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Linear Programming Graphic Method Academic Year 2019-20 B.Com(H) 2 nd Year Section A and B Prepared By: Hansika Khurana Assistant Professor Department of Commerce Gargi College

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Page 1: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

LinearProgrammingGraphicMethodAcademicYear2019-20B.Com(H)2ndYearSectionAandB

PreparedBy:

HansikaKhuranaAssistantProfessor

DepartmentofCommerceGargiCollege

Page 2: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

WhendoweuseLPP?•  Businessproblemswheretherearemanyrequirementsbutlimitedresources

•  Someexamplesare:a.  FormulatingDiets–withoptimumcostwithout

compromisingonminimumnutritionalvalueb.  Manufacturing–withoptimumcostwithoutcompromising

onprofitmarginsc.  Transportation–tofindtheleastexpensivewayto

transportshipmentsfromoneplacetoanotherd.  JobAssignments–topeopleforacertaintimeframe

withoutincurringtoomuchexpensee.  Production–tobedoneasperschedule,keepinginmind

thedemandandfluctuationsinprices

Page 3: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

BuildingBlocksofanLPPAsetofnon-negativerestrictions

ThismeansthatwhileobtainingtheanswertoanLPP,Iwouldwantpositivefigures,andnotnegativeones.Forinstance,ifwearesolvingforaminimumnumberofchairstobemanufacturedforaresultingprofit,weshouldideallynotbeproducingnegativenumberofchairs.Thenumbercaneitherbe0orpositive.

Asetoflinearconstraints

Theserepresenttherestrictionsorlimitationsofresourcesthatareusedwhilemanufacturing.Forinstance,machinescanworktillacertaincapacity,labourcanworkforasetnumberofhours,rawmaterialcanbeprocuredforafixednumberofunits,etc.

Anobjectivefunction Thisisthefunctionthatwehavetomaximizeorminimize,keepinginmindtheconstraints.Forinstance,weneedtomaximizeprofitfromproductionoftablesandchairs,keepinginmindmachinecapacityandlabouravailability.

Page 4: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

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.

Page 5: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

FormulationofLPP•  Firstrefertovideosentonclassgroup.

Page 6: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

Example1–Page3.6•  Refervideosentonclassgroup

Example2–Page3.7•  Tryyourselfusingsamemethod(thisisaminimizationcase)

Page 7: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

•  Refervideosentonclassgroup

Example3–Page3.8

Page 8: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

LPPGraphsRefervideossentonclassgroup

Page 9: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

ExceptionstoLPPGraphsFirstrefertovideossentonclassgroup1.  InfeasibleSolutionWhenthereisnocommonshadedareai.e.thereisnopointwhichsatisfiesallconstraints,thesolutionissaidtobeinfeasible.

Page 10: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

2.  UnboundedSolutionThecommonshadedareadoesnothaveend-pointsi.e.itcannotbeenclosedinanyarea,thenitisunbounded.Maybeavalueexistsintheunboundedregion,whicharenotcornerpoints,thatcangiveabettersolution.

Page 11: Linear Programming · Formulation of LPP • First refer to video sent on class group. Example 1 – Page 3.6 • Refer video sent on class group Example 2 – Page 3.7 • Try yourself

3.  MultipleOptimalTheremightbetwodifferentequations,whichwhensolved,yieldthesameanswer.Thatis,whenmorethanonesolutionexists4.  RedundancyIfweremoveaparticularconstraint(equation)andyettheboundedarearemainsthesame,thenthatconstraintisredundanti.e.itmakesnodifferencetothesolution.Removingthatconstraintwillnotchangethesolution


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