Generalized Network Flow

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.

Slide 1

Generalized Network Flow

LP Formulation of GNFP

Aji

Ni

uxbxx

xc

ijijij

iAijj

jijiAjijij

Ajiijij

),(

st

min

),(:),(:

),(

Note: the flows are usually not integral in GNFP

Slide 2

Generalized Network Flow

GNFP Example: Paper Recycling Problem

• Three types of paper plus fresh wood

• Minimize use of fresh wood subject to:

Paper Type Demand Min. Fresh. Supply Yield Can Make1 3475 0 4000 0.85 1,22 1223 575 1600 0.90 1,2,33 2260 300 1000 0.80 2,3

Slide 3

Generalized Network Flow

Formulation as GNFP: Transportation Subproblem

1a

2a

3a

1b

2b

3b

F

ij = 0.85

ij =0.90

ij =0.80

0

575

300

cij=1

slide 4

Generalized Network Flow

Formulation as GNFP: Supplies and Demands

1a

2a

3a

1b

2b

3b

F

4000

1600

1000

-3475

-1223

-2260

?

slide 5

Generalized Network Flow

Supply of Fresh Wood

• Add arc (F, F) with multiplier FF .

• Flow Out = xF1b + xF2b + xF3b + xFF• Flow In = FF xFF• Out – In = xF1b + xF2b + xF3b + (1- FF) xFF• Let bF = 0 and FF = 2.

• 0 = xF1b + xF2b + xF3b + (-1) xFF• xFF= xF1b + xF2b + xF3b

Slide 6

Generalized Network Flow

Supply of Wood Type 1

• Add arc (1a, 1a) with multiplier 1a1a.

• Flow Out = x1a1a + x1a1b + x1a2b

• Flow In = 1a1a x1a1a

• 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

Slide 7

Generalized Network Flow

Formulation as GNFP: Slack Arcs

1a

2a

3a

1b

2b

3b

F

0

575

300

= 2

= 0.5

= 0.5

= 0.5 slide 8