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Unformatted text preview: Copyright © 2007 by Robert Sedgewick and Kevin Wayne. Linear Programming Reference: The Allocation of Resources by Linear Programming, Scientific American, by Bob Bland • brewer’s problem • simplex algorithm • implementation • solving LPs • linear programming 2 Linear Programming What is it? ! Quintessential tool for optimal allocation of scarce resources, among a number of competing activities. ! Powerful and general problemsolving method that encompasses: – shortest path, network flow, MST, matching, assignment... – Ax = b, 2person zero sum games Why significant? ! Widely applicable problemsolving model ! Dominates world of industry. ! Fast commercial solvers available: CPLEX, OSL. ! Powerful modeling languages available: AMPL, GAMS. ! Ranked among most important scientific advances of 20 th century. see ORF 307 Ex: Delta claims that LP saves $100 million per year. 3 Applications Agriculture. Diet problem. Computer science. Compiler register allocation, data mining. Electrical engineering. VLSI design, optimal clocking. Energy. Blending petroleum products. Economics. Equilibrium theory, twoperson zerosum games. Environment. Water quality management. Finance. Portfolio optimization. Logistics. Supplychain management. Management. Hotel yield management. Marketing. Direct mail advertising. Manufacturing. Production line balancing, cutting stock. Medicine. Radioactive seed placement in cancer treatment. Operations research. Airline crew assignment, vehicle routing. Physics. Ground states of 3D Ising spin glasses . Plasma physics. Optimal stellarator design. Telecommunication. Network design, Internet routing. Sports. Scheduling ACC basketball, handicapping horse races. 4 brewer’s problem simplex algorithm implementation solving LPs linear programming 5 Toy LP example: Brewer’s problem Small brewery produces ale and beer. ! Production limited by scarce resources: corn, hops, barley malt. ! Recipes for ale and beer require different proportions of resources. Brewer’s problem: choose product mix to maximize profits. corn (lbs) hops (oz) malt (lbs) profit ($) available 480 160 1190 ale (1 barrel) 5 4 35 13 beer (1 barrel) 15 4 20 23 all ale (34 barrels) 179 136 1190 442 all beer (32 barrels) 480 128 640 736 20 barrels ale 20 barrels beer 400 160 1100 720 12 barrels ale 28 barrels beer 480 160 980 800 more profitable product mix? >800 ? 6 Brewer’s problem: mathematical formulation ale beer maximize 13A + 23B profit subject to the constraints 5A + 15B ! 480 corn 4A + 4B ! 160 hops 35A + 20B ! 1190 malt A " B " Small brewery produces ale and beer. ! Production limited by scarce resources: corn, hops, barley malt. ! Recipes for ale and beer require different proportions of resources....
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 Spring '08
 KEVINWAYNE
 Linear Programming, Optimization, Simplex algorithm

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