# NF1 - Topics Terminology and Notation Network diagrams...

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8/14/04 J. Bard and J. W. Barnes Operations Research Models and Methods Copyright 2004 - All rights reserved Lecture 4 – Network Flow Programming Topics Terminology and Notation Network diagrams Generic problems (TP, AP, SPT, STP, MF) LP formulations Finding solutions with Excel add-in

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2 Network Optimization Network flow programming (NFP) is a special case of  linear programming Important to identify problems that can be modeled as  networks because: (1) Network representations make optimization models  easier to visualize and explain (1) Very efficient algorithms are available
3 Example of (Distribution) Network 8 5 6 4 2 7 3 1 (6) (3) (5) (7) (4) (2) (4) (5) (5) (6) (4) (7) (6) (3) [-150] [200] [-300] [200] [-200] [-200] (2) (2) (7) [-250] [700] [external flow] (cost) lower = 0, upper = 200 1 2 3 4 5 6 7 10 8 9 11 12 13 14 15 16 17

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4 Terminology • Nodes and arcs • Arc flow • Upper and lower bounds • Cost • Gains • External flow • Optimal flow
5 Network Flow Problems Pure Minimum Cost Flow Problem Generalized Minimum Cost Flow Problem Linear Program Transportation Problem Assignment Problem Shortest Path Problem Maximum Flow Problem Less general models More general models

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6 Transportation Problem We wish to ship goods (a single commodity) from            m  warehouses to  n  destinations at minimum cost. Warehouse i has s i  units available  i  = 1, …,  m  and destination j  has a demand of  d j j  = 1, …, n   Goal:   Ship the goods from warehouses to destinations            at minimum cost. Example: Plants Supply Markets Demand San Francisco 350 New York 325 Los Angeles 600 Chicago 300 Austin 275 Unit Shipping Costs From/To NY Chicago Austin SF 2.5 1.7 1.8 LA NA 1.8 1.4
7 • Total supply = 950, total demand = 900   • Transportation problem is defined on a bipartite network • Arcs only go from supply nodes to destination nodes; to handle excess  supply create a  dummy  destination with a demand of 50 • The min-cost flow network for this transportation problem is given by SF LA NY CHI AUS DUM [350] [600] [-50] [-275] [-300] [-325] (2.5) (1.7) (1.8) (0) (M) (1.8) (1.4) (0)

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8 Costs on arcs to dummy destination = 0 (In some settings it would be necessary   to include a nonzero warehousing cost.) The objective coefficient on the LA   NY arc is  M . This denotes a large value and effectively prohibits use of this arc (could eliminate arc). We are assured of integer solutions because technological matrix  A  is totally unimodular.  (important in some applications) Modeling Issues
9 The LP formulation of the transportation problem with  m sources and  n  destinations is given by: Min m =1    n =1 c ij x ij s.t.

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