1. 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.
Generalized Network Flow

2. LP Formulation of GNFP Note: the flows are usually not integral
in GNFP
Generalized Network Flow

3. GNFP Example: Paper Recycling Problem
Three types of paper plus fresh wood
Minimize use of fresh wood subject to:
Generalized Network Flow

4. Formulation as GNFP: Transportation Subproblem
ij = 0.85
ij =0.80 1a 1b
ij =0.90 2a 2b F
cij=1 3a 3b
Generalized Network Flow

5. Formulation as GNFP: Supplies and Demands1b
-3475
4000 2a 2b F
1600
-1223 ? 3a 3b
1000
-2260
Generalized Network Flow

6. 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
Generalized Network Flow

7. 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
Generalized Network Flow

8. Formulation as GNFP: Slack Arcs
 = 0.5 1a 1b
 = 2 2a 2b F
 = 0.5 3a 3b
 = 0.5
Generalized Network Flow