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mincostnew

# mincostnew - Informatics and Mathematical Modelling...

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1 Jesper Larsen & Jens Clausen Informatics and Mathematical Modelling / Operations Research The Min Cost Flow Problem Jesper Larsen & Jens Clausen Informatics and Mathematical Modelling Technical University of Denmark

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2 Jesper Larsen & Jens Clausen Informatics and Mathematical Modelling / Operations Research Min Cost Flow - Terminology We consider a digraph G = ( V ( G ) , E ( G )) in which each edge e has a capacity u e ∈ R + and a unit transportation cost c e ∈ R . Each vertex v furthermore has a demand b v ∈ R . If b v 0 then v is called a sink , and if b v < 0 then v is called a source . We assume that b ( V ) = v V b v = 0 .
3 Jesper Larsen & Jens Clausen Informatics and Mathematical Modelling / Operations Research Min Cost Flow - Example 2 10 3 8 1 15 4 15 5 15 1 15 1 15 Cost Capacity Demand 5 Demand 10 Capacity 5 Capacity 10

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4 Jesper Larsen & Jens Clausen Informatics and Mathematical Modelling / Operations Research Min Cost Flow - Definition The Min Cost Flow problem consists in supplying the sinks from the sources by a flow in the cheapest possible way: min e E c e x e f x ( v ) = b v v V 0 x vw u vw ( u, w ) E where f x ( v ) = ( w,v ) E x wv - ( w,v ) E x vw .
5 Jesper Larsen & Jens Clausen Informatics and Mathematical Modelling / Operations Research Special cases Numerous flow problems can be stated as a Min Cost Flow problem: 1. The Transportation Problem 2. The Shortest Path Problem 3. The Max Flow Problem

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mincostnew - Informatics and Mathematical Modelling...

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