Transcription of Linear Programming: Chapter 5 Duality - Robert J. Vanderbei
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Linear Programming: Chapter 5 Duality - Robert J. Vanderbei
Linear Programming: Chapter 5 DualityRobert J. VanderbeiOctober 17, 2007 Operations Research and Financial EngineeringPrinceton UniversityPrinceton, NJ 08544 rvdbResource AllocationRecall the resource allocation problem (m= 2,n= 3):maximizec1x1+c2x2+c3x3subject toa11x1+a12x2+a13x3 b1a21x1+a22x2+a23x3 b2x1, x2, x3 0,wherecj=profit per unit of productjproducedbi=units of raw materialion handaij=units raw materialirequired to produce1unit of Up ShopIf we produce one unit less of productj, then we free up: a1junits of raw material1and a2junits of raw these unused raw materials fory1andy2dollars/unityieldsa1jy1+ interested if this exceeds lost profit on each productj:a1jy1+a2jy2 cj,j= 1,2, a buyer offering to purchase our entire to above constraints, buyer wants to minimize cost:minimizeb1y1+b2y2subject toa11y1+a21y2 c1a12y1+a22y2 c2a13y1+a23y2 c3y1, y2 Problem:maximizen j=1cjxjsubject ton j=1aijxj bii= 1,2, .. , mxj 0j= 1,2, .. , n,Has a Dual:minimizem i=1biyisubject tom i=1yiaij cjj= 1,2.
Linear Programming: Chapter 5 Duality Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544
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