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Lecture17

Lecture17 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 17 Dr. Ted Ralphs

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IE316 Lecture 17 1 Reading for This Lecture Bertsimas 7.1-7.3
IE316 Lecture 17 2 Network Flow Problems Networks are used to model systems in which a commodity or commodities must be transported from one or 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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IE316 Lecture 17 3 Undirected Graphs An undirected graph G = ( N, E ) consists of A 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.
IE316 Lecture 17 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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IE316 Lecture 17 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 arcs .
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Lecture17 - Advanced Operations Research Techniques IE316...

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