AnintroductiontoLinearProgramming
EricBentzen
OperationsManagement
CopenhagenBusinessSchool
August2014
.
.
1 Content
1. 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 7.1 The Managerial Perspective ................................................................................................................ 25 8. The Dual Problem ........................................................................................................................................ 26 8.1 The Model ............................................................................................................................................ 28 8.2 The Managerial Perspective ................................................................................................................ 30 9. A linear programming application: Investment .......................................................................................... 31 9.1 The model ............................................................................................................................................ 33 10. Finale note ................................................................................................................................................. 35 2 1.Introduction
Decisionmakinginabusinessareveryoftenrestrictedbythelimitationofavailableresourcesandat
the same time a business manager has to meet specified goals. In this paper quantitative analysis is
importanceforthemanagerialdecisionprocess.Butkeepinmindthatquantitativeanalysiscannever
providetheentireanswerforallstrategicdecisions.Useitwithcareanduseitasasystematicwayof
workingthroughacomplexmanagerialdecisionprocess.Decisionmodelingisascientificapproachto
managerial decision making where we develop a mathematical model of a real‐world problem. The
modelshouldbesuchthatthedecision‐makingprocessisnotaffectedbypersonalbias,emotionsand
guesswork.
2.Theprocessofquantitativeanalysis
Quantitativeanalysisisascientificmethodthatcanhelpinthemanagerialdecisionmakingprocess.The
decisionmodelingprocessinvolvesthreedistinctsteps:
Definingtheproblem
Formulation
Developingamodel
Acquiringthedata
Solution
Developingasolution
Testingthesolution
Interpretation
Analyzingtheresults
Implementingtheresults
3 2.1
Formulation
Inthisparteachpartofamanagerialproblemistranslatedandexpressedintermsofamathematical
model.Oneverycommonpitfallisthattheproblemcannotbeformulatedbecausetheproblemistoo
complex (then one could break down the problem into smaller pieces), another that technical or
managerial information are not available (often there has not been formulated an objective of the
managerialproblem).Theaiminformulationistoensurethatthemathematicalmodeladdressesallthe
relevant issues to the managerial problem at hand. Formulation of the problem should further be
classifiedintoa)Definingtheproblem,b)Developingamodel,andc)Acquiringthedata

Definingtheproblemisperhapsthemostimportantproblem.Whenthemanagerialproblemis
difficulttoquantify,itsometimesmaybenecessarytodevelopspecific,measurableobjectives.
Oneobjectiveinmanagerialeconomicscouldbethatofmaximizingtheprofitandanothercould
bethatofminimizingthecosts.Sometimestherearemultipleconflictinggoals,andifthisisthe
case,onecouldsolveitbyminimizingthedistancebetweenthegoals.

Developingthemodelisthestepwheredifferenttypesofmodelscanbespecified.Thesemodels
areexpressedasequationsorinequalitieswithoneormorevariablesandparameters.
Usethefollowingthreestepproceduretodefineandidentify




Decisionvariables,whichrepresenttheunknownentitiesinamanagerialproblem.
DecisionvariablescouldbecarsoftypeI(X1)andcarsoftypeII(X2).
Objectivefunction,whichstatesthegoalofthemanagerialproblem,canbequantifiedin
afunctionlike
Maxprofit=(50000€percartypeI)*(numberofcarstypeIproduced)+
(60000€percartypeII)*(numberofcarstypeIIproduced)
Ifweintroduce
DecisionvariableX1=numberofcarstypeIand
DecisionvariableX2=numberofcarstypeII,
wehavetheobjectivefunction
Max.profit=(50000€percartypeI)*(X1)+
(60000€percartypeII)*(X2)
Othertypes ofcommon managerial problemsaremaximizing(profit,efficiency,...)or
minimizing(cost,time,labor,...).
Constraints can be classified into technical and economical constraints. A technical
constraint could be the space or number or workers that are available. An economic
constraintcouldbetheamountofavailablemoneytoinvestinaproject.
Acquiringinputdataisthestepwhereinputdatatobeusedinthe modelareobtained.This
detailedaswellastechnicalinformationarecollectedandimplementedintothemodel.Youwill
have to use knowledge from the basis of managerial economics. Obtaining accurate data is
essential,improperdata willresultinmisleading results.For realworldproblems,collecting
accuratedatacanbeoneofthemostdifficultandchallengingaspectsofdecisionmodeling.
4 2.2
Solution
Itisthestepwherethemathematicalexpressionsfromyourformulationprocessaresolvedtoidentify
anoptimalsolutionofthemanagerialeconomicsproblem.Becausewecanusesoftwarepackagestofind
asolutionourfocushasshiftedawayfromdetailedalgorithm(thesimplexalgorithm)andtowardsthe
bestofthesepackages.Wedistinguishbetweena)developingasolutionandb)testingthesolution.

Developingasolutionisthestepthatinvolvestheuseofanalgorithmthatconsistsofaseriesof
procedures,andtheaccuracyofthesolutiondependsontheaccuracyoftheinputdataandthe
model.Donotwastetimewithspecializedprogrammingskills.Insteaditismucheasiertousea
standardsoftwareprogramthatisabletohandleandsolvetheproblem.Forsmallproblems
withonlytwodecisionvariablesyoucandoagraphicalanalysis,andforlargerproblemsyou
willhavetousetheanalyticalsolution.

Testingthesolutionmeansthatonehastobesurethatnewdatafromanothersourcebehaves
inasimilarwayastheoriginaldata.Onehastocheckdataaswellasthemodelitselftomake
surethattheyreallyrepresentthemanagerialeconomicproblem.
2.3
Interpretation
Afterasolutionhasbeenfoundyouhavetotakeacloserlookatsolutionandalldecisionvariables.The
mostimportantjobforanymanagerialeconomistswouldbetoaskthefollowingquestion:"whatif".
The"whatif"questionhastobeaskedsothatyoucanhaveanideaofhowsensitivethesolutionis.We
distinguishbetweena)analyzingtheresultsandb)implementingtheresults.
Analyzingtheresultsbeginswithdeterminingtheimplicationsofthesolution.Itisimportanttoaskthe
followingquestion:“Whatistheinterpretationoftheresults?”.Andnextquestiontoaskis“Whatifwe
change one or more of the input data?”. This sensitivity analysis is important because it shows how
sensitivethesolutionistochanges.
Implementationisthestepthatmostoftenisneglectedinrealworld.Mostpeopleforgetthispartand
veryoftenawellspecifiedandpreparedplanaremissingthelastandfinalstep:implementationinthe
organization.
5 3.Practicalapplications
Anumberofpracticalapplicationshavedocumentedtheuseofthequantitativeprocessaswellasthe
useofamorecomplexmathematicalformulation.Inthefollowingwewillonlybelookingatapplications
thatuseaspecificmathematicalmodelcalledlinearprogramming.Thelinearprogrammingmodelisa
simpleandimportantmathematicaloptimizationmodel,anditcansometimesbeappliedtoanumber
ofeconomicproblems.
Acoupleofoldapplicationsare:

The linear programming model has been used for bank asset management and it has been
implemented to include comprehensive risk constraints, various policy considerations,
economicandinstitutionalrealitiesofthemarketplace,andavarietyofdifferentdynamiceffects
whichmustbeconsideredinordertomakeoptimalassetmanagementdecisions.Conceptual
problemsconcerningtheformulationofthebank'sgoalswereconsideredaswell.1

A linear programming formulation of Shell's distribution network between four sources of
productandalargenumberoftransshipmentpointsandterminalshasbeenimplemented,with
emphasisoncostlogisticsandrankingofvariousalternatives.2

Critical to an airline's operation is the effective use of its reservations inventory. American
Airlinesbeganresearchintheearly1960sinmanagingrevenuefromthisinventory.Becauseof
the problem's size and difficulty, American Airlines Decision Technologies has developed a
seriesofORmodelsthateffectivelyreducethelargeproblemtothreemuchsmallerandfarmore
manageable sub problems: overbooking, discount allocation, and traffic management. The
results of the sub problem solutions are combined to determine the final inventory levels.
AmericanAirlinesestimatesthequantifiablebenefitat$1.4billionoverthelastthreeyearsand
expectsanannualrevenuecontributionofover$500milliontocontinueintothefuture.3

Alinearprogrammingmodelwithseveralobjectiveswasdevelopedinchoosingmediaplans.4
1CohenandHammer(1967):"LinearProgrammingandOptimalBankAssetManagementDecisions".Journalof
Finance,147‐165.
2
Zierer, Mitchell, and White (1976): "Practical applications of linear programming to Shell's distribution
problems".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. 6 Andacoupleofrecentapplicationsare:

JandeWitCompanyimplementedadecision‐supportsystembasedonlinearprogrammingasa
production‐planning and trade tool for the management of its lily flower business. The LP
maximizesthefarm'stotalcontributionmargin,subjecttosuchconstraintsasmarketdefined
sales limits, market requirements, characteristics of the production cycle duration, technical
requirements, bulb inventory, and greenhouse limitations. The main decision variable to be
calculatedisthenumberofflowerbedsinaspecificgreenhouse,fromaspecificbulbbatch,ofa
specific variety, for a specific purpose, taking into consideration planting and expected
harvestingweeks.Between1999and2000,companyrevenuegrew26percent,salesincreased
14.8percentforpotsofliliesand29.3percentforbunchesoflilies,costsfellfrom87.9to84.7
percentofsales,incomefromoperationsincreased60percent,returnonowner'sequitywent
from 15.1 to 22.5 percent, and best quality cut lilies jumped from 11 to 61 percent of the
quantitiessold.5

U.S.CoastGuardusedlinearprogramminginanextensivepreventativemaintenanceprogram
for its Sikorsky HH60J helicopters based on helicopter flight time. The model must consider
differentmaintenancetypes,maintenancecapacityandvariousoperationalrequirements.6

RecentreformofEuropeanagriculturalpolicyhasresultedinsubstantialchangestothecriteria
bywhichpremiapaymentsaremade.Beeffarmers,whohavebeenparticularlydependenton
premia payments to maintain margins, must re‐evaluate their systems to identify optimal
systems in these new circumstances. A linear programming model has been used to identify
optimalbeefproductionsystemsinIreland.Theobjectivefunctionmaximizesfarmgrossmargin
andthemodelisprimarilyconstrainedbyanimalnutritionalrequirements.7

SchedulingtheItalianMajorFootballLeague(theso‐called"SerieA")consistsinfindingforthat
league a double round robin tournament schedule that takes into account both typical
requirements such as conditions on home‐away matches and specific requests of the Italian
FootballAssociationsuchastwin‐schedulesforteamsbelongingtothesamehome‐town.The
model takes into account specific cable television companies requirements and satisfying
various otheroperationalconstraintswhile minimizingthetotalnumberofviolationsonthe
home‐awaymatchesconditions.8

AlinearprogrammingmodelwasimplementedusingActivityBasedCostingforcalculatingunit
product cost, and dynamic Activity Based Management for assessing the feasibility of
prospective production plans. The model is used to optimize the business plan at a steel
manufacturer.9
5Filho,José,Neto,andMaarten(2002):"OptimizationoftheProductionPlanningandTradeofLilyFlowersatJan
deWitCompany".Interfaces,35‐46.
6HahnandNewmanb(2008):"SchedulingUnitedStatesCoastGuardhelicopterdeploymentandmaintenanceat
ClearwaterAirStation,Florida".Computers&OperationsResearch35(2008)1829—1843.
7Crosson,O'Kiely,O'MaraandWallace(2006):"ThedevelopmentofamathematicalmodeltoinvestigateIrish
beefproductionsystems".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. 7 
Minnesota's Nutrition Coordination Center used linear programming to estimate content of
commercialfoodproducts.10

LinearProgrammingwasusedtodecidethedailyroutesofloggingtrucksinforestry.Aspects
suchaspickupanddeliverywithsplitpickups,multipleproducts,timewindows,severaltime
periods,multipledepots,driverchangesandaheterogeneoustruckfleet.11

Airlineseatisthemostperishablecommodityintheworld.Eachtimeanairlinertakesoffwith
anemptyseat,arevenueopportunityislost.DeltaAirLinesusedaLPmodelwithmorethan
40,000constraintsand60,000variablestosolvethisemptyseatproblem.Theysavedmorethan
$220,000perday.12

AllocationoftraincapacityamongmultipletravelsegmentsonanIndianRailwaystrainroute
withseveralstops.Duetohistoricalandsocialreasons,IndianRailwayssplitsitstraincapacity
based on user and type of travel. The determination of the optimal split of such capacity is
nontrivial.TheirLPmodelwasapplied17IndianRailwaystrains,andtheyincreasedrevenue
from2.6to29.3percentinrevenue,6.7to30.8percentinloadfactors,and8.4to29percentin
passengerscarried.13
Intheshippingandtransportationindustry,thereareseveraltypesofstandardcontainerswith
different dimensions and different associated costs. Investigation of the multiple container
loadingcostminimizationproblem,wheretheobjectiveistoloadproductsofvarioustypesinto
containersofvarioussizessoastominimizethetotalcost.14
This paper describes an integer linear programming model conceived as an alternative to a
traditionalmaterialrequirementsplanning(MRP)systemforextendingtheconceptofsupply
chainsynchronisationupstreaminamulti‐tiersupplychain.Inthismodel,weassumethereis
an incumbent application for transmitting original equipment manufacturer (OEM)
requirementstofirst‐,second‐andthird‐tiersuppliers.15


10Westrich,Altmann,andPotthoff(1988):"Minnesota'sNutritionCoordinatingCenterUsesMathematical
OptimizationtoEstimateFoodNutrientValues".Interfaces,86‐99.
11Flisberg,P.,B.Lidén,M.Rönnqvist(2009):"Ahybridmethodbasedonlinearprogrammingandtabusearch
forroutingofloggingtrucks".Computers&OperationsResearch,1122‐1144.
12Subramanian,R.(1994):“Coldstart:FleetAssignmentatDeltaAirLines”.Interfaces,104‐120.
13GopalakrishnanandNarayan(2010):“CapacityManagementonLong‐DistancePassengerTrainsofIndian
Railways”.Interfaces,291‐302.
14ChanHouChea,WeiliHuanga,AndrewLima,WenbinZhu(2011):“Themultiplecontainerloadingcost
minimizationproblem”.EuropeanJournalofOperationalResearch,501‐511.
15
Mula,J.,A.C.Lyons,J.E.Hernández,R.Poler(2014):“Anintegerlinearprogrammingmodeltosupportcustomer‐
drivenmaterialplanninginsynchronised,multi‐tiersupplychains”.InternationalJournalofProductionResearch,
14,4267‐4278. 8 Thesepracticalapplicationsbelongstoaclassofbusinessproblemsclassifiedasallocationproblems
andundercertainconditionsasolutioncanbeachievedbythelinearprogrammingmodel.
4.Linearprogrammingdefined16
We define a linear programming problem as an allocation problem wherein the values of decision
variablesmustbedeterminedtomeetagoal,underasetoflimitationsbasedonavailableresources.
Linearprogrammingmodels
A linear programming problem may be defined as the problem of
maximizing(orminimizing)alinearobjectivefunctionsubjecttoa
setoflinearconstraints.
Theconstraintsmaybeequalitiesorinequalities.
ALinearProgrammingmodelisbasedonthefollowingproperties:
1.Proportionality
Thismeansthatthecontributionofeachdecisionvariabletothevalueof
the objective function and the left hand side of constraints are directly
proportional to the level of the decision variable. In other words we are
talkingaboutconstantreturnstoscale.
2.Nonnegativity
Decisionvariablesarenotallowedtobenegative.Thismeansthatsolutions
variables of say costs cannot be negative, or the number of workers
allocatedtoajobcannotbenegative.
3.Additivity
Theobjectivefunctionorthefunctionofthelefthandsideofafunctional
constraintisthesumoftheindividualcontributionsofeachvariable.
4.Divisibility
Decisionvariablesareallowedtofractionalvaluesequaltoorabovezero.
Fractionalvaluesforthedecisionvariables,suchas1.25areallowed.
5.Certainty
This assumption asserts that the objective and constraints coefficients of
the LP model are deterministic. This means that they are known fixed
constants.
16WeusetheDantzig(1963)specification.Dantzig,G.B.(1963):LinearProgrammingandExtensions.Princeton
UniversityPress.
9 4.1
Themanagerialperspective
It is important to realize that linear programming is not a panacea. Instead linear programming is a
mathematicaltoolthatsometimesapproximatesamanagerialproblemquitewell.
Ifwetakeacloserlookattheassumptionsunderlyingthelinearprogrammingmodelwecanveryeasily
specifyapplicationswheretheassumptionsarenotmeet.

Insteadofhavingconstantreturnstoscalewecouldhaveincreasingordecreasingreturnsto
scale.17

Ifweareunsureoftheexactnumberofresourcesthatareneededinamanufacturingproblem
thenthecertaintyassumptionwouldbeviolated.

Data that are used in a linear programming model are uncertain because they cannot be
measuredpreciselyandbecausetheycanfluctuateindifferentunpredictableways.Justthinkof
machinebreakdowns,absenceofworkersorpowerfailures,etc.Thecertaintyassumption‐a
rare occurrence in real life, where data are more likely to be presented by probabilistic
distributions. If the standard deviations of these distributions are sufficiently small, then the
approximationisacceptable.18

Theprofitorthecostapproximatesanuncertainamount.Theactualvaluedependsoncurrent
priceofrawmaterials,defectsduringmanufacturingorchangingininventorycosts,etc.
To sum up one should have in mind that uncertainty in the data is one reason why models are just
approximate.
17Decreasingreturnstoscalecanbetreatedbyintroducinganewdecisionvariablethatcoversthedecreasing
returnstoscale.Inthiscasewewouldhavetwodecisionvariablesonethattakecareofconstantreturnstoscale
andanothertotakeofdecreasingreturnstoscale.Decreasingandincreasingreturnstoscalecanalsobesolved
bynonlinearprogramming.
18Largestandarddeviationscanbeaccountedbyapplyingsensitivityanalysistotheoptimalsolution. 10 5.Asimpleproductionproblem
5.1
Problemdescription
A large manufacturing company that specializes in personal computers must decide what types and
quantitiesofoutputtomanufactureforeachday'sproduction.Wejustlookattwokindsofpersonal
computers‐thesmallandthebig‐fromwhichwemayselecttheproductmix.
Themanufactoryhasthefollowingfreeresourcesavailablefortheproduction:

numberoflaborhours600

numberofprocessinghours800
Frompastexperiencetheycansell

nomorethan100bigcomputersperday.
Theresourcesavailablearerelatedtothetwoalternativeoutputsasfollows.Eachunitofsmallpersonal
computersproducedrequire2.5hoursoflaborand4.5hoursofprocessing,whileeachunitofthebig
personalcomputersproducedrequires3.5hoursoflaborand5.5hoursofprocessing.
Theprofitperunitofsmallpersonalcomputersmanufacturedis€15,whiletheprofitperunitofthebig
personalcomputersmanufacturedis€28.
Howmanyofeachtypeofpersonalcomputersshouldthemanufactoryproducetomaximizeprofit?
Decisionvariables
Weintroducetwodecisionvariables.Letourdecisionvariablesbe
X1=numberofsmallpersonalcomputersmanufactured
X2=numberofbigpersonalcomputersmanufactured
Objectivefunction
The profit per day is equal to Z the profit from the production of each type of personal computers
manufactured.Withaprofitperunitof€15and€28wehavethetotalprofitequalto:
15X1astheprofitfromproductionofsmallpersonalcomputersand28X2astheprofitfromproduction
ofbigpersonalcomputers.
Themanufactoryhasastheirobjectivetomaximizethetotalprofitfromproducingpersonalcomputers
whichmeansthatourobjectivefunctionisequalto:
MaxZ=15X1+28X2
OnefirstandfeasiblesolutionisnottoproduceanythingthatisX1=0andX2=0whichyieldsatotalprofit
ofzero.ButanypositivesolutionsofX1andX2givemoreproductionandgreaterprofit.IfX1=2andX2=3
11 then the profit is equal to Z=2(15) +3(28) = €114. Unfortunately the maximization of the objective
functionhastomeetacoupleofconstraints,andwehavetotaketheseconstraintsintoconsiderationin
ordertofindanoptimalsolution.
NegativevaluesofX1andX2makenomanagerialsense.
Constraints
Fromtheproblemwehavethreerestrictions:

Laborhours:600hours

Processinghours:800hours

Salesrestriction:nomorethan100bigcomputers
Weformulatethethreeconstraintsas:

Laborhours
Thenumberofhourstoproduce1smallcomputeris2.5hoursandtoproduce1bigcomputeris4.5
hours.TotalhoursrequiredtoproduceX1andX2computersis2.5X1+4.5X2.
With600hoursavailabletoproduceX1andX2weformulatethelaborconstraintas:
2.5X1+4.5X2≤600

Processinghours
Numberofprocessinghourstohandle1smallcomputeris3.5hoursandtohandle1bigcomputeris5.5
hours.Thetotalprocessinghoursrequiredis3.5X1+5.5X2.
With800hoursavailabletohandleX1andX2weformulatetheprocessingconstraintas:
3.5X1+5.5X2≤800

Salesrestriction
Only100bigcomputerscanbesold.Thelimitationonsalescanbeformulatedas:
1X2≤100
12 5.2
Themodel
Wesummarizetheproblemasfollows.WeintroduceX1unitsofsmallcomputersandX2unitsofbig
computerssuchthatthetotalprofitZismaximized19withrespecttoavailableresources:
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
Summary
Theproductionexampleaboveisanoptimizationproblemwherewemaximizealinearfunctionofthe
decisionvariablessubjecttoasetofconstraints.Eachconstraintmustbealinearequationorlinear
inequality. We have formulated the production example as an allocation problem in which limited
resourcesareallocatedtoanumberofeconomicactivities.AgeneralLPmodelcanbewrittenasfollows:
MaximizeZ=
c₁x₁+c₂x₂+...+cnxn
Subjectto
a₁₁x₁+a₁₂x₂+...+a1nxn≤b₁
a₂₁x₁+a₂₂x₂+...+a2nxn≤b₂
...
am1xn+am2x2+...+amnxn≤bm
x1,x2,…,xn≥0
In the general LP model we have m activities whose levels are represented by n decision variables,
x₁,x₂,...,xn.
Eachunitofactivityjuseanamountaijofresourcei,andthemresourcesaregivenbyb₁,b₂,...,bm.
19Theobjectivefunctionasspecifiedmaximizestheprofit.Rememberthatprofitisequaltoincomeminus
expenditure.Ifourobjectivefunctionwasspecifiedasminimizationofcostswecoulddothateasilyjustby
multiplyingourobjectivefunctionwithminusone.Hereweusethefactthatcostisequaltoexpenditureminus
income.
13 6.Aserviceproblem
6.1
Problemdescription
Aninsurancecompanyisintroducingtwonewproductlines:normalriskinsurancemortgageAand
expanded insurance mortgage B. The expected profit is 35€ per unit on mortgage A and 50€ on
mortgageB.
Managerofthemortgagedepartmenthasestablishedsalesquotasforthenewproductlinestomaximize
totalexpectedprofit.
Furtherinformationis:
Mortgage
Department
Work‐hoursperunit
Normalrisk
Underwriting
3
Administration
1
Claims
4
Expanded
risk
Work‐hours
available
5
1,300
2
500
3
900
Decisionvariables
Weintroduce2variables.Let
x₁=numberofnormalriskinsurancemortgageA
x₂=numberofexpandedinsurancemortgageB
Objectivefunction
Theinsurancecompanyseekstomaximizethetotalprofit.Alinearobjectivefunctioncanbespecified
whichmaximizethetotalprofit.Wehave
MaxZ=profitperunitfortotalinsurance
14 TheprofitfromeachmortgageAwas35€wecanformulatethisas35x₁astheprofitfrommortgageA
andtheprofitfromeachmortgageBwas50€whichgivesusaprofitfrommortgageBequalto50x₂.
Thuswecanformulatethetotalprofitas
Z=35x₁+50x₂€
Theprofitfromlet’ssay5mortgageAand3mortgageBwouldgiveusatotalprofitequalto
Z=5(35)+3(50)=325€.
Constraints
Fromtheproblemspecificationwehavethreerestrictionsonthenumberofworkinghoursavailable.

Underwriting:1,300hours

Administration:500hours

Claims:900hours
Weformulatethethreeconstraintsas:

Underwriting
Thenumberofhoursrequiredtohandle1unitofmortgageAis3hoursandtohandle1unitof
mortgageBis5hours.Thusthetotalhoursrequiredtohandlex₁andx₂mortgagesis3x₁+5x₂
With1,300hoursavailabletohandlex₁andx₂weformulatetheUnderwritingconstraintas:
3x₁+5x₂≤1,300

Administration
Numberofhoursrequiredtohandle1unitofmortgageAis1hourand1unitofmortgageBis2
hours.Thetotalhoursrequiredis1x₁+2x₂.
With500hoursavailabletohandlex₁andx₂weformulatetheAdministrationconstraintas:
1x₁+2x₂≤500

Claims
Numberofhoursrequiredtohandle1unitofmortgageAis4hoursand1unitofmortgageBis
3hours.Thetotalhoursrequiredis4x₁+3x₂.
With900hoursavailabletohandlex₁andx₂weformulatetheClaimconstraintas:
4x₁+3x₂≤900
15 6.2
Themodel
Wesummarizetheinsuranceproblemasfollows.Weintroducex₁unitsofmortgageAandx₂unitsof
mortgageBsuchthatthetotalprofitZismaximizedwithrespecttoavailableresources:
MaximizeZ= 35x₁+50x₂
Subjecttoconstraints
Underwriting
3x₁+5x₂≤1,300
Administration
1x₁+2x₂≤500
Claim
4x₁+3x₂≤900
and
X₁,X₂≥0
6.3
Performingtheanalysis
GraphicalSolution
ThegraphicalsolutiontoLP‐problemscanonlybeshownwithtwovariables.Webeginwithamapthat
expressesonlynonnegativeconstraints.
16 We continue with the three constraints and graph them each by each, and ends with the objective
function.Thefinalgraphicalsolutionhasafeasibleregion.Wedefinethefeasibleregionfortheproblem
bounded by the three constraints and the assumptions of nonnegativity. Points where two or more
constraintsintersectarecalledcornerpoints(orcornersolution).
Ifwetakealookatthefourcornersolutionswehave:
0:
(x1=0,x2=0) →Z=0
1:
(x₁=225,x₂=0)
2:
(x₁=60,x₂=220)→Z=13,100
3:
(x₁=0,x₂=250)
→Z=7,875
→Z=12,500
Fromthisweseethatthesolutionthathasitsmaximumatcornersolution2wherex₁=60andx₂=220
whichgivesasolutionequaltoZ=13,100.AtthispointthevalueofZismaximizedandtheotherthree
cornersolutionshaveaZvaluelessthancornersolution2.
17 6.4
Utilizationofresources
Attheoptimalsolutionjustfoundwehadx₁=60andx₂=220.Atthatcornersolutionwecanseethatnot
allavailableresourcesareused.

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,but
theycanbereallocatedbymanagementforotherpurposes‐ifpossible.
The20hoursavailablearecalledslackorunusedhoursandwenamevariables₁asslackvariable
number1.Becausethevalueofslackvariable1isnotequaltozerowesaythattheconstraintis
notbinding.

TheAdministrationconstraint1x₁+2x₂≤500
utilize1(60)+2(220)=500
If we add an extra variable, slackvariable number s₂ , to the constraint we can write the
constraintasanequality1x₁+2x₂+1s₂=500
Thedifferencebetweenresourcesusedandthoseavailableequalsslackvariablenumber2
s₂=500‐500=0.
This means the department has used all available resources allocated at the Administration
department,andwesaythisconstraintisbindingandtheslackorunusedhoursareequalto
zero.
Ifaconstraintisbindingthereisnoslack.

ThefinalClaimconstraint4x₁+3x₂≤900
utilize4(60)+3(220)=900
If we add an extra variable, slackvariable number 3 s₃ , to the constraint we can write the
constraintasanequality4x₁+3x₂+1s₃=900
Thedifferencebetweenresourcesusedandthoseavailableequalss₃=900‐900=0.
ThismeansthedepartmenthasusedallavailableresourcesallocatedforClaims.Thisconstraint
isbindingandtheslackisequaltozero.
18 We have introduced an important part in linear programming and that is the introduction of slack
variables.Aswehaveseenaslackvariableisequaltounusedamountofaresource.
SlackVariable
Aslackvariablecontainsthedifferencebetween
theresourcesactuallyusedandthoseavailable.
6.5
TheManagerialPerspective
Fromtheabovesolutionandconclusionwithrespecttousedorunusedresourcesfurthershouldbe
mentioned.Whenallresourcesareusedonecouldaskwhatifwecouldhavemoreresourcesallocated,
and if we could choose freely between the two binding constraints (Administration and Claim) and
whichofthetwoshouldbethefirsttohavemoreresourcesallocated.Thebestwaytoallocatemore
resources would be to the department that utilizes the resources the best way. From an economic
perspectivethebestwayresourcesareallocatedarewithaneyetowhatthealternativesare.Iftheonly
alternatives are between the two binding constraints then the resources should be allocated to that
constraintthatincreasethevalueoftheobjectivefunctionmost.
IfoneextraresourceisallocatedtoUnderwritingwewouldhave1,301hoursavailable.Wehavefound
thattheslackvariables₁=20,andifweallocateonehourmoretoUnderwritingwewouldaddextrato
theslackvariables₁=20+1=21.Weintroduceanewvariable"shadowprice"andtheshadowpriceat
Underwritingequaltoλ₁=0.
IfoneextraresourceisallocatedtoAdministrationwenowhave501hoursavailable.Ourslackvariable
was equal to s₂=0 and with one extra hour we have 501 hours available and we end up with a new
optimalsolutionwherex₁=59.4andx₂=220.8andanewobjectivefunctionZ=13,119.Theincreasein
theobjectivefunctionisequalto19,andwehaveashadowpriceofλ₂=19.
IfoneextraresourceisallocatedtoClaimwenowhave901hoursavailable.With901hoursavailable
we end up with another optimal solution with x₁=60.4 and x₂=219.8 and a new objective function
Z=13,104.Theincreaseintheobjectivefunctionisequalto4,whichisequaltoourshadowprice,that
isλ₃=4.
ThebestwayoneextrahourcanbeusedshouldbeattheAdministrationdepartment.Theshadowprice
attheAdministrationequaltoλ₂=19andthereforeextrahoursallocatedtoAdministrationdepartment
increases the objective function more compared with the Claim department which only increase the
objectivefunctionwithλ₃=4.
19 Weintroduceanewvariable"unitworthofaresource"whichwewillcallaShadowPrice.
ShadowPrice
Ashadowpricecontainstheamounttheoptimal
objectivefunctionvaluechangesperunitincrease
intherighthandsidevalueoftheconstraint.
Wehavedemonstratedtheconnectionbetweenslackvariablesandshadowpricesandwesummarize
theresults:
Constraint
Slack
Shadowprice
Underwriting
20
0
Administration
0
19
Claim
0
4
20 7.Usingthecomputer
WecansolvetheaboveLP‐problemusingthecomputer.InthefollowingweuseaLPprogramthatis
availableatCopenhagenBusinessSchool.Weproceedasfollows.
Afterinstallationoftheprogram20youstarttheprogramandreceivethefollowingstartwindow.
AsyoucanseeyouhavetheoptionofchoosingbetweenaDanishorEnglishversion.Wecontinuewith
theEnglishversionandwebeginwithformulatinganewmodel.Youpushthestartbottom,andgetthe
followingpicture.
20AvailableviaLearnatCBS.
21 PushingtheNewmodelbottomweget
Weformulateanewmodelandcallit"Myownmodel".
Wemaximizetheobjectivefunctionandwehave2variablesand3constraints.
22 After pushing the OK bottom we can fill in all the coefficients from the LP‐problem, and we get the
followingpicture.
PushingtheOKbottomwehavethepossibilitytohavesolutionseitherasAnalyticalorGraphical.We
choosetheanalyticalsolution.
Theanalyticalsolutionhastheoptimalvalueswefoundinthegraphicalsolution,butwehavefurther
numbersthatrequireafewwords.
23 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.Thecoefficientvalueof
x1=35has"lower”range=25and"upper“range=66.6667.Theserangestellusthatthesolution
ofthedecisionvariablex1=60willremainthesamewithinthe"lower"and"upper"ranges.Ifwe
changethegivencoefficientfrom35to36thesolutionwillremainthesamex₁=60,butwehave
anewoptimalobjectivefunctionthatisincreasedwith60toanewvalueequalto13,160.We
have"lower"and"upper"rangesforx₂equalto(26.25;70).Ifthecoefficientofx₂ischanged
from50to51westillhavetheoptimalsolutionofx₂=220.Wecanchangethecoefficientfrom
50upto51andthevalueoftheobjectivefunctionincreaseswith220.

TheRighthandsideofthethreeconstraintshastheirownranges.Ifwelookatconstraint2(the
Administration constraint) we have a "Shadow price"=19. This shadow price will remain the
sameintheintervaloftheRighthandsidefrom(225;509).IfwechangetheRighthandside
from 500 to 501 the value of the objective function will change from 13,100 to
13,100+19=13,119.IfwechangeConstraint1from1,300to1,301weseewithaShadowprice
=0thatwedonotgetabettersolution.Theobjectivefunctionremainsthesame.

TheReducedcost21columnshowsthevalueofvariableswhenthesevariableshavesolutions
equaltozero.
21Reducedcost=[(changeinoptimalobjectivefunctionvalue)/(unitincreaseofvariable=0)]
24 Reducedcost
Thereducedcostofanunusedactivityistheamount
by which profits will decrease if one unit of this
activityisforcedintothesolution.
Obviously, a variable that already appears in the
optimalsolutionwillhaveazeroreducedcost.
Sometimestheobjectivefunctionisparalleltooneofthebindingconstraintsandwewillhavewhatis
called alternative optimum. Alternative optimum can be important because it gives management
anothersolutionwiththesamefinal valueofthe objective function.Iftheprofitfrom mortgageB is
changedfrom50€to70€wehaveanoptimalsolutionwithmortgageA=0andmortgageB=250.But
theobjectivefunctionisparalleltotheAdministrationconstraintintherangefromA=0andA=60.This
means management can include mortgage A in their product portfolio, and instead of having one
producttooffertheycanhavetwoproducts.
7.1
TheManagerialPerspective
Linearprogrammingisamathematicaltoolwhichsometimesfitorapproximatesamanagerialsituation.
Justlikeahammer.Ahammercanbeusedtohammernails,butitcanalsobeusedtohammerscrews,
holes,andbolts.Itisobviousthattheabovetasksaremoreefficientlydonebyascrewdriver,adrill,but
sometimesthereisnoappropriatetoolexceptfromthehammer.Inthiscasethehammeristhebest
available tool, and that is the same with linear programming. Linear programming is the best
mathematicaltooltodescribemanymanagerialsituations.
Amanagershouldnotgointoallthetechnicaldetailsofhowlinearprogrammingmodelsaresolved.
Insteadamanagercanhelpwiththeformulationofeconomiccoherenceandlatertranslatetheresults
inamanagerialcontentandtakeadecision.
25 8.TheDualProblem
Linear programming models have brought to economists the meaning of duality22. We know that
economistsareinterestedinproductionandcost,pricesandquantities,but,dualitypossessesameaning
that transcends linear programming and its economic interpretation. Most problems have a twofold
representationandwecallthistheduality.
Insteadoflookingatthemaximizationofnetprofit
weturnourfocustowardstheresourcesusedand
we now look at the minimization of the available
resources.
Intheinsuranceexamplethecompanyhadanumberofavailableworkinghours,butwedidnotuseall
thehours.Wefoundthat20hourswasnotusedintheUnderwritingdepartment.Thequestioniswhat
theeconomicvalueassociatedwiththelasthourisworth?Thelessweuseofavailableworkinghours
thebetter.Anyfreeworkinghourcanbeusedforanotherpurpose.
Ifwetakeacloserlookattheworkinghoursandweconnectthesetothedepartmentswecanswitchto
anotherproblemwhereinsteadweminimizetheuseofavailableresources.Intheinsurancecaseour
original problem was the maximization of the profit per unit for total insurance. The model was
formulatedwithrespecttoavailableworkinghoursatthreedepartments.
We could alternatively formulate the same problem by focusing on the minimization of available
workinghourswithrespecttohavingtheprofitperunitfortotalinsuranceaslargeaspossible.
Weformulatethedualproblemasfollows.
Wehave1,300Underwritinghours,500Administrationhours,and900Claimshours.Thisinformation
cannowbeusedtoformulateourobjectiveistominimizetheuseofavailableworkinghours.
Decisionvariables
Weintroducethreedecisionvariables.Let
λ₁=numberofhoursatUnderwritingdepartment
λ₂=numberofhoursatAdministrationdepartment
λ₃=numberofhoursatClaimsdepartment
Objectivefunction
22
ThewordcanbefoundintheworksbyEulerandLegendrein1750wheretheywereinterestedinmethodsfor
solvingdifferentialequations.LaterBoole(1859)wrote:“Thereexistsinpartialdifferentialequationsaremarkable
duality,invirtueofwhicheachequationstandsconnectedwithsomeotherequationofthesameorderbyrelationsof
aperfectlyreciprocalcharacter”.Boole,G.B.ATreatiseOnDifferentialEquations.NewYork,ChelseaPub.Co.,5th,
1859.
26 Ourobjectiveistominimizethenumberofworkinghoursusedatthethreedepartments.Wespecify
ourlinearobjectivefunctionby
MinimizeZ'=1,300λ₁+500λ₂+900λ₃hours
Constraints
Inouroriginalproblem(theprimalproblem)wemaximizedthetotalprofit.Inthedualproblemwenow
minimizetheavailableworkinghourswithrespecttohavingtheprofitfromthetwomortgagesaslarge
aspossible.
Fromtheinsurancecasethemanagerhasestablishedquotasforthenewproductlines:

normalriskinsurance

expandedinsurance
IfwelookattheMortgageA=normalriskquotasweseethat

thesalesquotais3hoursattheUnderwritingdepartment,

1houratAdministration

4hoursatClaimsdepartment
Wealsoknowthatthe expectedprofitis35€perunitof normalrisk.Wecan now formulatethis as
constraintsinthedualLPmodelasfollows:
WewanttohaveMortgageAaslargeaspossible(thelargerthebetter):

MortgageA:
3λ₁+1λ₂+4λ₃≥35
AlsowewantmortgageBaslargeaspossible(again,thelargerthebetter):

MortgageB:
5λ₁+2λ₂+3λ₃≥50
27 8.1
TheModel
ThefinaldualLP‐problemtosolveis:
MinimizeZ’= 1,300λ₁+500λ₂ +900λ3
Subjecttoconstraints
MortgageA: 3λ₁+1λ₂+4λ3≥35
MortgageB: 5λ₁+2λ₂+3λ3≥50
and
λ₁,λ₂,λ3≥0
Usingthecomputerwehave
28 Ifwesolvethisminimizationproblemwefindthefollowingoptimalsolution:
FirstweseethattheoptimalsolutionofZ'=13,100whichisidenticaltotheoptimalsolutionofthe
originalprimalproblemwhereZ=13,100.
Ascanbeseenthevaluesofthethreedecisionvariablesareλ₁=0,λ₂=19,λ₃=4
Rememberfromourpresentationaboveintheprimalproblemthatthethreeshadowpriceswhere:
{0,19,4}.Thisisanimportantobservation.
We have the correspondence between the original
problems‐theprimalproblem‐tothedualproblem,
thatshadowpricesintheprimalproblemareequal
tosolutionvaluesinthedualproblem.
The shadow prices in the dual problem are equal to solution values in the primal problem. Shadow
pricesinthedualproblemhavesolutionvaluesequalto{60,220}.Thesevaluesareequaltothesolution
valuesfromtheprimalproblem{60,220}.
Wealsohaveareducedcostthatisnotequaltozero.Weseeinthisdualsolutionthatthereducedcost
ofvariable1isequalto20andthisisexactlyequaltothevalueoftheslackvalueintheprimalproblem.
Dualprice
Thedualpriceofaconstraintmeasurestherateat
whichthesolutionvalueimprovesastheright‐hand
sideisincreased.
29 8.2
TheManagerialPerspective
Thereisadirectlinkbetweenaprimalandadualproblem.Dualityisnothingelsethanthespecification
ofoneproblemfromtwodifferentpointsofview.Onceaproblemisspecified,werefertoitastheprimal
problem. A second, associated problem very often exists and it is called the dual specification. In
managerial economics we are often more interested in the dual problem because it contains mostly
economicinformation.Sometimesamanagerislessconcernedabouttheprofitfromasolutionthanthe
use of available resources. This is because a manager has more control over the use of available
resourcescomparedwiththeprofits.Thedualsolutiongivesamanagertheinformationaboutthevalue
ofresources,andthisinturnisimportantwhenhehastodecideifmoreresourcesshouldbeallocated
andhowmuchtopayfortheseextraresources.
TheprofitofMortgageA=35andMortgageB=50appearascoefficientsintheobjectivefunctioninthe
primal problem. And in the dual problem they are constraints coefficients we seek to be as large as
possible.
Tofullyunderstandthedualityrelationsinlinearprogramming,itisimportanttoknowhowtosetup
dualpairsofproblemsproperlyandhowtointerpretalltheircomponentsineconomicterms.
30 9.Alinearprogrammingapplication:Investment
Wehaveintroducedthegeneralstructureofthelinearprogrammingproblemandcharacteristicsofthe
solution.Nowletustakealookataninvestmentproblem.
Acompanyhas12mill.€forinvestmentwithinthecompany.Therearefivepossibleprojectsunder
consideration.
Purchasing
Expected
return
Maximum
investment
(mill.of€)
Improvedmaterials‐handlingequipment
15%
3
Automatingpackagingoperations
10%
5
Purchasing raw materials in anticipation of price
increase
18%
6
8%
4
20%
1
Payingupoutstandingnotes
Additionalpromotionforanewproductline
There is no minimum investment required for any project. Project 1 and 2 are classified as capital
expenditureprojects,projects3and5arespeculativeinvestmentprojects,andproject4isafinancial
project.Thecompanyhasthefollowingrequirements:

Investmentincapitalexpendituresmustatleastbe40%ofthetotal

Theinvestmentinspeculativeprojectsmustbenomorethan50%oftheamountusedtopay
notes.
Decisionvariables
The decision variables must specify the amount of money invested in each of the alternatives. We
introduce6decisionvariablesthatspecifytheamountinvestedineachofthealternatives.
Let
x₁=investmentinImprovedmaterials‐handlingequipment
x₂=investmentinAutomatingpackagingoperations
x3=investmentinPurchasingrawmaterialsinanticipationofpriceincrease
x4=investmentinPayingupoutstandingnotes
x5=investmentinAdditionalpromotionforanewproductline
31 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
32 9.1
Themodel
Wesummarizetheportfolioselectionproblemasfollows.
Weintroducedecisionvariablesx₁,...,x5
MaxZ=0.15x₁+0.10x₂+0.18x₃+0.08x₄+0.20x₅
Subjecttoconstraints
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
33 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
3
Project2
3
Project3
1
Project4
4
Project5
1
Total
12
From the output we see that the shadow price on cash is equal to 0.1, this means a 10% return on
additionalcash;Theshadowpriceonmaxproject1isequalto0.05,thismeans5%additionalreturnfor
additionalinvestmentpermittedinproject1;project4hasshadowpriceequalto0.02,thismeans2%
foradditionalproject4funds;project5hasshadowpriceequalto0.02,thismeans2%foradditional
project 5 funds; notes has shadow price equal to 0.08, this means 8% additional return for each
percentagepointlessthan50%permitted.
34 10.Finalenote
Inthisnoteyouhavebeenintroducedtotheideaofbuildingmanagerialmodelsinascientificway.Most
oftheassumptionsareveryoftenrequirementsthatarenevermeetinpractice,butasanapproximation
ofanuncertainworldyouhaveatoolthatpointstowardsonefeasiblemanagerialdirection.Withthe
useofsensitivityanalysisseveralotherdirectionscanbeincludedinthemanagerialproposal.Andwith
thedualyouhavetheinformationofthevalueofresources.
35