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Unformatted text preview: Lecture 4 – Network Flow Programming Topics • Terminology and Notation • Network diagrams • Generic problems (TP, AP, SPP, STP, MF) • LP formulations • Finding solutions with Excel addin 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 (2) Very efficient algorithms are available 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 Terminology • Nodes and arcs • Arc flow (variables) • Upper and lower bounds • Cost • Gains (and losses) • External flow (supply an demand) • Optimal flow 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 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: Warehouse 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 1.8 1.4 • 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 we can create a dummy destination with a demand of 50 and 0 shipment cost • The mincost flow network for this transportation problem is given by SF LA NY CHI AUS [350] [600] [275] [300] [325] (2.5) (1.7) (1.8) (0) ( M ) (1.8) (1.4) DUM [50] (0) • 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....
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 Spring '08
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 Shortest path problem, shortest path, Flow network, Maximum flow problem, xij

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