An introduction to Linear Programming - Tyllesentyllesen.dk/onewebmedia/lp.pdf · An introduction...

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AnintroductiontoLinearProgramming

EricBentzen

OperationsManagement

CopenhagenBusinessSchool

August2014

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Content1. Introduction ................................................................................................................................................... 3

2. The process of quantitative analysis ............................................................................................................. 3

2.1 Formulation ........................................................................................................................................... 4

2.2 Solution .................................................................................................................................................. 5

2.3 Interpretation ........................................................................................................................................ 5

3. Practical applications ..................................................................................................................................... 6

4. Linear programming defined ......................................................................................................................... 9

4.1 The managerial perspective ................................................................................................................ 10

5. A simple production problem ...................................................................................................................... 11

5.1 Problem description ............................................................................................................................ 11

5.2 The model ............................................................................................................................................ 13

5.3 Summary .............................................................................................................................................. 13

6. A service problem ........................................................................................................................................ 14

6.1 Problem description ............................................................................................................................ 14

6.2 The model ............................................................................................................................................ 16

6.3 Performing the analysis ....................................................................................................................... 16

6.4 Utilization of resources ........................................................................................................................ 18

6.5 The Managerial Perspective ................................................................................................................ 19

7. Using the computer ..................................................................................................................................... 21


8. The Dual Problem ........................................................................................................................................ 26

8.1 The Model ............................................................................................................................................ 28


9. A linear programming application: Investment .......................................................................................... 31

9.1 The model ............................................................................................................................................ 33

10. Finale note ................................................................................................................................................. 35

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1.IntroductionDecisionmakinginabusinessareveryoftenrestrictedbythelimitationofavailableresourcesandatthesametimeabusinessmanagerhas tomeet specifiedgoals. In thispaperquantitativeanalysis isimportanceforthemanagerialdecisionprocess.Butkeepinmindthatquantitativeanalysiscanneverprovidetheentireanswerforallstrategicdecisions.Useitwithcareanduseitasasystematicwayofworkingthroughacomplexmanagerialdecisionprocess.Decisionmodelingisascientificapproachtomanagerial decisionmakingwherewedevelop amathematicalmodel of a real‐worldproblem.Themodelshouldbesuchthatthedecision‐makingprocessisnotaffectedbypersonalbias,emotionsandguesswork.

2.TheprocessofquantitativeanalysisQuantitativeanalysisisascientificmethodthatcanhelpinthemanagerialdecisionmakingprocess.Thedecisionmodelingprocessinvolvesthreedistinctsteps:

Formulation

Solution

Interpretation

Definingtheproblem

Acquiringthedata

Developingamodel

Developingasolution

Testingthesolution

Analyzingtheresults

Implementingtheresults

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2.1 Formulation

Inthisparteachpartofamanagerialproblemistranslatedandexpressedintermsofamathematicalmodel.Oneverycommonpitfallisthattheproblemcannotbeformulatedbecausetheproblemistoocomplex (then one could break down the problem into smaller pieces), another that technical ormanagerial information are not available (often there has not been formulated an objective of themanagerialproblem).Theaiminformulationistoensurethatthemathematicalmodeladdressesalltherelevant issues to the managerial problem at hand. Formulation of the problem should further beclassifiedintoa)Definingtheproblem,b)Developingamodel,andc)Acquiringthedata

Definingtheproblemisperhapsthemostimportantproblem.Whenthemanagerialproblemisdifficulttoquantify,itsometimesmaybenecessarytodevelopspecific,measurableobjectives.Oneobjectiveinmanagerialeconomicscouldbethatofmaximizingtheprofitandanothercouldbethatofminimizingthecosts.Sometimestherearemultipleconflictinggoals,andifthisisthecase,onecouldsolveitbyminimizingthedistancebetweenthegoals.

Developingthemodelisthestepwheredifferenttypesofmodelscanbespecified.Thesemodelsareexpressedasequationsorinequalitieswithoneormorevariablesandparameters.

Usethefollowingthreestepproceduretodefineandidentify

Decisionvariables,whichrepresenttheunknownentitiesinamanagerialproblem.DecisionvariablescouldbecarsoftypeI(X1)andcarsoftypeII(X2).

Objectivefunction,whichstatesthegoalofthemanagerialproblem,canbequantifiedinafunctionlikeMaxprofit=(50000€percartypeI)*(numberofcarstypeIproduced)+

(60000€percartypeII)*(numberofcarstypeIIproduced)

IfweintroduceDecisionvariableX1=numberofcarstypeIandDecisionvariableX2=numberofcarstypeII,wehavetheobjectivefunction

Max.profit=(50000€percartypeI)*(X1)+

(60000€percartypeII)*(X2)

Othertypesofcommonmanagerialproblemsaremaximizing(profit,efficiency, ...)orminimizing(cost,time,labor,...).

Constraints can be classified into technical and economical constraints. A technicalconstraint couldbe the spaceornumberorworkers that are available.Aneconomicconstraintcouldbetheamountofavailablemoneytoinvestinaproject.

Acquiring inputdata is thestepwhere inputdatatobeused inthemodelareobtained.Thisdetailedaswellastechnicalinformationarecollectedandimplementedintothemodel.Youwillhave to use knowledge from the basis ofmanagerial economics. Obtaining accurate data isessential, improperdatawillresult inmisleadingresults.Forrealworldproblems,collectingaccuratedatacanbeoneofthemostdifficultandchallengingaspectsofdecisionmodeling.

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2.2 SolutionItisthestepwherethemathematicalexpressionsfromyourformulationprocessaresolvedtoidentifyanoptimalsolutionofthemanagerialeconomicsproblem.Becausewecanusesoftwarepackagestofindasolutionourfocushasshiftedawayfromdetailedalgorithm(thesimplexalgorithm)andtowardsthebestofthesepackages.Wedistinguishbetweena)developingasolutionandb)testingthesolution.

Developingasolutionisthestepthatinvolvestheuseofanalgorithmthatconsistsofaseriesofprocedures,andtheaccuracyofthesolutiondependsontheaccuracyoftheinputdataandthemodel.Donotwastetimewithspecializedprogrammingskills.Insteaditismucheasiertouseastandardsoftwareprogramthatisabletohandleandsolvetheproblem.Forsmallproblemswithonlytwodecisionvariablesyoucandoagraphicalanalysis,andforlargerproblemsyouwillhavetousetheanalyticalsolution.

Testingthesolutionmeansthatonehastobesurethatnewdatafromanothersourcebehavesinasimilarwayastheoriginaldata.Onehastocheckdataaswellasthemodelitselftomakesurethattheyreallyrepresentthemanagerialeconomicproblem.

2.3 InterpretationAfterasolutionhasbeenfoundyouhavetotakeacloserlookatsolutionandalldecisionvariables.Themostimportantjobforanymanagerialeconomistswouldbetoaskthefollowingquestion:"whatif".

The"whatif"questionhastobeaskedsothatyoucanhaveanideaofhowsensitivethesolutionis.Wedistinguishbetweena)analyzingtheresultsandb)implementingtheresults.

Analyzingtheresultsbeginswithdeterminingtheimplicationsofthesolution.Itisimportanttoaskthefollowingquestion:“Whatistheinterpretationoftheresults?”.Andnextquestiontoaskis“Whatifwechangeoneormoreof the inputdata?”. This sensitivity analysis is importantbecause it showshowsensitivethesolutionistochanges.

Implementationisthestepthatmostoftenisneglectedinrealworld.Mostpeopleforgetthispartandveryoftenawellspecifiedandpreparedplanaremissingthelastandfinalstep:implementationintheorganization.

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3.Practicalapplications

Anumberofpracticalapplicationshavedocumentedtheuseofthequantitativeprocessaswellastheuseofamorecomplexmathematicalformulation.Inthefollowingwewillonlybelookingatapplicationsthatuseaspecificmathematicalmodelcalledlinearprogramming.Thelinearprogrammingmodelisasimpleandimportantmathematicaloptimizationmodel,anditcansometimesbeappliedtoanumberofeconomicproblems.

Acoupleofoldapplicationsare:

The linear programmingmodel has been used for bank assetmanagement and it has beenimplemented to include comprehensive risk constraints, various policy considerations,economicandinstitutionalrealitiesofthemarketplace,andavarietyofdifferentdynamiceffectswhichmustbeconsideredinordertomakeoptimalassetmanagementdecisions.Conceptualproblemsconcerningtheformulationofthebank'sgoalswereconsideredaswell.1

A linear programming formulation of Shell's distribution network between four sources ofproductandalargenumberoftransshipmentpointsandterminalshasbeenimplemented,withemphasisoncostlogisticsandrankingofvariousalternatives.2

Critical to an airline's operation is the effective use of its reservations inventory. AmericanAirlinesbeganresearchintheearly1960sinmanagingrevenuefromthisinventory.Becauseofthe problem's size and difficulty, American Airlines Decision Technologies has developed aseriesofORmodelsthateffectivelyreducethelargeproblemtothreemuchsmallerandfarmoremanageable sub problems: overbooking, discount allocation, and traffic management. Theresults of the sub problem solutions are combined to determine the final inventory levels.AmericanAirlinesestimatesthequantifiablebenefitat$1.4billionoverthelastthreeyearsandexpectsanannualrevenuecontributionofover$500milliontocontinueintothefuture.3

Alinearprogrammingmodelwithseveralobjectiveswasdevelopedinchoosingmediaplans.4

1CohenandHammer(1967):"LinearProgrammingandOptimalBankAssetManagementDecisions".JournalofFinance,147‐165.

2 Zierer, Mitchell, and White (1976): "Practical applications of linear programming to Shell's distributionproblems".Interfaces,13‐26.

3Smith.,Leimkuhler,andDarrow(1992):"YieldManagementatAmericanAirlines".Interfaces,8‐31.

4 Charnes, Cooper, Devoe, Learner, and Reinecke (1968): "A Goal Programming Model for Media Planning".ManagementScience,14,B423‐B430.

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Andacoupleofrecentapplicationsare:

JandeWitCompanyimplementedadecision‐supportsystembasedonlinearprogrammingasaproduction‐planning and trade tool for the management of its lily flower business. The LPmaximizesthefarm'stotalcontributionmargin,subjecttosuchconstraintsasmarketdefinedsales limits,market requirements, characteristics of the production cycle duration, technicalrequirements, bulb inventory, and greenhouse limitations. Themain decision variable to becalculatedisthenumberofflowerbedsinaspecificgreenhouse,fromaspecificbulbbatch,ofaspecific variety, for a specific purpose, taking into consideration planting and expectedharvestingweeks.Between1999and2000,companyrevenuegrew26percent,salesincreased14.8percentforpotsofliliesand29.3percentforbunchesoflilies,costsfellfrom87.9to84.7percentofsales,incomefromoperationsincreased60percent,returnonowner'sequitywentfrom 15.1 to 22.5 percent, and best quality cut lilies jumped from 11 to 61 percent of thequantitiessold.5

U.S.CoastGuardusedlinearprogramminginanextensivepreventativemaintenanceprogramfor its SikorskyHH60Jhelicoptersbasedonhelicopter flight time.Themodelmust considerdifferentmaintenancetypes,maintenancecapacityandvariousoperationalrequirements.6

RecentreformofEuropeanagriculturalpolicyhasresultedinsubstantialchangestothecriteriabywhichpremiapaymentsaremade.Beeffarmers,whohavebeenparticularlydependentonpremia payments to maintain margins, must re‐evaluate their systems to identify optimalsystems in thesenewcircumstances.A linearprogrammingmodelhasbeenused to identifyoptimalbeefproductionsystemsinIreland.Theobjectivefunctionmaximizesfarmgrossmarginandthemodelisprimarilyconstrainedbyanimalnutritionalrequirements.7

SchedulingtheItalianMajorFootballLeague(theso‐called"SerieA")consistsinfindingforthatleague a double round robin tournament schedule that takes into account both typicalrequirements such as conditionsonhome‐awaymatches and specific requests of the ItalianFootballAssociationsuchastwin‐schedulesforteamsbelongingtothesamehome‐town.Themodel takes into account specific cable television companies requirements and satisfyingvariousotheroperationalconstraintswhileminimizingthetotalnumberofviolationsonthehome‐awaymatchesconditions.8

AlinearprogrammingmodelwasimplementedusingActivityBasedCostingforcalculatingunitproduct cost, and dynamic Activity Based Management for assessing the feasibility ofprospective production plans. The model is used to optimize the business plan at a steelmanufacturer.9

5Filho,José,Neto,andMaarten(2002):"OptimizationoftheProductionPlanningandTradeofLilyFlowersatJandeWitCompany".Interfaces,35‐46.

6HahnandNewmanb(2008):"SchedulingUnitedStatesCoastGuardhelicopterdeploymentandmaintenanceatClearwaterAirStation,Florida".Computers&OperationsResearch35(2008)1829—1843.

7Crosson,O'Kiely,O'MaraandWallace(2006):"ThedevelopmentofamathematicalmodeltoinvestigateIrishbeefproductionsystems".AgriculturalSystems,349‐370.

8 Croce and Oliveri (2006): "Scheduling the Italian Football League: an ILP‐based approach". Computers &OperationsResearch,1963‐1974.

9SingerandDonoso(2006):"Strategicdecision‐makingatasteelmanufacturerassistedbylinearprogramming".JournalofBusinessResearch,387‐390.

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Minnesota's Nutrition Coordination Center used linear programming to estimate content ofcommercialfoodproducts.10

LinearProgrammingwasusedtodecidethedailyroutesofloggingtrucksinforestry.Aspectssuchaspickupanddeliverywithsplitpickups,multipleproducts,timewindows,severaltimeperiods,multipledepots,driverchangesandaheterogeneoustruckfleet.11

Airlineseatisthemostperishablecommodityintheworld.Eachtimeanairlinertakesoffwithanemptyseat,arevenueopportunityislost.DeltaAirLinesusedaLPmodelwithmorethan40,000constraintsand60,000variablestosolvethisemptyseatproblem.Theysavedmorethan$220,000perday.12

AllocationoftraincapacityamongmultipletravelsegmentsonanIndianRailwaystrainroutewithseveralstops.Duetohistoricalandsocialreasons,IndianRailwayssplitsitstraincapacitybased on user and type of travel. The determination of the optimal split of such capacity isnontrivial.TheirLPmodelwasapplied17IndianRailwaystrains,andtheyincreasedrevenuefrom2.6to29.3percentinrevenue,6.7to30.8percentinloadfactors,and8.4to29percentinpassengerscarried.13

Intheshippingandtransportationindustry,thereareseveraltypesofstandardcontainerswithdifferent dimensions and different associated costs. Investigation of the multiple containerloadingcostminimizationproblem,wheretheobjectiveistoloadproductsofvarioustypesintocontainersofvarioussizessoastominimizethetotalcost.14

This paper describes an integer linearprogrammingmodel conceived as an alternative to atraditionalmaterialrequirementsplanning(MRP)systemforextendingtheconceptofsupplychainsynchronisationupstreaminamulti‐tiersupplychain.Inthismodel,weassumethereisan incumbent application for transmitting original equipment manufacturer (OEM)requirementstofirst‐,second‐andthird‐tiersuppliers.15

10Westrich,Altmann,andPotthoff(1988):"Minnesota'sNutritionCoordinatingCenterUsesMathematicalOptimizationtoEstimateFoodNutrientValues".Interfaces,86‐99.

11Flisberg,P.,B.Lidén,M.Rönnqvist(2009):"Ahybridmethodbasedonlinearprogrammingandtabusearchforroutingofloggingtrucks".Computers&OperationsResearch,1122‐1144.

12Subramanian,R.(1994):“Coldstart:FleetAssignmentatDeltaAirLines”.Interfaces,104‐120.

13GopalakrishnanandNarayan(2010):“CapacityManagementonLong‐DistancePassengerTrainsofIndianRailways”.Interfaces,291‐302.

14ChanHouChea,WeiliHuanga,AndrewLima,WenbinZhu(2011):“Themultiplecontainerloadingcostminimizationproblem”.EuropeanJournalofOperationalResearch,501‐511.15 Mula,J.,A.C.Lyons,J.E.Hernández,R.Poler(2014):“Anintegerlinearprogrammingmodeltosupportcustomer‐drivenmaterialplanninginsynchronised,multi‐tiersupplychains”.InternationalJournalofProductionResearch,

14,4267‐4278.

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Thesepracticalapplicationsbelongstoaclassofbusinessproblemsclassifiedasallocationproblemsandundercertainconditionsasolutioncanbeachievedbythelinearprogrammingmodel.

4.Linearprogrammingdefined16We define a linear programming problem as an allocation problemwherein the values of decisionvariablesmustbedeterminedtomeetagoal,underasetoflimitationsbasedonavailableresources.

Linearprogrammingmodels

A linear programming problemmay be defined as the problemofmaximizing(orminimizing)alinearobjectivefunctionsubjecttoasetoflinearconstraints.

Theconstraintsmaybeequalitiesorinequalities.

ALinearProgrammingmodelisbasedonthefollowingproperties:

1.Proportionality Thismeansthatthecontributionofeachdecisionvariabletothevalueofthe objective function and the left hand side of constraints are directlyproportional to the level of thedecision variable. In otherwordswe aretalkingaboutconstantreturnstoscale.

2.Nonnegativity Decisionvariablesarenotallowedtobenegative.Thismeansthatsolutionsvariables of say costs cannot be negative, or the number of workersallocatedtoajobcannotbenegative.

3.Additivity Theobjectivefunctionorthefunctionofthelefthandsideofafunctionalconstraintisthesumoftheindividualcontributionsofeachvariable.

4.Divisibility Decisionvariablesareallowedtofractionalvaluesequaltoorabovezero.Fractionalvaluesforthedecisionvariables,suchas1.25areallowed.

5.Certainty Thisassumptionasserts that theobjectiveandconstraintscoefficientsofthe LP model are deterministic. This means that they are known fixedconstants.

16WeusetheDantzig(1963)specification.Dantzig,G.B.(1963):LinearProgrammingandExtensions.PrincetonUniversityPress.

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4.1 Themanagerialperspective

It is important torealize that linearprogramming isnotapanacea. Instead linearprogramming isamathematicaltoolthatsometimesapproximatesamanagerialproblemquitewell.

Ifwetakeacloserlookattheassumptionsunderlyingthelinearprogrammingmodelwecanveryeasilyspecifyapplicationswheretheassumptionsarenotmeet.

Insteadofhavingconstantreturnstoscalewecouldhaveincreasingordecreasingreturnstoscale.17

Ifweareunsureoftheexactnumberofresourcesthatareneededinamanufacturingproblemthenthecertaintyassumptionwouldbeviolated.

Data that are used in a linear programming model are uncertain because they cannot bemeasuredpreciselyandbecausetheycanfluctuateindifferentunpredictableways.Justthinkofmachinebreakdowns,absenceofworkersorpowerfailures,etc.Thecertaintyassumption‐arare occurrence in real life, where data are more likely to be presented by probabilisticdistributions. If thestandarddeviationsof thesedistributionsaresufficientlysmall, then theapproximationisacceptable.18

Theprofitorthecostapproximatesanuncertainamount.Theactualvaluedependsoncurrentpriceofrawmaterials,defectsduringmanufacturingorchangingininventorycosts,etc.

Tosumuponeshouldhave inmind thatuncertainty in thedata isonereasonwhymodelsare justapproximate.

17Decreasingreturnstoscalecanbetreatedbyintroducinganewdecisionvariablethatcoversthedecreasingreturnstoscale.Inthiscasewewouldhavetwodecisionvariablesonethattakecareofconstantreturnstoscaleandanothertotakeofdecreasingreturnstoscale.Decreasingandincreasingreturnstoscalecanalsobesolvedbynonlinearprogramming.

18Largestandarddeviationscanbeaccountedbyapplyingsensitivityanalysistotheoptimalsolution.

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5.Asimpleproductionproblem

5.1 ProblemdescriptionA largemanufacturingcompany that specializes inpersonal computersmustdecidewhat typesandquantitiesofoutputtomanufactureforeachday'sproduction.Wejustlookattwokindsofpersonalcomputers‐thesmallandthebig‐fromwhichwemayselecttheproductmix.

Themanufactoryhasthefollowingfreeresourcesavailablefortheproduction:

numberoflaborhours600

numberofprocessinghours800

Frompastexperiencetheycansell

nomorethan100bigcomputersperday.

Theresourcesavailablearerelatedtothetwoalternativeoutputsasfollows.Eachunitofsmallpersonalcomputersproducedrequire2.5hoursoflaborand4.5hoursofprocessing,whileeachunitofthebigpersonalcomputersproducedrequires3.5hoursoflaborand5.5hoursofprocessing.

Theprofitperunitofsmallpersonalcomputersmanufacturedis€15,whiletheprofitperunitofthebigpersonalcomputersmanufacturedis€28.

Howmanyofeachtypeofpersonalcomputersshouldthemanufactoryproducetomaximizeprofit?

Decisionvariables

Weintroducetwodecisionvariables.Letourdecisionvariablesbe

X1=numberofsmallpersonalcomputersmanufactured

X2=numberofbigpersonalcomputersmanufactured

Objectivefunction

Theprofit per day is equal to Z the profit from theproduction of each type of personal computersmanufactured.Withaprofitperunitof€15and€28wehavethetotalprofitequalto:

15X1astheprofitfromproductionofsmallpersonalcomputersand28X2astheprofitfromproductionofbigpersonalcomputers.

Themanufactoryhasastheirobjectivetomaximizethetotalprofitfromproducingpersonalcomputerswhichmeansthatourobjectivefunctionisequalto:

MaxZ=15X1+28X2

OnefirstandfeasiblesolutionisnottoproduceanythingthatisX1=0andX2=0whichyieldsatotalprofitofzero.ButanypositivesolutionsofX1andX2givemoreproductionandgreaterprofit.IfX1=2andX2=3

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then theprofit is equal to Z=2(15) +3(28) =€114.Unfortunately themaximization of the objectivefunctionhastomeetacoupleofconstraints,andwehavetotaketheseconstraintsintoconsiderationinordertofindanoptimalsolution.

NegativevaluesofX1andX2makenomanagerialsense.

Constraints

Fromtheproblemwehavethreerestrictions:

Laborhours:600hours

Processinghours:800hours

Salesrestriction:nomorethan100bigcomputers

Weformulatethethreeconstraintsas:

Laborhours

Thenumberofhourstoproduce1smallcomputeris2.5hoursandtoproduce1bigcomputeris4.5hours.TotalhoursrequiredtoproduceX1andX2computersis2.5X1+4.5X2.

With600hoursavailabletoproduceX1andX2weformulatethelaborconstraintas:

2.5X1+4.5X2≤600

Processinghours

Numberofprocessinghourstohandle1smallcomputeris3.5hoursandtohandle1bigcomputeris5.5hours.Thetotalprocessinghoursrequiredis3.5X1+5.5X2.

With800hoursavailabletohandleX1andX2weformulatetheprocessingconstraintas:

3.5X1+5.5X2≤800

Salesrestriction

Only100bigcomputerscanbesold.Thelimitationonsalescanbeformulatedas:

1X2≤100

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5.2 ThemodelWesummarizetheproblemasfollows.WeintroduceX1unitsofsmallcomputersandX2unitsofbigcomputerssuchthatthetotalprofitZismaximized19withrespecttoavailableresources:

MaximizeZ= 15x₁+28x₂

Subjecttoconstraints

Labor 2.5x₁+4.5x₂ ≤600

Processing 3.5x₁+5.5x₂ ≤800

Sales 1x₂ ≤100

and

X₁,X₂≥0

5.3 SummaryTheproductionexampleaboveisanoptimizationproblemwherewemaximizealinearfunctionofthedecisionvariablessubjecttoasetofconstraints.Eachconstraintmustbea linearequationor linearinequality.We have formulated the production example as an allocation problem in which limitedresourcesareallocatedtoanumberofeconomicactivities.AgeneralLPmodelcanbewrittenasfollows:

MaximizeZ= c₁x₁+c₂x₂+...+cnxn

Subjectto

a₁₁x₁+a₁₂x₂+...+a1nxn≤b₁

a₂₁x₁+a₂₂x₂+...+a2nxn≤b₂

...

am1xn+am2x2+...+amnxn≤bm

x1,x2,…,xn≥0

In thegeneralLPmodelwehavemactivitieswhose levels are representedbyndecisionvariables,x₁,x₂,...,xn.

Eachunitofactivityjuseanamountaijofresourcei,andthemresourcesaregivenbyb₁,b₂,...,bm.

19Theobjectivefunctionasspecifiedmaximizestheprofit.Rememberthatprofitisequaltoincomeminusexpenditure.Ifourobjectivefunctionwasspecifiedasminimizationofcostswecoulddothateasilyjustbymultiplyingourobjectivefunctionwithminusone.Hereweusethefactthatcostisequaltoexpenditureminusincome.

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6.Aserviceproblem

6.1 ProblemdescriptionAninsurancecompanyis introducingtwonewproduct lines:normalriskinsurancemortgageAandexpanded insurance mortgage B. The expected profit is 35€ per unit on mortgage A and 50€ onmortgageB.

Managerofthemortgagedepartmenthasestablishedsalesquotasforthenewproductlinestomaximizetotalexpectedprofit.

Furtherinformationis:

Mortgage Work‐hoursperunit

Department

Underwriting

Administration

Claims

Normalrisk

3

1

4

Expandedrisk

5

2

3

Work‐hoursavailable

1,300

500

900

Decisionvariables

Weintroduce2variables.Let

x₁=numberofnormalriskinsurancemortgageA

x₂=numberofexpandedinsurancemortgageB

Objectivefunction

Theinsurancecompanyseekstomaximizethetotalprofit.Alinearobjectivefunctioncanbespecifiedwhichmaximizethetotalprofit.Wehave

MaxZ=profitperunitfortotalinsurance

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TheprofitfromeachmortgageAwas35€wecanformulatethisas35x₁astheprofitfrommortgageAandtheprofitfromeachmortgageBwas50€whichgivesusaprofitfrommortgageBequalto50x₂.Thuswecanformulatethetotalprofitas

Z=35x₁+50x₂€

Theprofitfromlet’ssay5mortgageAand3mortgageBwouldgiveusatotalprofitequalto

Z=5(35)+3(50)=325€.

Constraints

Fromtheproblemspecificationwehavethreerestrictionsonthenumberofworkinghoursavailable.

Underwriting:1,300hours

Administration:500hours

Claims:900hours

Weformulatethethreeconstraintsas:

Underwriting

Thenumberofhoursrequiredtohandle1unitofmortgageAis3hoursandtohandle1unitofmortgageBis5hours.Thusthetotalhoursrequiredtohandlex₁andx₂mortgagesis3x₁+5x₂

With1,300hoursavailabletohandlex₁andx₂weformulatetheUnderwritingconstraintas:

3x₁+5x₂≤1,300

Administration

Numberofhoursrequiredtohandle1unitofmortgageAis1hourand1unitofmortgageBis2hours.Thetotalhoursrequiredis1x₁+2x₂.

With500hoursavailabletohandlex₁andx₂weformulatetheAdministrationconstraintas:

1x₁+2x₂≤500

Claims

Numberofhoursrequiredtohandle1unitofmortgageAis4hoursand1unitofmortgageBis3hours.Thetotalhoursrequiredis4x₁+3x₂.

With900hoursavailabletohandlex₁andx₂weformulatetheClaimconstraintas:

4x₁+3x₂≤900

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6.2 ThemodelWesummarizetheinsuranceproblemasfollows.Weintroducex₁unitsofmortgageAandx₂unitsofmortgageBsuchthatthetotalprofitZismaximizedwithrespecttoavailableresources:

MaximizeZ= 35x₁+50x₂


Underwriting 3x₁+5x₂≤1,300

Administration 1x₁+2x₂≤500

Claim 4x₁+3x₂≤900

and

X₁,X₂≥0

6.3 PerformingtheanalysisGraphicalSolution

ThegraphicalsolutiontoLP‐problemscanonlybeshownwithtwovariables.Webeginwithamapthatexpressesonlynonnegativeconstraints.

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We continuewith the three constraints and graph them each by each, and endswith the objectivefunction.Thefinalgraphicalsolutionhasafeasibleregion.Wedefinethefeasibleregionfortheproblemboundedby the three constraints and the assumptionsof nonnegativity. Pointswhere twoormoreconstraintsintersectarecalledcornerpoints(orcornersolution).

Ifwetakealookatthefourcornersolutionswehave:

0: (x1=0,x2=0) →Z=0

1: (x₁=225,x₂=0) →Z=7,875

2: (x₁=60,x₂=220)→Z=13,100

3: (x₁=0,x₂=250) →Z=12,500

Fromthisweseethatthesolutionthathasitsmaximumatcornersolution2wherex₁=60andx₂=220whichgivesasolutionequaltoZ=13,100.AtthispointthevalueofZismaximizedandtheotherthreecornersolutionshaveaZvaluelessthancornersolution2.

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6.4 UtilizationofresourcesAttheoptimalsolutionjustfoundwehadx₁=60andx₂=220.Atthatcornersolutionwecanseethatnotallavailableresourcesareused.

TheUnderwritingconstraint3x₁+5x₂≤1,300

utilize3(60)+5(220)=1,280<1,300

Ifweaddanextravariable,s₁,totheconstraintwecanwritetheconstraintasanequalityas

3x₁+5x₂+1s₁=1,300

Thedifferencebetweentheresourcesactuallyusedandthoseavailableequals

s₁=1,300‐1,280=20

Thismeansthedepartmenthas20hoursavailablewhichhasnotbeenusedforthepurpose,buttheycanbereallocatedbymanagementforotherpurposes‐ifpossible.

The20hoursavailablearecalledslackorunusedhoursandwenamevariables₁asslackvariablenumber1.Becausethevalueofslackvariable1isnotequaltozerowesaythattheconstraintisnotbinding.

TheAdministrationconstraint1x₁+2x₂≤500

utilize1(60)+2(220)=500

If we add an extra variable, slackvariable number s₂ , to the constraint we can write theconstraintasanequality1x₁+2x₂+1s₂=500

Thedifferencebetweenresourcesusedandthoseavailableequalsslackvariablenumber2

s₂=500‐500=0.

Thismeans the department has used all available resources allocated at the Administrationdepartment,andwesaythisconstraintisbindingandtheslackorunusedhoursareequaltozero.

Ifaconstraintisbindingthereisnoslack.

ThefinalClaimconstraint4x₁+3x₂≤900

utilize4(60)+3(220)=900

Ifwe add an extra variable, slackvariable number 3 s₃ , to the constraintwe canwrite theconstraintasanequality4x₁+3x₂+1s₃=900

Thedifferencebetweenresourcesusedandthoseavailableequalss₃=900‐900=0.

ThismeansthedepartmenthasusedallavailableresourcesallocatedforClaims.Thisconstraintisbindingandtheslackisequaltozero.

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Wehave introduced an important part in linear programming and that is the introduction of slackvariables.Aswehaveseenaslackvariableisequaltounusedamountofaresource.

SlackVariable

Aslackvariablecontainsthedifferencebetween

theresourcesactuallyusedandthoseavailable.

6.5 TheManagerialPerspectiveFromtheabovesolutionandconclusionwithrespecttousedorunusedresourcesfurthershouldbementioned.Whenallresourcesareusedonecouldaskwhatifwecouldhavemoreresourcesallocated,and ifwe could choose freelybetween the twobinding constraints (Administration andClaim) andwhichofthetwoshouldbethefirsttohavemoreresourcesallocated.Thebestwaytoallocatemoreresourceswould be to the department that utilizes the resources the bestway. From an economicperspectivethebestwayresourcesareallocatedarewithaneyetowhatthealternativesare.Iftheonlyalternativesarebetween the twobindingconstraints then theresources shouldbeallocated to thatconstraintthatincreasethevalueoftheobjectivefunctionmost.

IfoneextraresourceisallocatedtoUnderwritingwewouldhave1,301hoursavailable.Wehavefoundthattheslackvariables₁=20,andifweallocateonehourmoretoUnderwritingwewouldaddextratotheslackvariables₁=20+1=21.Weintroduceanewvariable"shadowprice"andtheshadowpriceatUnderwritingequaltoλ₁=0.

IfoneextraresourceisallocatedtoAdministrationwenowhave501hoursavailable.Ourslackvariablewasequal tos₂=0andwithoneextrahourwehave501hoursavailableandweendupwithanewoptimalsolutionwherex₁=59.4andx₂=220.8andanewobjectivefunctionZ=13,119.Theincreaseintheobjectivefunctionisequalto19,andwehaveashadowpriceofλ₂=19.

IfoneextraresourceisallocatedtoClaimwenowhave901hoursavailable.With901hoursavailablewe end upwith another optimal solutionwith x₁=60.4 and x₂=219.8 and a new objective functionZ=13,104.Theincreaseintheobjectivefunctionisequalto4,whichisequaltoourshadowprice,thatisλ₃=4.

ThebestwayoneextrahourcanbeusedshouldbeattheAdministrationdepartment.TheshadowpriceattheAdministrationequaltoλ₂=19andthereforeextrahoursallocatedtoAdministrationdepartmentincreases theobjective functionmorecomparedwiththeClaimdepartmentwhichonly increasetheobjectivefunctionwithλ₃=4.

20

Weintroduceanewvariable"unitworthofaresource"whichwewillcallaShadowPrice.

ShadowPrice

Ashadowpricecontainstheamounttheoptimal

objectivefunctionvaluechangesperunitincrease

intherighthandsidevalueoftheconstraint.

Wehavedemonstratedtheconnectionbetweenslackvariablesandshadowpricesandwesummarizetheresults:

Constraint Slack Shadowprice

Underwriting

Administration

Claim

20

0

0

0

19

4

21

7.UsingthecomputerWecansolvetheaboveLP‐problemusingthecomputer.InthefollowingweuseaLPprogramthatisavailableatCopenhagenBusinessSchool.Weproceedasfollows.

Afterinstallationoftheprogram20youstarttheprogramandreceivethefollowingstartwindow.

AsyoucanseeyouhavetheoptionofchoosingbetweenaDanishorEnglishversion.WecontinuewiththeEnglishversionandwebeginwithformulatinganewmodel.Youpushthestartbottom,andgetthefollowingpicture.

20AvailableviaLearnatCBS.

22

PushingtheNewmodelbottomweget

Weformulateanewmodelandcallit"Myownmodel".

Wemaximizetheobjectivefunctionandwehave2variablesand3constraints.

23

Afterpushing theOKbottomwecan fill in all the coefficients from theLP‐problem, andweget thefollowingpicture.

PushingtheOKbottomwehavethepossibilitytohavesolutionseitherasAnalyticalorGraphical.Wechoosetheanalyticalsolution.

Theanalyticalsolutionhastheoptimalvalueswefoundinthegraphicalsolution,butwehavefurthernumbersthatrequireafewwords.

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OPTIMAL SOLUTION OBJECTIVE FUNCTION 13100.0000 PARTIAL SENSITIVITY ANALYSIS SOLUTION REDUCED OBJECTIVE FUNCTION RANGES VARIABLE VALUE COST LOWER GIVEN UPPER __________________________________________________________________________________ Mortgage A 60.0000 0.0000 25.0000 35.0000 66.6667 Mortgage B 220.0000 0.0000 26.2500 50.0000 70.0000 SHADOW SLACK RIGHT HAND SIDE RANGES CONSTRAINT TYPE PRICE LOWER GIVEN UPPER __________________________________________________________________________________ Underwriting ≤ 0.0000 20.0000 1280.0000 1300.0000 +INFINITY Administration ≤ 19.0000 0.0000 225.0000 500.0000 509.0909 Claim ≤ 4.0000 0.0000 750.0000 900.0000 1000.0000

TheoptimalsolutionZequals13,100

Thetwodecisionvaluesarex1=60andx2=220.

Theobjectivefunctionforeachdecisionvariablealsocomputesranges.Thecoefficientvalueofx1=35has"lower”range=25and"upper“range=66.6667.Theserangestellusthatthesolutionofthedecisionvariablex1=60willremainthesamewithinthe"lower"and"upper"ranges.Ifwechangethegivencoefficientfrom35to36thesolutionwillremainthesamex₁=60,butwehaveanewoptimalobjectivefunctionthatisincreasedwith60toanewvalueequalto13,160.Wehave"lower"and"upper"rangesforx₂equalto(26.25;70).Ifthecoefficientofx₂ischangedfrom50to51westillhavetheoptimalsolutionofx₂=220.Wecanchangethecoefficientfrom50upto51andthevalueoftheobjectivefunctionincreaseswith220.

TheRighthandsideofthethreeconstraintshastheirownranges.Ifwelookatconstraint2(theAdministrationconstraint)wehavea"Shadowprice"=19.Thisshadowpricewill remainthesameintheintervaloftheRighthandsidefrom(225;509).IfwechangetheRighthandsidefrom 500 to 501 the value of the objective function will change from 13,100 to13,100+19=13,119.IfwechangeConstraint1from1,300to1,301weseewithaShadowprice=0thatwedonotgetabettersolution.Theobjectivefunctionremainsthesame.

TheReducedcost21columnshowsthevalueofvariableswhenthesevariableshavesolutionsequaltozero.

21Reducedcost=[(changeinoptimalobjectivefunctionvalue)/(unitincreaseofvariable=0)]

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Reducedcost

Thereducedcostofanunusedactivityistheamountby which profits will decrease if one unit of thisactivityisforcedintothesolution.

Obviously, a variable that already appears in theoptimalsolutionwillhaveazeroreducedcost.

Sometimestheobjectivefunctionisparalleltooneofthebindingconstraintsandwewillhavewhatiscalled alternative optimum. Alternative optimum can be important because it gives managementanothersolutionwiththesamefinalvalueof theobjective function. If theprofit frommortgageB ischangedfrom50€to70€wehaveanoptimalsolutionwithmortgageA=0andmortgageB=250.ButtheobjectivefunctionisparalleltotheAdministrationconstraintintherangefromA=0andA=60.Thismeans management can include mortgage A in their product portfolio, and instead of having oneproducttooffertheycanhavetwoproducts.

7.1 TheManagerialPerspectiveLinearprogrammingisamathematicaltoolwhichsometimesfitorapproximatesamanagerialsituation.Justlikeahammer.Ahammercanbeusedtohammernails,butitcanalsobeusedtohammerscrews,holes,andbolts.Itisobviousthattheabovetasksaremoreefficientlydonebyascrewdriver,adrill,butsometimesthereisnoappropriatetoolexceptfromthehammer.Inthiscasethehammeristhebestavailable tool, and that is the same with linear programming. Linear programming is the bestmathematicaltooltodescribemanymanagerialsituations.

Amanagershouldnotgointoallthetechnicaldetailsofhowlinearprogrammingmodelsaresolved.Insteadamanagercanhelpwiththeformulationofeconomiccoherenceandlatertranslatetheresultsinamanagerialcontentandtakeadecision.

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8.TheDualProblemLinear programming models have brought to economists the meaning of duality22. We know thateconomistsareinterestedinproductionandcost,pricesandquantities,but,dualitypossessesameaningthat transcends linearprogrammingand itseconomic interpretation.Mostproblemshavea twofoldrepresentationandwecallthistheduality.

Insteadoflookingatthemaximizationofnetprofitweturnourfocustowardstheresourcesusedandwe now look at theminimization of the availableresources.

Intheinsuranceexamplethecompanyhadanumberofavailableworkinghours,butwedidnotuseallthehours.Wefoundthat20hourswasnotusedintheUnderwritingdepartment.Thequestioniswhattheeconomicvalueassociatedwiththelasthourisworth?Thelessweuseofavailableworkinghoursthebetter.Anyfreeworkinghourcanbeusedforanotherpurpose.

Ifwetakeacloserlookattheworkinghoursandweconnectthesetothedepartmentswecanswitchtoanotherproblemwhereinsteadweminimizetheuseofavailableresources.Intheinsurancecaseouroriginal problem was the maximization of the profit per unit for total insurance. The model wasformulatedwithrespecttoavailableworkinghoursatthreedepartments.

We could alternatively formulate the same problem by focusing on the minimization of availableworkinghourswithrespecttohavingtheprofitperunitfortotalinsuranceaslargeaspossible.

Weformulatethedualproblemasfollows.

Wehave1,300Underwritinghours,500Administrationhours,and900Claimshours.Thisinformationcannowbeusedtoformulateourobjectiveistominimizetheuseofavailableworkinghours.

Decisionvariables

Weintroducethreedecisionvariables.Let

λ₁=numberofhoursatUnderwritingdepartment

λ₂=numberofhoursatAdministrationdepartment

λ₃=numberofhoursatClaimsdepartment

Objectivefunction

22 ThewordcanbefoundintheworksbyEulerandLegendrein1750wheretheywereinterestedinmethodsforsolvingdifferentialequations.LaterBoole(1859)wrote:“Thereexistsinpartialdifferentialequationsaremarkableduality,invirtueofwhicheachequationstandsconnectedwithsomeotherequationofthesameorderbyrelationsofaperfectlyreciprocalcharacter”.Boole,G.B.ATreatiseOnDifferentialEquations.NewYork,ChelseaPub.Co.,5th,1859.

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Ourobjectiveistominimizethenumberofworkinghoursusedatthethreedepartments.Wespecifyourlinearobjectivefunctionby

MinimizeZ'=1,300λ₁+500λ₂+900λ₃hours

Constraints

Inouroriginalproblem(theprimalproblem)wemaximizedthetotalprofit.Inthedualproblemwenowminimizetheavailableworkinghourswithrespecttohavingtheprofitfromthetwomortgagesaslargeaspossible.

Fromtheinsurancecasethemanagerhasestablishedquotasforthenewproductlines:

normalriskinsurance

expandedinsurance

IfwelookattheMortgageA=normalriskquotasweseethat

thesalesquotais3hoursattheUnderwritingdepartment,

1houratAdministration

4hoursatClaimsdepartment

Wealsoknowthattheexpectedprofit is35€perunitofnormalrisk.WecannowformulatethisasconstraintsinthedualLPmodelasfollows:

WewanttohaveMortgageAaslargeaspossible(thelargerthebetter):

MortgageA: 3λ₁+1λ₂+4λ₃≥35

AlsowewantmortgageBaslargeaspossible(again,thelargerthebetter):

MortgageB: 5λ₁+2λ₂+3λ₃≥50

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8.1 TheModelThefinaldualLP‐problemtosolveis:

MinimizeZ’= 1,300λ₁+500λ₂ +900λ3


MortgageA: 3λ₁+1λ₂+4λ3≥35

MortgageB: 5λ₁+2λ₂+3λ3≥50

and

λ₁,λ₂,λ3≥0

Usingthecomputerwehave

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Ifwesolvethisminimizationproblemwefindthefollowingoptimalsolution:

FirstweseethattheoptimalsolutionofZ'=13,100whichisidenticaltotheoptimalsolutionoftheoriginalprimalproblemwhereZ=13,100.

Ascanbeseenthevaluesofthethreedecisionvariablesareλ₁=0,λ₂=19,λ₃=4

Rememberfromourpresentationaboveintheprimalproblemthatthethreeshadowpriceswhere:

{0,19,4}.Thisisanimportantobservation.

Wehave thecorrespondencebetween theoriginalproblems‐theprimalproblem‐tothedualproblem,thatshadowpricesintheprimalproblemareequaltosolutionvaluesinthedualproblem.

Theshadowprices in thedualproblemareequal to solutionvalues in theprimalproblem.Shadowpricesinthedualproblemhavesolutionvaluesequalto{60,220}.Thesevaluesareequaltothesolutionvaluesfromtheprimalproblem{60,220}.

Wealsohaveareducedcostthatisnotequaltozero.Weseeinthisdualsolutionthatthereducedcostofvariable1isequalto20andthisisexactlyequaltothevalueoftheslackvalueintheprimalproblem.

Dualprice

Thedualpriceofaconstraintmeasurestherateatwhichthesolutionvalueimprovesastheright‐handsideisincreased.

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8.2 TheManagerialPerspectiveThereisadirectlinkbetweenaprimalandadualproblem.Dualityisnothingelsethanthespecificationofoneproblemfromtwodifferentpointsofview.Onceaproblemisspecified,werefertoitastheprimalproblem. A second, associated problem very often exists and it is called the dual specification. Inmanagerial economicsweareoftenmore interested in thedualproblembecause it containsmostlyeconomicinformation.Sometimesamanagerislessconcernedabouttheprofitfromasolutionthantheuse of available resources. This is because a manager has more control over the use of availableresourcescomparedwiththeprofits.Thedualsolutiongivesamanagertheinformationaboutthevalueofresources,andthisinturnisimportantwhenhehastodecideifmoreresourcesshouldbeallocatedandhowmuchtopayfortheseextraresources.

TheprofitofMortgageA=35andMortgageB=50appearascoefficientsintheobjectivefunctionintheprimalproblem.And in thedualproblemtheyareconstraintscoefficientsweseek tobeas largeaspossible.

Tofullyunderstandthedualityrelationsinlinearprogramming,itisimportanttoknowhowtosetupdualpairsofproblemsproperlyandhowtointerpretalltheircomponentsineconomicterms.

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9.Alinearprogrammingapplication:InvestmentWehaveintroducedthegeneralstructureofthelinearprogrammingproblemandcharacteristicsofthesolution.Nowletustakealookataninvestmentproblem.

Acompanyhas12mill.€forinvestmentwithinthecompany.Therearefivepossibleprojectsunderconsideration.

Purchasing Expectedreturn

Maximuminvestment(mill.of€)

Improvedmaterials‐handlingequipment

Automatingpackagingoperations

Purchasing raw materials in anticipation of priceincrease

Payingupoutstandingnotes

Additionalpromotionforanewproductline

15%

10%

18%

8%

20%

3

5

6

4

1

There is nominimum investment required for any project. Project 1 and 2 are classified as capitalexpenditureprojects,projects3and5arespeculativeinvestmentprojects,andproject4isafinancialproject.Thecompanyhasthefollowingrequirements:

Investmentincapitalexpendituresmustatleastbe40%ofthetotal

Theinvestmentinspeculativeprojectsmustbenomorethan50%oftheamountusedtopaynotes.

Decisionvariables

The decision variablesmust specify the amount ofmoney invested in each of the alternatives.Weintroduce6decisionvariablesthatspecifytheamountinvestedineachofthealternatives.

Let x₁=investmentinImprovedmaterials‐handlingequipment

x₂=investmentinAutomatingpackagingoperations

x3=investmentinPurchasingrawmaterialsinanticipationofpriceincrease

x4=investmentinPayingupoutstandingnotes

x5=investmentinAdditionalpromotionforanewproductline

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Objectivefunction

Determinehowmuchtoinvestineachprojectsoastomaximizeannualreturn.Wehave

MaxZ=annualreturn

Z=0.15x₁+0.10x₂+0.18x₃+0.08x₄+0.20x₅

Constraints

Fromtheproblemspecificationwehaveanumberofconstraints

Availablecash(mill.€) 1x₁+1x₂+1x₃+1x₄+1x₅≤12

Maxproject1 1x₁≤3

Maxproject2 1x2≤5

Maxproject3 1x3≤6

Maxproject4 1x4≤4

Maxproject5 1x5≤1

Spec.50%notes 1x3+1x5≤0.5x4

Capitalexpenditureatleast40% (1x1+1x2)/(1x1+1x2+1x3+1x4+1x5)≥0.4

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9.1 ThemodelWesummarizetheportfolioselectionproblemasfollows.

Weintroducedecisionvariablesx₁,...,x5

MaxZ=0.15x₁+0.10x₂+0.18x₃+0.08x₄+0.20x₅


Availablecash(mill.€) 1x₁+1x₂+1x₃+1x₄+1x₅≤12

Maxproject1 1x₁≤3

Maxproject2 1x2≤5

Maxproject3 1x3≤6

Maxproject4 1x4≤4

Maxproject5 1x5≤1

Spec.50%notes 1x3–1x4+1x5≤0

Capitalexp.atleast40% 0.6x1+0.6x2‐0.4x3‐0.4x4‐0.4x5≥0

and

x₁,x₂,x₃,x₄,x₅,x6≥0

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Thesolutiontotheportfolioproblemgivesthefollowingoptimalsolution:

OPTIMAL SOLUTION OBJECTIVE FUNCTION 1.4500 PARTIAL SENSITIVITY ANALYSIS SOLUTION REDUCED OBJECTIVE FUNCTION RANGES VARIABLE VALUE COST LOWER GIVEN UPPER __________________________________________________________________________________ Project 1 3.0000 0.0000 0.1000 0.1500 +INFINITY Project 2 3.0000 0.0000 0.0000 0.1000 0.1133 Project 3 1.0000 0.0000 0.1400 0.1800 0.2000 Project 4 4.0000 0.0000 0.0600 0.0800 +INFINITY Project 5 1.0000 0.0000 0.1800 0.2000 +INFINITY SHADOW SLACK RIGHT HAND SIDE RANGES CONSTRAINT TYPE PRICE LOWER GIVEN UPPER __________________________________________________________________________________ Cash ≤ 0.1000 0.0000 10.0000 12.0000 14.0000 Max proj 1 ≤ 0.0500 0.0000 1.0000 3.0000 6.0000 Max proj 2 ≤ 0.0000 2.0000 3.0000 5.0000 +INFINITY Max proj 3 ≤ 0.0000 5.0000 1.0000 6.0000 +INFINITY Max proj 4 ≤ 0.0200 0.0000 2.6667 4.0000 4.8000 Max proj 5 ≤ 0.0200 0.0000 0.0000 1.0000 2.0000 Notes ≤ 0.0800 0.0000 -1.0000 0.0000 1.2000 Cap. Exp. ≥ 0.0000 -1.2000 -INFINITY 0.0000 1.2000

Theoptimalsolutiongivesavalueoftheobjectivefunctionequalto1.45.

Thismeansanannualreturn=1.45mill€ona12mill€investment.

Fromtheoutputwecanfindthevaluesofthedecisionvariables.

Variable Investment(inmill.€)

Project1

Project2

Project3

Project4

Project5

3

3

1

4

1

Total 12

From theoutputwesee that the shadowpriceoncash is equal to0.1, thismeansa10%returnonadditionalcash;Theshadowpriceonmaxproject1isequalto0.05,thismeans5%additionalreturnforadditionalinvestmentpermittedinproject1;project4hasshadowpriceequalto0.02,thismeans2%foradditionalproject4funds;project5hasshadowpriceequalto0.02,thismeans2%foradditionalproject 5 funds; notes has shadow price equal to 0.08, this means 8% additional return for eachpercentagepointlessthan50%permitted.

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10.FinalenoteInthisnoteyouhavebeenintroducedtotheideaofbuildingmanagerialmodelsinascientificway.Mostoftheassumptionsareveryoftenrequirementsthatarenevermeetinpractice,butasanapproximationofanuncertainworldyouhaveatoolthatpointstowardsonefeasiblemanagerialdirection.Withtheuseofsensitivityanalysisseveralotherdirectionscanbeincludedinthemanagerialproposal.Andwiththedualyouhavetheinformationofthevalueofresources.

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