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ENGM 732 Formalization of Network Flows Network Flow Models

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

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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]

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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)

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

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

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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)

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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)

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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,

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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 )

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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)

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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)

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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 )

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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

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Algebraic Model

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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 )

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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)

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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)

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Primal / Dual Review

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Example

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Max-flow/min-cut theorem Theorem: For each network with one source and one sink, the maximum flow from the source to the destination is equal to the minimal.

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