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Generalized Network Flow (GNF) ProblemPowerPoint Presentation

Generalized Network Flow (GNF) Problem

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Generalized Network Flow (GNF) Problem

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- Each arc (i, j) has a multiplier ij
- If 1 unit of flow leaves node i on arc (i, j), then ij will arrive node j.
- When ij< 1 the arc is said to be lossy.
- When ij> 1 the arc is said to be gainy.
- cij, ij and uij apply to the amount of flow leaving node i.

Note: the flows are usually not integral

in GNFP

- Three types of paper plus fresh wood
- Minimize use of fresh wood subject to:

ij = 0.85

ij =0.90

cij=1

ij =0.80

1a

1b

2a

2b

F

3a

3b

1a

1b

-3475

4000

2a

2b

F

1600

-1223

?

3a

3b

1000

-2260

- Add arc (F, F) with multiplier FF .
- Flow Out = xF1b + xF2b + xF3b + xFF
- Flow In = FFxFF
- Out – In = xF1b + xF2b + xF3b + (1-FF)xFF
- Let bF = 0 and FF = 2.
- 0 = xF1b + xF2b + xF3b + (-1)xFF
- xFF= xF1b + xF2b + xF3b

- Add arc (1a, 1a) with multiplier 1a1a.
- Flow Out = x1a1a + x1a1b + x1a2b
- Flow In = 1a1ax1a1a
- Out – In = x1a1b + x1a2b + (1-1a1a)x1a1a
- Let b1a = 4000 and 1a1a = 0.5.
- x1a1b + x1a2b + (0.5)x1a1a= 4000
- Unused supply of wood type 1 = x1a1a

= 0.5

1a

1b

= 2

2a

2b

F

= 0.5

3a

3b

= 0.5