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22multicommodityflows1

# 22multicommodityflows1 - 15.082 and 6.855 The...

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1 15.082 and 6.855 The Multicommodity Flow Problem

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2 On the Multicommodity Flow Problem O-D version K origin-destination pairs of nodes (s 1 , t 1 ), (s 2 , t 2 ), …, (s K , t K ) Network G = (N, A) d k = amount of flow that must be sent from s k to t k . u ij = capacity on (i,j) shared by all commodities k ij c = cost of sending 1 unit of commodity k in (i,j) k ij x = flow of commodity k in (i,j)
3 A Linear Multicommodity Flow Problem 5 units good 1 5 units good 1 2 units good 2 2 units good 2 \$1 \$1 1 2 3 4 5 6 \$5 \$1 \$6 \$1 \$1 u 25 = 5 Quick exercise: determine the optimal multicommodity flow.

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4 A Linear Multicommodity Flow Problem 1 2 3 4 5 6 5 units good 1 5 units good 1 2 units good 2 2 units good 2 \$1 \$5 \$1 \$1 u 25 = 5 \$1 \$1 \$6 222 32 25 56 2 xxx = == 111 12 25 54 3 = 1 14 2 x =
5 The Multicommodity Flow LP ∑∑ (,) kk ij ij ij A k cx Min ≤∈ for all ( , ) k ij ij k xu i jA Bundle constraints Supply/ demand constraints = −= − ∈ if if 0 ij ji k k jj di s xx d i t otherwise ≥∀ ∈ 0( , ), k ij xi j A k K

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6 Assumptions (for now) ± Homogeneous goods. Each unit flow of commodity k on (i,j) uses up one unit of capacity on (i,j). ± No congestion. Cost is linear in the flow on (i,j) until capacity is totally used up. ± Fractional flows . Flows are permitted to be fractional. ± OD pairs . Usually a commodity has a single origin and single destination.
7 Application areas Type of Network Nodes Arcs Flow Communic. Networks O-D pairs for messages Transmission lines message routing Computer Networks storage dev. or computers Transmission lines data, messages Railway Networks yard and junction pts. Tracks Trains Distribution Networks plants warehouses,. .. highways railway tracks etc. trucks, trains, etc

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8 On Fractional Flows ± In general, multicommodity flow problems have fractional flows, even if all data is integral.
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22multicommodityflows1 - 15.082 and 6.855 The...

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