Transcription of Linear Programming: Theory and Applications
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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].
This paper will cover the main concepts in linear programming, including examples when appropriate. First, in Section 1 we will explore simple prop-erties, basic de nitions and theories of linear programs. In order to illustrate some applicationsof linear programming,we will explain simpli ed \real-world" examples in Section 2.
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