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Unformatted text preview: IE 495 Lecture 24 November 28, 2000 Reading for This Lecture Primary Â¡ Bazaraa, Sherali, and Sheti, Chapter 2. Â¡ Chvatal, Chapters 6 and 7. Linear Programming Introduction Consider again the system Ax = b, A âˆˆ R m Ã— n , b âˆˆ R m . In this problem, there are either Â¡ no solutions Â¡ one solution Â¡ infinitely many solutions (if n > m) The problem of linear programming is min c T x s.t. Ax = b x â‰¥ 0 Applications of Linear Programming Linear programming is central to much of operations research. Many resource allocation problems can be described as linear programs. Example: The Diet Problem Â¡ We have a set of nutrients with RDAs. Â¡ We have a set of available foods. Â¡ We have preference constraints which limit the intake of particular foods. Â¡ We want to minimize our cost. Convex Sets A set S is convex â‡” x 1 , x 2 âˆˆ S, Î» âˆˆ [0,1] â‡’ Î» x 1 + ( 1 Î»29 x 2 âˆˆ S If x = Î£Î» i x i , where Î» i â‰¥ and Î£Î» i = 1 , then x is a convex combination of the x i 's . If the positivity restriction on Î» is removed, then y is an affine combination of the x i 's ....
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This note was uploaded on 08/06/2008 for the course IE 495 taught by Professor Linderoth during the Fall '08 term at Lehigh University .
 Fall '08
 Linderoth
 Operations Research

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