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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.

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Generalized Network Flow (GNF) Problem

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  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.

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

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

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

  5. Formulation as GNFP: Supplies and Demands 1a 1b -3475 4000 2a 2b F 1600 -1223 ? 3a 3b 1000 -2260

  6. 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

  7. 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

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

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