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7/27/2019 9-Geometry of LP
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DSC1007BusinessAnalytics
TheGeometryofLinearProgramming
Na#onalUniversityofSingapore
Lecture8
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OriginofOptimizationModeling Optimization:What?When?Why? Optimizationprocess:
OptimizationModel:3majorcomponents Objective Decisionvariables Constraints
Standardformulationofoptimizationmodels Examples:
production,transportation,investment
2
PracticalProblem
OptimizationModel
Solution
Recap:LinearOptimization
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Howtovisualizealinearprogramgeometrically? Decisionvariables? Constraints? Objective?
HowtoNindtheoptimalsolutiongeometrically? Anexample:
3
GeometryofLinearProgramming
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Eachdimensionrepresentsadecisionvariableann-dimensionalspace
Eachpointinthespacerepresentsaparticularsolution
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X1
X2
0
GeometryofLPDecisionVariables
(2, 3)3
2
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Constraint:x10 Inequalityconstraintx10deNinesahalfspace
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X1
X2
X1 0
GeometryofLPConstraints
0
x1 = 0
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Constraint:x20 Inequalityconstraintx20deNinesahalfspace
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X1
X2
0
X2 0
GeometryofLPConstraints
x2 = 0
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Constraint:2x1+x23 Inequalityconstraint2x1+x23deNinesahalfspace
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X1
X2
0
2x1 + x23
GeometryofLPConstraints
2x2 +x2 = 3(0, 3)
(1.5, 0)
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Constraint:x1+2x23 Inequalityconstraintx1+2x23deNinesahalfspace
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X1
X2
0
x1 + 2x23
GeometryofLPConstraints
(3, 0)
(0, 1.5)
x2 +2x2 = 3
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TheintersectionofalltheconstraintsFeasibleRegion
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X1
X2
0
GeometryofLPConstraints
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FeasibleRegion:Thesetofalltheallowedsolutions;aregion(polygon)boundedbytheconstraints EachequalityconstraintdeNinesaline EachinequalityconstraintdeNinesahalf-space
ExtremePoints:Cornerpointsontheboundaryofthefeasibleregion.E.g.,(0,0),(1.5,0),(0,1.5),and(1,1)
Infeasibleproblem:Aproblemwithanemptyfeasibleregion Redundantconstraint:Addingorremovingtheconstraintdoes
notaffectthefeasibleregion
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X1
X2
0
GeometryofLPConstraints
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Objective:maximizex1+x2 Isoquant:Alineonwhichallpointshavethesameobjective
value;allpointsareequallygoodontheobjectivefunction.
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X1
X2
0
x1
+ x2
= 0
GeometryofLPObjective
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Isoquant:x1+x2=1
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X1
X2
0
x1
+ x2
= 1
GeometryofLPObjective
(1, 0)
(0, 1)
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Isoquant:x1+x2=1.5
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X1
X2
0
x1
+ x2
= 1.5
GeometryofLPObjective
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Isoquant:x1+x2=3
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X1
X2
0
x1
+ x2
= 3
GeometryofLPObjective
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OptimalSolution:Thebestfeasiblesolution Theorem:ForanyfeasibleLPwithaNiniteoptimalsolution,
thereexistsanoptimalsolutionthatisanextremepoint
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X1
X2
0
Maxx1
+ x2
OptimalSolution:
(1,1)
GeometryofLPOptimalSolution
x1 + x2= 2
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OptimalsolutionsmayOTbeunique
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X1
X2
0
Max 2x1
+ x2
GeometryofLPOptimalSolution
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OptimalsolutionmayOTbeNinite
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X1
X2
0
Maxx1
+ x2
X1 - 2X2 2
-X1 + 2X2 2
GeometryofLPOptimalSolution
(2, 0)
(0, 1)
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Binding(oractive)constraints:TheconstraintsthataresatisNiedatequalityattheoptimalsolution
AllequalityconstraintsarebindingbydeNinition Non-binding(orinactive)constraintsaresatisNiedat
strictinequalityattheoptimalsolution
Theinequalitylevel(=RHSLHS)isknownastheslack BindingconstraintshavezeroslackbydeNinition
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GeometryofLPOptimalSolution
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Activeconstraintsaretheonesthatpassthroughtheoptimalsolution. Inactiveconstraintsaretheonesthatdonotpassthroughtheoptimal
solution.
Whyarebindingconstraintsimportant?
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Constraints Binding? Slack
X1 + 2X2 3 Yes 0
2X1 + X2 3 Yes 0
X1 0 No 1
X2 0 No 1
GeometryofLPOptimalSolution
Optimal
(1, 1)
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Decisionvariables
Eachdecisionvariableformsadimension
ndecisionvariablesdeNineann-dimensionalspace Asolutionisapointinthespace
Constraints EachequalityconstraintdeNinesahyper-plane(lineina2-Dspace) EachinequalityconstraintdeNinesahalf-space AllconstraintscollectivelydeNinethefeasibleregion
Objective TheobjectivefunctiondeNinesisoquantsandadirectioninthespace
ToNindingtheoptimalsolution,pushalongthedirectiondeNinedbytheobjectiveuntilwereachtheboundaryofthefeasibleregion
Attheoptimalsolution,someconstraintsarebinding(oractive)whileothersarenot
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GeometryofLPSummary
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AinNinitenumberoffeasiblesolutions
ANinitenumberofextremepoints Howmany? (+ )
Anextremepointistheoptimalsolution
TheSimplexalgorithmdevelopedbyDantzig
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GeometryofLPBeneNits
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