s02lec11 - 15.053 Network Models Thursday, March 14 Linear...

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1 15.053 Thursday, March 14 z Introduction to Network Flows z Handouts: Lecture Notes 2 Network Models z Linear Programming models that exhibit a very special structure z Can use this structure to dramatically reduce computational complexity z First widespread application of LP to problems of industrial logistics z Addresses huge number of diverse applications 3 Notation and Terminology Note: Network terminology is not (and never will be) standardized. The same concept may be denoted in many different ways. Called: NETWORK directed graph digraph graph 2 1 4 3 Class Handouts (Ahuja, Magnanti, Orlin) Node set N = {1,2,3,4} Arc Set Network G = (N,A) {(1,2),(1,3),(3,2),(3,4),(2,4)} A={1-2,1-3,3-2,3-4,2-4} Graph G = (V,E) Edge set: Also Seen Vertex set V = {1,2,3,4} 4 Directed and Undirected Networks 2 3 4 1 a b c d e An Undirected Graph 2 3 4 1 a b c d e A Directed Graph The field of Network Optimization concerns optimization problems on networks Networks are used to transport commodities physical goods (products, liquids) communication electricity, etc. Applications Physical analog Physical analog Flow Communication systems phone exchanges, computers, transmission facilities, satellites Cables, fiber optic links, microwave relay links Voice messages, Data, Video transmissions Hydraulic systems Pumping stations Reservoirs, Lakes Pipelines Water, Gas, Oil, Hydraulic fluids Integrated computer circuits Gates, registers, processors Wires Electrical current Mechanical systems Joints Rods, Beams, Springs Heat, Energy Transportation systems Intersections, Airports, Rail yards Highways, Airline routes Railbeds Passengers, freight, vehicles, operators An Overview of Some Applications of Network Optimization 6 Examples of terms. 2 3 4 Directed Path . Example: 1, 2, 3, 4 (or 1, a, 2, b, 3, e) Directions are important. Two paths a-b-e (or 1-2-3-4) and a-c-d-e (or 1-2-5-3-4) a b c 1 Cycle or circuit (or loop) 1, 2, 3, 1. (or 1, a, 2, b, 3, e) Note that directions are ignored. Directed Cycle: (1, 2, 3, 4, 1) or 1, a, 2, b, 3, c, 4, d, 1 Directions are important. Cycles (loops): a-b-c-d (or 1-2-3-4-1) b-a-d-c (or 3-2-1-4-3) e-b-a (or 1-3-2-1) c-d-e (or 3-4-1-3) 5 d e 2 3 4 a b c d 1 e Path: Example: 5, 2, 3, 4. (or 5, c, 2, b, 3, e, 4) Note that directions are ignored.
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7 More Definitions 2 1 4 3 A network is connected if every node can be reached from every other node by following a sequence of arcs in which direction is ignored.
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This note was uploaded on 12/20/2011 for the course BUS 15.053 taught by Professor Prof.jamesorlin during the Spring '05 term at MIT.

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s02lec11 - 15.053 Network Models Thursday, March 14 Linear...

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