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Lecture16

Lecture16 - Introduction to Mathematical Programming IE406...

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Introduction to Mathematical Programming IE406 Lecture 16 Dr. Ted Ralphs

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IE406 Lecture 16 1 Reading for This Lecture Bertsimas 7.1-7.3
IE406 Lecture 16 2 Network Flow Problems Networks are used to model systems in which a commodity or commodities must be transported from one or more supply points to one or more demand points along defined pathways. These models occur naturally in many contexts. Transportation Logistics Telecommunications Network flow problems are defined on graphs that define the structure of the pathways in the network.

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IE406 Lecture 16 3 Undirected Graphs An undirected graph G = ( N, E ) consists of A finite set of nodes N representing the supply and demand points. A set E of unordered pairs of nodes called edges representing the pathways joining pairs of nodes. We say that the edge { i, j } is incident to nodes i and j and i and j are its endpoints . The degree of a node is the number of edges incident to it. The degree of a graph is the maximum of the degrees of its nodes.
IE406 Lecture 16 4 Basic Definitions (Undirected) A walk is a finite sequence of nodes i 1 , . . . , i t such that { i k , i k +1 } ∈ E k = 1 , 2 , . . . , t - 1 . A walk is called a path if it has no repeated nodes. A cycle is a path with i 1 = i t , t > 2 . An undirected graph is said to be connected if for every pair of nodes i and j , there is a path from i to j . For undirected graphs, our convention will be to denote | N | = n and | E | = m .

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IE406 Lecture 16 5 Directed Graphs A directed graph G = ( N, A ) consists of a set of N nodes and a set A of ordered pairs of nodes called
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