lp - Linear Programming see ORF 307 Linear Programming What...

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Robert Sedgewick and Kevin Wayne • Copyright © 2006 • http://www.Princeton.EDU/~cos226 Linear Programming Reference: The Allocation of Resources by Linear Programming, Scientific American, by Bob Bland 2 Linear Programming What is it? ! Quintessential tool for optimal allocation of scarce resources, among a number of competing activities. ! Powerful and general problem-solving method that encompasses: – shortest path, network flow, MST, matching – Ax = b, 2-person zero sum games Why significant? ! Fast commercial solvers: CPLEX, OSL. ! Powerful modeling languages: AMPL, GAMS. ! Ranked among most important scientific advances of 20 th century. ! Widely applicable and dominates world of industry. see ORF 307 Ex: Delta claims saving $100 million per year using LP 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, two-person zero-sum games. Environment. Water quality management. Finance. Portfolio optimization. Logistics. Supply-chain 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 3-D Ising spin glasses . Plasma physics. Optimal stellarator design. Telecommunication. Network design, Internet routing. Sports. Scheduling ACC basketball, handicapping horse races. 4 Brewery Problem: A Toy LP Example 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. How can brewer maximize profits? ! Devote all resources to ale: 34 barrels of ale ! $442. ! Devote all resources to beer: 32 barrels of beer ! $736. ! 7.5 barrels of ale, 29.5 barrels of beer ! $776. ! 12 barrels of ale, 28 barrels of beer ! $800. Beverage Corn (pounds) Malt (pounds) Hops (ounces) Beer (barrel) 15 20 4 Ale (barrel) 5 35 4 Profit ($) 23 13 Limit 480 1190 160
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Brewery Problem max 13 A + 23 B s. t. 5 A + 15 B " 480 4 A + 4 B " 160 35 A + 20 B " 1190 A , B # 0 Ale Beer Corn Hops Malt Profit 6 Brewery Problem: Feasible Region Ale Beer (34, 0) (0, 32) Corn 5A + 15B " 480 Hops 4A + 4B " 160 Malt 35A + 20B " 1190 (12, 28) (26, 14) (0, 0) 7 Brewery Problem: Objective Function 13A + 23B = $800 13A + 23B = $1600 13A + 23B = $442 (34, 0) (0, 32) (12, 28) (26, 14) (0, 0) P r o f i t Ale Beer 8 (34, 0) (0, 32) (12, 28) (0, 0) (26, 14) Brewery Problem: Geometry Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at an extreme point . extreme point
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lp - Linear Programming see ORF 307 Linear Programming What...

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