Lecture 14 - Network

Lecture 14 - Network - 540:311DETERMINISTICMODELSIN...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 14: Chapter 8.1‐8.3 Class Mee±ng: Mon April 18 th 10:20‐11:40am Prof. W. Art Chaovalitwongse
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DescripCon Many important op@miza@on problems can be analyzed by means of graphical or network representa@on. The following network models will be discussed: 8.2 Shortest path problems 8.3 Maximum flow problems 8.5 Minimum cost network flow problems 8.6 Minimum spanning tree problems
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8.1 Basic Defni@ons A graph or network is defned by two sets oF symbols: • Nodes: A set oF points or ver@ces ( V ) are called nodes oF a graph or network. • Arcs: An arc consists oF an ordered pair oF ver@ces and represents a possible direc@on oF mo@on that may occur between ver@ces. 1 2 Nodes 1 2 Arc
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Nota@on and Terminology Network terminology. Node set N = {1, 2, 3, 4} = V Vertices Network G = (N, A) = (V,E) Arc Set A = {(1,2), (1,3), (3,2), (3,4), (2,4)} E Edges 2 3 4 1 a b c d e An Undirected Graph or Undirected Network 2 3 4 1 a b c d e A Directed Graph or Directed Network In an undirected graph, (i,j) = (j,i)
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Chain: A sequence of arcs such that every arc has exactly one vertex in common with the previous arc is called a chain. 1 2 Common vertex between two arcs Path: A path is a chain in which the terminal node of each arc is iden@cal to the ini@al node of next arc. For example in the ±gure below (1,2)‐(2,3)‐(4,3) is a chain but not a path; (1,2)‐(2,3)‐(3,4) is a chain and a path, which represents a way to travel from node 1 to node 4. 2 3 1 4
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Some Good Applica@ons Transporta@on Transporta@on of goods over transporta@on networks Scheduling of fleets of airplanes: @me/space networks Manufacturing Scheduling of goods for manufacturing Flow of manufactured items within inventory systems Communica@ons Design and expansion of communica@on systems Flow of informa@on across networks Personnel Assignment Assignment of crews to airline schedules Assignment of drivers to vehicles
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Communica@on Networks
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Example: Communica@on Network Modeling Given the topology and traffic matrix in an IP network, which link weights should be used such that the cost is minimized? Input: graph G(R,L) R is the set of routers L is the set of unidirec@onal links C l is the capacity of link l Input: traffic matrix M i,j is traffic load from router i to j Output: se‘ng of the link weights w l is weight on unidirec@onal link l P i,j,l is frac@on of traffic from i to j traversing link l
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Supply Chain Networks
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8.2 Shortest Path Problems Assume that each arc in the network has a length associated with it. Suppose we start with a par@cular node. The problem of Fnding the shortest path from node 1 to any other node in the network is called a shortest path problem. 2 3 4 5 6 2 4 2 1 3 4 2 3 2 1 Consider a network G = (N, A) in which there is an origin node s and a destination node t.
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This note was uploaded on 09/20/2011 for the course ENG 300 taught by Professor Albin during the Fall '11 term at Rutgers.

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Lecture 14 - Network - 540:311DETERMINISTICMODELSIN...

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