21_Multicommodity_Flows_1

21_Multicommodity_Flows_1 - 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)
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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 $5 $1 $6 $1 $1 $1 $1 u 25 = 5 2 2 2 32 25 56 2 x x x = = = 1 1 1 12 25 54 3 x x x = = = 1 14 2 x =
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5 The Multicommodity Flow LP 2200 0 ( , ) , k ij x i j A k K i ( , ) k k ij ij i j A k c x Min 5 5 for all ( , ) k ij ij k x u i j A Bundle constraints Supply/ demand constraints = - = - if if 0 k k k k ij ji k k j j d i s x x d i t otherwise
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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.
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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
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21_Multicommodity_Flows_1 - 15.082 and 6.855 The...

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