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.

lineariscostiffh

fhfH

k

m

kk

k

m

kk

,

)()(

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

11

)()(

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,

gainswithbfaf

networkpurebff

iMk

kkMk

k

iMk

kMk

k

TiOi

TiOi

,

,

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

ffnode

ffffnode

ffffnode

fffnode

:

:

:

:

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

iMk

kkMk

k

k

m

kk

f

cf

bfaf

ts

fhMin

TiOi

..

0f

cf

bAf s.t.

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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fff

ff

fff

ffff

fff

st

ffffffffMin

876

54321

876

54

532

6431

721

87654321

21

52133

0

5

0

0

3

01131215

,,

,,,,

Primal / Dual Review

6

5

2434

43

2

1

21

21

x

x

xx

st

xxMax

43

34

6524

31

21

321

yy

yy

st

yyyMin

Example

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876

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532

6431

721

87654321

21

52133

0

5

0

0

3

01131215

,,

,,,,

0

1

1

3

1

2

1

5

53

5

51

52

43

42

32

31

21

41

st

Min