Chapter 7 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 7:...

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Unformatted text preview: MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 7: The Network Simplex Method 1 / 19 In this topic, we develop the details of the simplex method applied to network flow problems (known as the network simplex method). This can lead to a much faster algorithm than standard simplex method. 2 / 19 Consider the uncapacitated minimum cost flow problem: min c x s.t. Ax = b x ≥ A is the node-arc incidence matrix of a directed graph G ( V , E ) where n = | V | and m = | E | . The size of the matrix A is n × m as opposed to earlier conventions. b is the supply vector and c is the cost vector Assumptions: 1 ∑ i ∈ V b i = 0 else the flow is infeasible. 2 Graph G ( V , E ) is connected else the network can be decomposed into smaller networks that can be solved independently. 3 / 19 Trees and Basic Feasible Solutions Network flow problems can be solved very efficiently because the basis matrices have a special structure that can be described nicely in terms of the network. To understand this structure, we analyze the node-arc incidence matrix. Each column of the incidence matrix A has exactly two nonzero entries, one equal to +1 and one equal to -1, indicating the start and end node of the arc. Adding up the rows of A gives the zero vector, indicating that the rows of A are linearly dependent....
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Chapter 7 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 7:...

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