linear programming · formulation of lpp • first refer to video sent on class group. example 1...
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LinearProgrammingGraphicMethodAcademicYear2019-20B.Com(H)2ndYearSectionAandB
PreparedBy:
HansikaKhuranaAssistantProfessor
DepartmentofCommerceGargiCollege
WhendoweuseLPP?• Businessproblemswheretherearemanyrequirementsbutlimitedresources
• Someexamplesare:a. FormulatingDiets–withoptimumcostwithout
compromisingonminimumnutritionalvalueb. Manufacturing–withoptimumcostwithoutcompromising
onprofitmarginsc. Transportation–tofindtheleastexpensivewayto
transportshipmentsfromoneplacetoanotherd. JobAssignments–topeopleforacertaintimeframe
withoutincurringtoomuchexpensee. Production–tobedoneasperschedule,keepinginmind
thedemandandfluctuationsinprices
BuildingBlocksofanLPPAsetofnon-negativerestrictions
ThismeansthatwhileobtainingtheanswertoanLPP,Iwouldwantpositivefigures,andnotnegativeones.Forinstance,ifwearesolvingforaminimumnumberofchairstobemanufacturedforaresultingprofit,weshouldideallynotbeproducingnegativenumberofchairs.Thenumbercaneitherbe0orpositive.
Asetoflinearconstraints
Theserepresenttherestrictionsorlimitationsofresourcesthatareusedwhilemanufacturing.Forinstance,machinescanworktillacertaincapacity,labourcanworkforasetnumberofhours,rawmaterialcanbeprocuredforafixednumberofunits,etc.
Anobjectivefunction Thisisthefunctionthatwehavetomaximizeorminimize,keepinginmindtheconstraints.Forinstance,weneedtomaximizeprofitfromproductionoftablesandchairs,keepinginmindmachinecapacityandlabouravailability.
BasicAssumptionsofanLPPProportionality ThismeansthatifImultiplyaconstant“K”with2,theresultis2K.
UsingthisinLPP,ifavariableismultipliedwithaconstant,thenthecontribution/profitwillalsogetmultipliedwiththesameconstant.Inreality,thisisnotalwaystrue.WehavestudiedtheLawofDiminishingReturnsinEconomics,whichiscontrarytothisassumption.
Additivity Thismeansthatthetotalvalueofthefunctionisexactlythesumofitsparts.Thismeansthatifafunctionhasthreeparts,a,2band4c,thevalueofthefunctionwillexactlybea+2b+4c.Inreality,thisisalsonotalwaystrue.Forinstance,mixingonecupofmilkwithonecupofwaterwillnotresultinexactlytwocupsofthemixture,butlesser,becausemilkandwaterarepartiallymiscible.
Divisibility Sincewehaveanon-negativeconstraint,thisassumptionmeansthatwecandividethevalueandproduceanyfraction,otherthannegativevalues.Again,inreality,thisisnotdesirable.Forinstance,aftersolvingtheLPP,wegetvalueof“X”as1/3,whereXrepresentsnumberofchairs.Itisnotpossibletoproduce1/3ofachair.
Certainty Thismeansthatalleventsandparametersareknowntouswithcertainty.However,thisisnotalsoalwaystrueinreality,aswearedealingwithproblemsofthefuture,andnotthepast.TheproblemwillalwaysaskyoutodetermineHowmanyofeachshouldbeproducedratherthanHowmanywereproduced.
FormulationofLPP• Firstrefertovideosentonclassgroup.
Example1–Page3.6• Refervideosentonclassgroup
Example2–Page3.7• Tryyourselfusingsamemethod(thisisaminimizationcase)
• Refervideosentonclassgroup
Example3–Page3.8
LPPGraphsRefervideossentonclassgroup
ExceptionstoLPPGraphsFirstrefertovideossentonclassgroup1. InfeasibleSolutionWhenthereisnocommonshadedareai.e.thereisnopointwhichsatisfiesallconstraints,thesolutionissaidtobeinfeasible.
2. UnboundedSolutionThecommonshadedareadoesnothaveend-pointsi.e.itcannotbeenclosedinanyarea,thenitisunbounded.Maybeavalueexistsintheunboundedregion,whicharenotcornerpoints,thatcangiveabettersolution.
3. MultipleOptimalTheremightbetwodifferentequations,whichwhensolved,yieldthesameanswer.Thatis,whenmorethanonesolutionexists4. RedundancyIfweremoveaparticularconstraint(equation)andyettheboundedarearemainsthesame,thenthatconstraintisredundanti.e.itmakesnodifferencetothesolution.Removingthatconstraintwillnotchangethesolution