engm 732 network flow programming network flow models
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ENGM 732Network Flow Programming
Network Flow Models
Transportation Models(Flow, Cost)
[External Flow]
[2]
[4]
[3] [-3]
[-3]
[-3](3,3)(1,1)
(0,4)
(2,2)
(3,3)(0,3)
(0,4)
1
2
3
4
5
6
Transportation Models(Flow, Cost)
[External Flow]
[2]
[4]
[3] [-3]
[-3]
[-3](3,3)(1,1)
(0,4)
(2,2)
(3,3)(0,3)
(0,4)
1
2
3
4
5
6
Properties1. All arcs have infinite
capacity.2. All nodes have nonzero
fixed external flows.3. The sum of the external
flows over all nodes is zero.
Assignment Models(Flow, Cost)
[External Flow]
[1]
[1]
[1] [-1]
[-1]
[-1](0,4)(1,1)
(1,3)
(1,2)
(0,4)
(0,2)
1
2
3
4
5
6
(0,8
)
Assignment Models(Flow, Cost)
[External Flow]
[1]
[1]
[1] [-1]
[-1]
[-1](0,4)(1,1)
(1,3)
(1,2)
(0,4)
(0,2)
1
2
3
4
5
6
(0,8
)Assignment1. All demands and supplies
are unity.2. Find the one-to-one
pairing of the two sets that minimizes the sum of the pairing costs.
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)
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)
Shortest Path1. One node is the source.2. One node is the sink.3. Optimal path is the
sequence of arcs such that the sum of the arc costs on the path are minimized.
Maximum Flow Models(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Maximum Flow Models(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Maximal Flow1. Capacity is only relevant
parameter.2. Find maximal flow from
source to sink.
Maximum Flow Models(Flow, Capacity)
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Maximal Flow1. Capacity is only relevant
parameter.2. Find maximal flow from
source to sink.
Maximum Flow Models(Flow, Capacity)[External Flow]
(0,3
)
(2,2)(5,7)
(0,8
)
(3,6)
1
2
3
4
5
6
(6,8)
(3,3)
(4,4)
(4,10)
Maximal Flow1. Capacity is only relevant
parameter.2. Find maximal flow from
source to sink.
S S
[M] [-M]
Network with Gains
2
1
3
4
[3] [-3]
(Flow, capacity, gain, cost)[External Flow]
(1,2,.5,3)
(2,2,.5,2)
(0,2
,1,1
)
(1,2
,1,-1
)
(0,4,2,5)
(1.5,4,2,1)
Relationships
Less General More General
Assignment
Transpor- tation
ShortestPath
Pure MinCost Flow
Maximal Flow
GeneralMin Cost
Flow
Linear Program
Network with Slack External Flow
2
1
3
4
[2,2,1] [-1]
[External Flow, max slack external flow, slack cost]
[0,-2,1]
[-1,2,-2]
Pure Min Cost FlowConsider K-Chair Corp.
Plant Cost / Chair Max Production Min Production 1 $5 500 0 2 7 750 400 3 3 1000 500 4 4 250 250
Wood comes from 1 of 2 suppliers and K-Chair agrees to buy 8 tons (800 chairs at 20 lbs / chair) from each supplier. Cost is $0.10 per lbs from supplier one and $0.075 per lbs from supplier 2. Transportation costs follow.
P1 P2 P3 P4Supplier 1 0.01 0.02 0.04 0.04Supplier 2 0.04 0.03 0.02 0.02
Pure Min Cost FlowChairs are sold in NY, Houston, San Francisco, and Chicago. Transportation costs From each plant to each city follows
NY H SF CP1 1 1 2 0P2 3 6 7 3P3 3 1 5 3P4 8 2 1 4
Selling price, maximum demand, and minimum demand follow
SP Max MinNY $20 2000 500H 15 400 100SF 20 1500 500C 18 1500 500
K-Chair (Supplier)
1
2
3
4
5
6
[800,M,2]
[800,M,1.5]
(lower, upper, cost)[Fixed, slack, cost]
(0,M
,.2)
K-Chair (Production)
1
2
3
4
5
6
[800,M,2]
[800,M,1.5]
(lower, upper, cost)[Fixed, slack, cost]
(0,M
,.2)
7
8
9
10
(400,750,7)
K-Chair (Shipping)
1
2
3
4
5
6
[800,M,2]
[800,M,1.5]
(lower, upper, cost)[Fixed, slack, cost]
(0,M
,.2)
7
8
9
10
(400,750,7)
NY
H
SF
C
(0,M,1)
(0,M,1)
(0,M,2)
(0,M,0)
K-Chair (Sales)
1
2
3
4
5
6
[800,M,2]
[800,M,1.5]
(lower, upper, cost)[Fixed, slack, cost]
(0,M
,.2)
7
8
9
10
(400,750,7)
NY
H
SF
C
(0,M,1)
(0,M,1)
(0,M,2)
(0,M,0)
[-500,-1500,-20]
[-100,-300,-15]
[-500,-1000,-20]
[-500,-1000,-18]
(0,500,5)
(500,1000,3)
(250,250,4)