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Linear Programming: Theory and Applications

LinearProgramming:TheoryandApplicationsC atherineLewisMay 11, 20081 Contents1 Introductionto a linearprogram?.. LinearProgrammingProblem.. LinearProgramming.. SetsandDirections..82 ..163 nitions..194 AnOutlineof theProof205 ExamplesWithConvex SetsandExtremePoints Precursorsto theSimplexMethod ..237 TheSimplexMethod In Practice258 Whatif thereis noinitialbasisin theSimplextableau? ..319 .. 'sRule.. [2].. Ruleto Use?..3910 Sensitivity .. Analysisfora costcoe cient .. Analysisfora right-hand-sidevalue..4111 CaseStudy:BusingChildrento .. Function.. Together.. Prices..5712 Conclusion5721 Introductionto LinearProgrammingLinearprogrammingwas developedduringWorldWar II, whena systemwithwhich to maximizethee ciencyof resourceswas of \Program-ming"was a militarytermthatreferredto activitiessuch as planningschedulese cientlyor deployingmenoptimally. GeorgeDantzig,a member of ,developedtheSimplexmethod of optimizationin 1947in ordertoprovidean e cient ,expertsfroma variety of elds,especiallymathematicsandeconomics,h ave developedthetheorybehind\linearprogrammi ng"andexploreditsapplications[1].

3 The Theory Behind Linear Programming 17 ... function for a maximization problem max z = min ( z): 1.4 The Linear Algebra of Linear Programming The example of a canonical linear programming problem from the introduction lends itself to a linear algebra-based interpretation. As a reminder, the form of

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