运筹学线性规划英文英国研究生用.docx
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运筹学线性规划英文英国研究生用.docx
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运筹学线性规划英文英国研究生用
MSCI535/536:
Optimisation
NotesonLinearProgramming
1.Introduction
LinearProgramming(LP)meansoptimisingagivenfunctionundercertainconstraints.Inthefirstpartofthiscourse,youwilldiscovertheversatilityofLPtechniques,theirlimitationsandtheinformationwhichcanbeobtainedfromaLinearProgrammingsolution.
2.LPExample
AfirmmanufacturestwoproductsP1andP2onthreemachinesM1,M2,M3.P1mustbeprocessedonM1andM2,P2isprocessedonallthreemachines.Themanufacturingtimesfortheproductsaregiveninthefollowingtable:
M1
M2
M3
P1
.25
.4
0
P2
.5
.2
.8
Manufacturingtimesinhours/unitproduct
Eachmachinecanbeusedforatotalof40hours/week.Ifthefirmobtainaprofitof£2/unitforproductP1and£3/unitforproductP2thenhowmuchofeachproductshouldbemadeeachweektomaximiseprofits?
Thisproblemcanbeformulatedinthefollowingway:
Letx1betheno.unitsofP1produced/week.
Letx2betheno.unitsofP2produced/week.
Thenwewishtomaximisetheprofitfunctionf=2x1+3x2
subjecttothemachinecapacityconstraints
.25x1+.5x240(Machine1)
.4x1+.2x240(Machine2)
.8x240(Machine3)
andalsototheconditions
x10,x20.
ThisformulationisanexampleofaLinearProgram(LP).
3.FormalDefinitions
ALinearProgramisaproblemofmaximising(orminimising)alinearfunctionofasetofvariableswhenthesevariablesarerequiredtosatisfyasetoflinearconstraints.
NotethatthewordProgramminginthiscontextsimplymeans'planning'.Thisisnottobeconfusedwiththe'programming'ofcomputers.LinearProgramminginvolvesoptimisingacertainfunction,whereasComputerProgrammingisinstructingacomputertoperformacertaintask.Ofcourse,onemightuseacomputerprogramtosolveanLPproblem.
Byalinearfunctionofasetofvariablesismeantanexpressionwhichcanbewrittenintheformc0+c1x1+c2x2+c3x3+...+cnxnwherec0,c1,c2,...,cnareconstants.
Inthepreviousexamplethelinearfunctionwasf=2x1+3x2.
Theconstantwhichmultipliesavariableiscalledthecoefficientofthevariable.Thusthecoefficientofx1inourexampleis2.
Thisfunctionwhichistobemaximised(orminimised)iscalledtheobjectiveorcostfunction.
Alinearconstraintisonewhichcanbewrittenas:
alinearfunction=constant,
alinearfunctionconstant,
oralinearfunctionconstant.
Inourexampletherewere3linearconstraintsallinvolving"".
Acompletelygenerallinearprogrammingproblemcouldthusbewritteninthefollowingway:
Max(orMin)c1x1+c2x2+...+cnxn
subjecttoa11x1+a12x2+...+a1nxn(R1)b1
..................
..................
am1x1+am2x2+...+amnxn(Rm)bm
wheretherelationships(R1)...(Rm)areeither‘=’,‘’,or‘’.
4.Formulation
Itisimpossibletogiveadetailedprocedureforformulatinglinearprogramswhichwillworkineverycase.However,thefollowingsimplestepscanbehelpfullyborneinmind.
Step1:
Identifythevariables.Thesearesometimesknownas‘activities’andrepresentthingsunderourcontrol.Inourexampleweusedx1andx2torepresenttheamountsofthetwoproductstobeproducedeachweek.Itisimportanttohavedefinedtheunitsofeachvariablecarefully,soyouunderstandpreciselywhatxi=1means.
Step2:
Identifythingswhicharetobeconstrained.Theseareoftencalled‘items’.Inourexamplethesewerethetimesavailableonthemachines.
Step3:
Writedowntheobjectivefunctionintermsofthevariables.
Wehadf=2x1+3x2
Step4:
Writedowntheconstraintswhichlimittheitems.
Wehad.25x1+.5x240
.4x1+.2x240
.8x240.
Step5:
Addinanyadditionalconstraints,suchasthoseinvolvingtheratiosofactivities.
Step6:
Don'tforgetanynon-negativityconstraints!
Wehadx10,x20.
5.ThreeExamplestoTry
1.Ametalproductscompanyproduceswastecans,filingcabinets,correspondenceboxesandlunchboxes.ThemanufactureoftheseproductsrequiressheetMetaloftwodifferentthicknesses,calledAandB,andofcourselabour(measuredinmanhours).Onewastecanrequires.6m2ofA,4manhoursandgenerates£1.98profit.Onefilingcabinetrequires1.0m2ofB,3manhours,andgenerates£44.95profit.Onecorrespondenceboxrequires.2m2ofA,2manhoursandgenerates£9.95profit.Onelunchboxrequires.3m2ofA,3manhours,andgenerates£2.98profit.Thisweektherewillbe22.5m2ofA,30.0m2ofBand190manhoursavailable.
Thecompanyalsohasarollingmachinewhichwillconvert1m2ofBto2.5m2ofAinonemanhouratacostof£1.
FormulatetheproblemofmaximisingprofitfortheweekasanLP.
2.Anamateurdramaticsocietyisputtingonashowinalocalhall.Theymustdecidehowmanyrowsofhardchairsandhowmanyrowsofeasychairstosetout.12hardchairscanbearrangedinonerowor10easychairs.Theyallow3ftforeachrowofhardchairsand3.5ftforeachrowofeasychairsandthehallis84ftlong.Thereare280hardchairsand210easychairsavailable.Theydecidetoprovideatleasttwiceasmanyrowsofhardchairsasrowsofeasychairs,butthehardchairsshouldnotbemorethan90%ofthetotalno.chairs.Theycharge£2forahardchairand£2.60foraneasychair.Theywanttodecidehowmanyrowsofeachtypetosetoutinordertomaximisetheirprofit.
FormulatethisproblemasanLP.
3.Threetypesofcrudepetroleumareavailableasfollows:
OctaneVapourLitresCostper
RatingPressureAvailableLitre
Crude110843200040p
Crude290102000024p
Crude37353800024p
Thesemaybeblendedtoformpremiumorregularpetrol.Premiumhasanoctaneratingofatleast100,avapourpressureofnomorethan6,andcanbesoldataprofit(ignoringthecrudes)of36pperlitre.Regularpetrolhasanoctaneratingofatleast80,avapourpressureof9orless,andcanbesoldataprofitof28pperlitre.
Howshouldthepetrolbeblendedtomaximiseprofits?
6.TheGraphicalMethodforLPswithTwoVariables
LPsinpracticeinvolvemanymorethan2variables.However,itisinstructivetoseehowthisspecialcasemaybesolvedgraphically.Wewillagainusethemanufacturingprobleminsection1whichwasformulatedas:
Maxf=2x1+3x2
subjectto0.25x1+0.5x240
(1)0.4x1+0.2x240
(2)
0.8x240(3)
andx10,x20.
Thesetofpoints(x1,x2)whichsatisfythefirstconstraintisillustratedinthedigramnoverleaf,asallthepointsbelowtheline0.25x1+0.5x2=40:
Constraints
(2)and(3)canbeillustratedinthesameway,aswellasthenon-negativityconditionsonx1andx2.Wethenfindthattheareasatisfyingalltheconstraintsistheoneshadedbelow.Theshadedareaisknownasthefeasibleregion:
Therearetwoimportantpointsworthnotinghere:
(i)Thefeasibleregionisaconvexpolygon,i.e.astraightsidedfigurewithnoindentations.
(ii)Giventhattheobjectivefunctionislinear,thepointrepresentingtheoptimalLPsolutionwillbeatavertex(cornerpoint)ofthepolygon.
Thefirstpointalwaysholdsiftherearetwovariables.(Iftherewerethreevariablesinstead,thenthefeasibleregionwouldbeaconvexpolyhedron.Iftherearefourormoredimensions,thefeasibleregionbecomesimpossibletovisualise,butitisstillcalledapolyhedronintheacademicliterature.)
Wemustnowfindvaluesofx1andx2withinthefeasibleregionwhichmaximisethefunctionf=2x1+3x2.Onewaytodothisistodraw“contourlines”asfollows.Forseveralspecificvaluesofthefunction,f,wedrawalineonthegraphshowingthesetofpoints(x1,x2)whichwillgivethatprofit:
Theselinesareallparallel,andthevalueofthefunctionincreasesasthelinemovesinthedirectionoftheboldarrow.
Thequestionnowbecomes:
‘Howfarcanthelinebemovedinthedirectionofthearrowandstillcontainatleastonepointinsidethefeasibleregion?
’Fromthefollowingdiagram,themaximumvalueoffis280(atpointD),sinceifthedottedlineweretobemovedanyfurtherinthedirectionofthearrowitwouldnolongerincludeapointwithinthefeasibleregion:
Theco-ordinatesofDmaynowbereadfromthegraph.Ifyoudrawthisaccuratelyongraphpaperyoushouldfindthatx1=80,x2=40.So,thefirmshouldproduce80unitsofP1and40unitsofP2,soastomakethemaximumpossibleweeklyprofitof£280.
Toobtainanaccurateanswer,weshouldreallycheckourvaluesmathematically.NoticethatDisthepointattheintersectionoflines
(1)and
(2).HencewemayfindDbysolvingthesimultaneousequations:
0.25x1+0.5x2=40
0.4x1+0.2x2=40.
Itcanbeeasilycheckedthatthisgivesx1=80,x2=40.
Thesameideageneralisestondimensions:
eachvertexofthefeasibleregionwillbedefinedbynsimultaneousequations.
7.TheSimplexMethod-GeometricDescription
ThegraphicalmethodofsolutionofLPsissuitableforproblemsinvolvingtwovariables,butweneedamethodwhichcanbeusedforproblemswithmorethantwovariables.TheSimplexMethodissuchamethod.
Inthetwovariableexample,theSimplexMethodproceedsinthefollowingway:
Startatanycorner(orvertex)ofthefeasibleregionandmovealonganedgetoaneighbouringcornerwheretheobjectivefunctionftakesahighervalue.Continuetomovetoneighbouringcornersuntiloneisreachedwhereitisimpossibletoincreasetheobjectivefunctionfbymovingtoanotherneighbouringcorner.Thisisthenthe
pointrepresentingthefinalsolutiontotheLPproblem.TheSimplexMethodisthusaprocedureformovingroundtheedges
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