ENGM 732 Formalization of Network Flows Network Flow Models

ENGM 732Formalization of Network Flows

Network Flow Models

Origin and Termination Lists

O = [O1, O2, O3, . . . , Om]

T = [T1, T2, T3, . . . , Tm]

Shortest Path (Flow, Cost)

[External Flow]

[1] [-1]

(0,3)

(0,5)(0,4)

(1,1)

2

1

4

53(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]

Flow

fk = flow into a nodefk

’= 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)

Cost

Cost may be associated with a flow in an arc.

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m

kk

k

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kk

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1

1

Capacity

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

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)

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)

n

isisi

m

kkk fhfhfH

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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 bi = 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

[ bi, bsi, his ](ck , hk)

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

[ bi ](ck , hk)

[3] [-5](1,2)

(4,-1)(3,5)

2

1

3

4(2,1) (3

,3)

[0]

[0]

1

2 5

4

35

8

7

6

(2-1)

(1,1)

(1,1)

Slack Node[ bi ]

(ck , hk)

[3] [-5](1,2)

(4,-1)(3,5)

2

1

3

4(2,1) (3

,3)

[0]

[0]

1

2 5

4

35

8

7

6

(2-1)

(1,1)

(1,1)

54

03

02

31

54

8532

6431

721

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Delete Nonzero Lower Bound

[3] [-3](fk,1,2)

2

1

3

4

[0]

[0]

1

2 5

4

3

[ bi](fk , ck , ck)

Delete Nonzero Lower Bound

[3] [-3](fk,1,2)

2

1

3

4

[0]

[0]

1

2 5

4

3

[ bi](fk , ck , ck)

[3] [-3](f’k,0,1)

2

1

3

4

[-1]

[+1]

1

2 5

4

3

Algebraic Model

0

1

k

kk

iMk

kkMk

k

k

m

kk

f

cf

bfaf

ts

fhMin

TiOi

..

Algebraic Model

0

1

k

kk

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kkMk

k

k

m

kk

f

cf

bfaf

ts

fhMin

TiOi

..

0f

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hf

Min

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

[ bi, bsi, his ](ck , hk)

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

[ bi ](ck , hk)

[3] [-5](1,2)

(2,-1)(3,5)

2

1

4

5(2,1) (5

,3)

[0]

[0]

1

2 5

4

357

6

(1,-1)

(2,1)

Example[ bi ]

(ck , hk)[3] [-5](1,2)

(2,-1)(3,5)

2

1

4

5(2,1) (5

,3)

[0]

[0]

1

2 5

4

357

6

(1,-1)

(2,1)

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0

0

3

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Primal / Dual Review

6

5

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2

1

21

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x

x

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st

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43

34

6524

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21

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Example

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0

1

1

3

1

2

1

5

53

5

51

52

43

42

32

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