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Generalized Network Flow (GNF) Problem. 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.

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generalized network flow gnf problem
Generalized Network Flow (GNF) Problem
  • 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.
lp formulation of gnfp
LP Formulation of GNFP

Note: the flows are usually not integral

in GNFP

gnfp example paper recycling problem
GNFP Example: Paper Recycling Problem
  • Three types of paper plus fresh wood
  • Minimize use of fresh wood subject to:
formulation as gnfp transportation subproblem

ij = 0.85

ij =0.90

cij=1

ij =0.80

Formulation as GNFP: Transportation Subproblem

1a

1b

2a

2b

F

3a

3b

formulation as gnfp supplies and demands
Formulation as GNFP: Supplies and Demands

1a

1b

-3475

4000

2a

2b

F

1600

-1223

?

3a

3b

1000

-2260

supply of fresh wood
Supply of Fresh Wood
  • 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
supply of wood type 1
Supply of Wood Type 1
  • 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
formulation as gnfp slack arcs
Formulation as GNFP: Slack Arcs

 = 0.5

1a

1b

 = 2

2a

2b

F

 = 0.5

3a

3b

 = 0.5