engm 732 formalization of network flows
DESCRIPTION
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). - PowerPoint PPT PresentationTRANSCRIPT
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)
54030231
54
8532
6431
721
ffnodeffffnode
ffffnodefffnode
::::
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
..
0fcf
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)
edunristrictffffffff
ffffffff
fffffff
stffffffffMin
876
54321
876
54
532
6431
721
87654321
2152133
05003
01131215
,,,,,,
Primal / Dual Review
65
2434
43
2
1
21
21
xx
xxst
xxMax
4334
6524
31
21
321
yyyy
styyyMin
Example
edunristrictffffffff
ffffffff
fffffff
stffffffffMin
876
54321
876
54
532
6431
721
87654321
2152133
05003
01131215
,,,,,,
01131215
53
5
51
52
43
42
32
31
21
41
stMin