [2,2,3,4,4,5,5,5]. 2. (0,4). (0,5). (0,3).

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ENGM 732 Formalization of Network Flows

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  1. ENGM 732Formalization of Network Flows Network Flow Models

  2. Origin and Termination Lists O = [O1, O2, O3, . . . , Om] T = [T1, T2, T3, . . . , Tm]

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

  4. Flow fk = flow into a node fk’= flow out of a node fk’= fk , flow in = flow out fk’= akfk , flow with gains f = [f1, f2, f3, . . . , fm]’ (flow is a column vector)

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

  6. Capacity , flow is restricted between upper and lower bounds ck<fk<ck

  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. fsi is allowable slack flow (positive or negative) hsi 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. fsi is allowable slack flow (positive or negative) hsi 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 bi = fixed external flow at node i. Then,

  10. Slack Node [0,2,-1] [ bi, bsi, his ] (ck, hk) 2 (3,5) (4,-1) 1 4 [3,1,1] [-5,0,0] (1,2) 1 4 3 (2,1) (3,3) 5 2 3 [0,-1,1]

  11. Slack Node [0,2,-1] [ bi ] (ck, hk) 2 (3,5) (4,-1) [0] 1 (2-1) 4 [3,1,1] 2 [-5,0,0] 6 (3,5) (4,-1) (1,2) 1 1 4 4 [3] 3 [-5] (1,1) (1,2) 1 4 5 3 7 (3,3) (2,1) (3,3) (2,1) 5 2 (1,1) 5 2 3 8 [0] 3 [0,-1,1]

  12. Slack Node [ bi ] (ck, hk) [0] (2-1) 2 6 (3,5) (4,-1) 1 4 [3] [-5] (1,1) (1,2) 1 4 5 3 7 (3,3) (2,1) 5 2 (1,1) 3 8 [0]

  13. Delete Nonzero Lower Bound [ bi] (fk , ck, ck) [0] 2 1 4 [3] [-3] (fk,1,2) 1 4 3 5 2 3 [0]

  14. Delete Nonzero Lower Bound [ bi] (fk , ck, ck) [0] [-1] 2 2 1 1 4 4 [3] [3] [-3] [-3] (f’k,0,1) (fk,1,2) 1 1 4 4 3 3 5 5 2 2 3 3 [0] [+1]

  15. Algebraic Model

  16. Algebraic Model

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

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

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

  20. Primal / Dual Review

  21. Example

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