The practical calculation of schedule variance | PMI

The cost performance index (CPI) is a measure of the conformance of the actual work completed (measured by its earned value) to the actual cost incurred: CPI = ... Learning Library Thepracticalcalculationofschedulevarianceintermsofschedule Tweet ConferencePaper Scheduling,CostManagement 19October2008 Warburton,RogerDavidHand|Kanabar,Vijay Howtocitethisarticle: Warburton,R.D.H.&Kanabar,V.(2008).Thepracticalcalculationofschedulevarianceintermsofschedule.PaperpresentedatPMI®GlobalCongress2008—NorthAmerica,Denver,CO.NewtownSquare,PA:ProjectManagementInstitute. Abstract Thevariationinaproject'sactualschedule,ascomparedtoitsplannedschedule,ismeasuredbyitsschedulevariance(SV),whichmeasuresthedifferencebetweentheearnedvalue(EV)(thevalueofworkactuallyperformed)andtheplannedvalue(PV),soSV=EV–PV.However,SVisexpressedasamonetaryunit(e.g.,indollars),whichmakesitdifficulttounderstandasavarianceintheschedule,whichshouldpresumablybemeasuredintimeunits,suchasdaysormonths.Severalauthorshaveproposeda“time-based”earnedschedule(ES),whichiseasiertointerpret. Unfortunately,therearenogenerallyavailableformulasforcalculatingES,whichmeansthatitisdifficulttocalculate.WeremedythisshortcomingbypresentingformulastocalculateES.Theformulasarerelativelystraightforward,andareillustratedbyapplicationsintwoindustries:constructionandsoftwaredevelopment.Theresultsandconclusionsfromthetwoindustriesarequitesimilar,andindicatethataslongasprojectlaborcurvesfollowthegeneral“S”shape,theformulasshouldhavewideapplicability. Thefirstusefulresultisthattheformulasshowhowthecostandschedulevariancesevolveovertime.Thatis,theydemonstratehowESperformsoverthelifeofaproject,andexplainitsbehavior.TheyalsoallowthecalculationofActualCosts,PlannedValue,andEarnedValueovertime.Therefore,onecanalsoeasilycalculatethecostperformanceindex,CPI(t),andscheduleperformanceindex,SPI(t),bothofwhicharealsofunctionsoftime. Theabilitytomakescheduleforecastswithoutperformingacompletebottom-upanalysishasbeenlongdesiredbyprojectmanagers.Thatdeficiencyiseliminated:projectmanagerscandevelopearlyscheduleestimates,andthentracktheschedule'sactualperformanceastheprojectevolves.Whilemuchworkremainstobedone,itappearsthattheformulasareausefulstartingpointandshowgreatpromise. Introduction Anbari(2003)describedthemajoraspectsoftheearnedvaluemethod(EVM),includingsomeextensionsnotfoundinthesimplifiedguidetoEVMterminologyprovidedinAGuidetotheProjectManagementBodyofKnowledge(PMBOK®Guide)(ProjectManagementInstitute,2000).EVMusescost,orsomeotherreasonablesubstitute,asthecommonmeasureofprojectperformanceforbothcostandscheduleparameters,andhaswideapplicabilitytobothpublicandprivatesectorprojects.Ontheotherhand,non-usersofEVMoftenindicatethatthemethodishardtouse(Fleming&Koppelman,2000;Kirn,2000). Thekeyconceptistheearnedvalue,EV,whichconvertsprojectaccomplishmentsfromphysicalunitsofmeasure(e.g.,milesofroadwayordeliverablescompleted)tofinancialunits(e.g.,dollarsorlaborhours).EVMalsodefinestheplannedvalue(PV,thetime-phasedbudgetbaseline)andtheactualcost(AC,thecumulativecostspenttoagivenpointintimetoaccomplishanactivity,workpackage,orproject).ACisdeterminedasthecosttoearntherelatedvalue. FromEVtoTV Aproject'scostperformanceismeasuredbycomparingEVtoAC,whilescheduleperformanceismeasuredbycomparingEVtoPV.Theschedulevariance,SV,isameasureoftheconformanceoftheactualprogresstotheplannedprogress:SV=EV–PV.AmajorcriticismofthestandardEVMisthattheschedulevarianceismeasuredincostunits,nottime.Thisissuehasbeenaddressedintwoways: ConvertingtheSVintotimeunits(Anbari,2003) Measuringthetimedelayonthecumulativecostcurve(Fleming&Koppelman,2000) ThefirstoftheaboveapproachesinvolvesdefiningtheaverageACpertimeperiod,calledthespendrate(ACRate),andtheaveragePVpertimeperiod,calledtheplannedvaluerate(PVRate).PVrateisdefinedasthebaselinebudgetatcompletion(BAC)dividedbythebaselinescheduleatcompletion(SAC).Thus,PVRate=BAC/SAC.TheusefulnessofPVRateisthatittranslatesSVintotimeunits.DividingSVbyPVRateconvertsSVintotimeunits,whichisreferredtoasTV,whereTV=SV/PVRate(Anbari,2003). Inthesecondapproach,TVismeasuredgraphicallybydrawingahorizontallinefromtheintersectionoftheEVcurvewiththestatusdatetothePVcurveandreadingthedistanceonthehorizontaltimeaxis(Fleming&Koppelman,2000).Bothoftheaboveapproacheshavethedesirableresultofdefiningtheschedulevarianceintimeunits,whichisamoreusefulcharacteristicofascheduleover-run. However,bothoftheaboveapproachessufferfromaseriousdrawback.Bothassumethatalloftheparametersareindependentoftime.Forexample,thePVRateiscalculatedasthetotalbudgetdividedbythetotalschedule(bothatcompletion),andisassumedtobeconstantoverthelifeoftheproject.WhenonedividesthecurrentSV(attime,t)bythePVRate,oneisassumingthattheaveragePVRateappliesforalltime. Theassumptionthattheparametersareindependentoftimeismanifestlynottrue.ThiscaneasilybeseenbyexaminingthebehaviorofSVattheendoftheproject.Asalloftheactivitiesarecompleted,thentheearnedvalueapproachestheplannedvalue.Moreprecisely,EV→PV,andSV=EV–PV→0.SVis,therefore,inherentlyafunctionoftime,andtoemphasizethefactwewilldenoteitasSV(t).Infact,itiseasytoseethatallquantitiesinEVMarefunctionsoftime. Acceptingthattheparametersarefunctionsoftimepresentsadilemma:Attheendoftheproject,thescheduleover-runisadefinitenumber,suchas6weeks,andismanifestlyaconstant.(Ofcourse,theschedulemayunder-run,butforsimplicitywewillrefertoanover-run.)Howdoweestimatetheconstant,eventual,finalscheduleover-runfromquantitiesthatvaryovertime?Anevenmoreintriguingquestionis,Canweestimatetheconstantvalueforthefinalscheduleover-runearlyintheproject?Thatis,givensomedataonCVandSV,canwepredicttheeventualfinalscheduleover-run?Finally,canweestimatetheover-runintimeunits? Forascheduleover-runpredictiontobereasonable,oneshouldexpectthatitsestimationremainconstantoverthelifeoftheproject.ThissuggeststhatTVshouldbeaconstant.Inwhichcase,PVRateshouldbeafunctionoftime,asindicatedinequation(1): KnowingthatSV(t)isinherentlyafunctionoftime,thenforTVtobeaconstant,PVRatemustalsobeafunctionoftime.Ifso,thenhowdoesonecalculatePVRate(t)?Ananalogousargumentshowsthatthecostvariancemustalsobeafunctionoftime,CV(t). Inthispaper,wewillmakesomeprogressinansweringthesequestionsbydevelopingamodelthatpredictsthebehavioroftheEVMparametersovertime.Thiscanevenbedoneearlyonintheproject'slife.Whatemergesisapredictionfortheeventualscheduleover-runfortheentireproject,andintimeunits. CPIandSPI Thecostperformanceindex(CPI)isameasureoftheconformanceoftheactualworkcompleted(measuredbyitsearnedvalue)totheactualcostincurred:CPI=EV/AC.Thescheduleperformanceindex(SPI)isameasureoftheconformanceofactualprogress(earnedvalue)totheplannedprogress:SPI=EV/PV.Inbothoftheaboveformulas,avalueof1.0indicatesthattheprojectperformanceisontarget.WhenCPIorSPIaregreaterthan1.0,thisindicatesbetter-than-plannedprojectperformance,whileCPIorSPIlessthan1.0indicatespoorer-than-plannedprojectperformance.TheformulasusedtocalculatetheCPIandSPIindicesaregenerallybasedoncumulativecosts. Theinverseoftheaboveformulasareusedinforecasting(Anbari,1980;Egan,1982;Cioffi,2002;Webster,2002).DividingtheforecastedbudgetbythecurrentCPIgivesapredictionofthefinalbudgetifperformancecontinuesatthesamerate.AsimilarcomputationcanbeperformedusingtheSPI. However,justastheschedulevarianceisafunctionoftime,SV(t),byanalogousreasoning,thescheduleperformanceindexmustbeafunctionoftimealso,SPI(t).Andofcourse,thisalsomeansthatthecostperformanceindexmustbeafunctionoftimeaswell,CPI(t). Graphsofthevariancesandperformanceindicesovertimeprovidevaluableinformationabouttrendsinprojectperformance.Whencorrectiveactionsareimplemented,thechangesinthebehavioroftheindexesrevealtheimpactofthechanges.Suchgraphscanbeveryeffectiveinprojectreviews.However,examiningandanalyzingthechangesintheparametersbegsthequestionofhowtheychangeovertime.Forexample,“WhatisthebaselinechangeinCPIovertime,andhowdoesitcomparetothemeasuredperformance?” CostandScheduleForecasting Theestimatedcosttocompletetheremainderoftheactivitiesforaprojectiscalledtheestimatetocomplete(ETC),whiletheestimateofthefinalcostatcompletioniscalledtheestimateatcompletion(EAC).MethodsforcalculatingEACdependontheassumptionsmadeaboutthefutureperformanceoftheprojectversusthehistorical,establishedperformancetodate.ThePMBOK®Guideprovidesthreeapproaches,basedonthreedifferentsetsofassumptions: Whentheassumptionsunderlyingtheoriginalestimateareflawed Whenpastperformanceisnotagoodpredictoroffutureperformance Whenpastperformanceisagoodpredictoroffutureperformance ManyformulashavebeenproposedtocalculatetheEAC,andunderavarietyofassumptions(Anbari,2003;Fleming&Koppelman,2000;Kerzner,2006).However,accordingtoAnbari(2003),“EVMhasnotbeenwidelyusedtoestimatethetotaltimeatcompletion,totalprojectduration,orschedule…basedonactualperformanceuptoagivenpointintheproject.”Usingreasonableassumptions,Anbariprovidedformulasfortheproject'stimeestimateatcompletion(TEAC)andtimevarianceatcompletion(TVAC),basedonthebaselinescheduleatcompletion(SAC)andtheactualperformanceuptoagivenpointintheproject(Anbari,2001,2002). ThedifficultywithalloftheaboveformulasisthattheyassumethatthevaluesforCPIandSPIareconstant.Evenifoneassumesthatthecurrentprojectperformanceisanexcellentpredictoroffutureperformance(assumption#3),onestillneedstoassumethat:1)thecurrentvaluesofCPIandSPIareconstant;and2)thattheyarerepresentativemeasuresoftheentirefutureperformanceoftheproject.WhengraphsofCPIandSPIarechangingovertime,whichistheusualcase,thecriticalquestionbecomes“HowdoCPIandSPIevolveovertime?” ThisimportanceofthisquestioncanbeseenbyreviewingthebehavioroftheSPI.AsnotedbyFlemingandKoppelman(2000)andKerzner(2006),attheendoftheproject,theSPIalwaysapproaches1.0.Thisisthesimpleresultofthecompletionofallproposeddeliverables;thatis,aseachactivitycompletes,theearnedvaluebecomesequaltotheplannedvalue.Thisistrueeveniftheprojectislate,inwhichcasetheSPIstillapproaches1.0,butaftertheplannedcompletiondate.WewouldliketoknowhowlatetheprojectisgoingtobebydecodingthebehaviorofSPI(t)overtime. ConstructionLaborCurves Wenowturntoanalyzinglaborcurvesthataretypicallyappliedintheconstructionindustry.Wideman(1994,2001,2004)providedlaborcurvesforaprofitablecivilcontractwhichwaspredominantlyformworkandconcreteplacing.ThedataisredrawninExhibit1,whichshowsahistogramoftheproductionworkforceoverthe38-weekprojectduration.Thereisaninitialperiodofbuild-up,aperiodofpeakloading,followedbyaperiodofprogressivedemobilizing. Exhibit1–Actual(Histogram)andModel(Trapezoidal)LaborCurvesforaConstructionProject Allen(1979)suggestedthatasimpletrapezoidalfigurecanbeusedasanexcellentapproximationtotheactuallaborloading.Thetrapezoidalprofile,whichisalsoshowninExhibit1,consistsofalinearramp-uptoapeakafter50%oftheschedule,aconstantpeakloadendingafter75%oftheschedule,andaramp-downtotheend.InExhibit1,theagreementbetweentheactualprojectdataandtheapproximateloadingisquitestriking.Infact,theestimateofthetotallaborfromthetrapezoidalcurveiswithin2%oftheactualvalue. Warburton(2008)developedamodelusingtheequationsforthetrapezoidalcurve,whichisdenotedasc(t).Thepeakvalueofthelaborcurveis,P,andtheprojectendsattime,Te.Theramp-upspanstheinterval[0,Te/2].UsingAllen'srule#5,(thepeaklaborloadingoccursfrom50%to75%oftheproject),theconstantsectionspanstheinterval[Te/2,3Te/4].Thus,theequationforthetrapezoidalcurveis(Warburton,2008): PlannedValue Theinstantaneousrateatwhichactivitiesareplannedtobecompletedisdefinedastheinstantaneousplannedvalue,PVI(t).Asactivitiesarestaffed,thereisnoguaranteethattheyarecompletedontime,andsoequation(3)representstheplannedcompletionofactivities.Therefore,thecurveforPVI(t)followsthesamecurveastheinstantaneouslaborrate,c(t).Traditionally,thePlannedValue,PV(t),isdefinedasthecumulativesumoftheprevious,instantaneousPVI(t),andsoisdefinedas: TheinstantaneousandcumulativeplannedvaluesareplottedinExhibit2(thedottedlines).Thetraditionalcumulativeversion(Exhibit2b)showsthetypical“S”curveassociatedwithcumulativecosts.AccordingtoinvestigationsbySinghandLakanathan(1992),theapplicationof“Scurves”forcashflowprojectionscanachieveaccuraciesofapproximately88–97%,andtheshapeoftheS-curvebudgetversustimeisaquickwaytojudgeperformance.Thisispreciselytheobjectiveofmacroestimationmethods:anearlyestimationofprojectperformancefromoverallsystemparameters. Exhibit2–PlannedValue(dottedline)andActualCost(solidline)forTrapezoidalLaborCurvea)InstantaneousLabor                                              b)CumulativeLabor TheWarburtonmodelassumesthatastheprojectproceedsnotallactivitieswillbecompletedontime,andoneassumesthatafractionoftheactivitiesthataresupposedtobecompletedattime,t,arerejectedforsomereasonandrequireextrawork.Projectmanagersshouldbeabletoestimatethisparameterearlyonintheprojectastheinitialactivitiesarecompletedandanyover-runsnoted.Softwareprojectdatasuggeststhaterrorratesremainconstantoverthelifeofaproject,sotheavailabilityofanearlyestimateoftherejectionrateisareasonableassumption. Thecompletionofrejectedactivitieswillbedelayed,anditisassumedthatthisdelayisaconstantamount,τ.Thatis,eachactivitythatisrejectedisdelayedthesameamount.Unliketherejectionrate,thereislittledataontheaveragedelayexperiencedbyindividualactivities.Therefore,thereasonablenessofthisassumptionneedstobeevaluatedbycomparingrealprojectdatawiththemodel. ActualCost TheinstantaneousActualCost,ACI(t),includestheworkperformedonboththeactivitiesthatweresuccessfullycompletedandthosethatwererejected.Attime,t,theactivitiesthatwererejectedattimet–τwillbesuccessfullycompleted.Therefore,theinstantaneousActualCostattime,t,is: Theparameter,α,representstheaverageproductoftherejectionrateandtheworkrequiredtofixtheproblem.Weseefromequation(5)thatαisthefractionalextraworkrequiredforeachplannedactivity.Usingthismodel,WarburtoncalculatesthecumulativeActualCost,AC(t),byintegrating,exactlyasinequation(4).ThecumulativeActualCostasafunctionoftimeisshowninExhibit2(solidline).TheActualCostincreasesasthecostoftherejectedactivitiesaccumulates.TheActualCostast→∞representsthetotalcostoftheproject,whichis(Warburton,2008): Thisisreasonable,becauseitsaysthatwhenthecostover-runineachactivityisonaverageafixedpercentage,theendresultisacostover-runfortheentireprojectbythesamepercentage.Itisinterestingtonotethatinthismodel,thetotalprojectcostdoesnotdependonthetimedelay,τ,butonlyonthefractionofactivitiesthatwererejected. EarnedValue TheEarnedValueisthevalue,orcost,ofthesuccessfullycompletedactivities.Ineachinterval,afraction,α,oftheactivitieswererejected,sotheremainingactivities(1–α)werecompletedandearnvalue.Also,previouslyrejectedactivitiesthatarecompletedinthisintervalalsoearnvalue.Therefore,theinstantaneousEarnedValueis: Exhibit3ashowstheinstantaneousEarnedValue(solidline)asafunctionoftimeascomparedtothePlannedValue(dottedline).TheEVisinitiallydelayedrelativetotheplannedvalue,buteventuallycatchesup.Theearnedvalueisslightlydelayedattheend,representingascheduleover-run. Exhibit3–PlannedValue(dotted)andEarnedValue(solid)forTrapezoidalLaborCurve                   a)Instantaneous                                                              b)Cumulative ThecumulativeEarnedValue(EV)isfoundbythesameprocessasforthecumulativeActualCost—integrating.Exhibit3bshowsthecumulativeEarnedValueasafunctionoftime(solidline)ascomparedtothecumulativePlannedValue(dottedline).Bothcurvesapproachthesamevalueast→∞becausethetotalnumberofactivitiesintheprojectremainsthesame.Theearnedvalueisdelayedbecausesomeoftheactivitieswererejectedandcreditfortheirworkwasonlyearnedwhentheywerefinallycompletedafterthedelaytime. Ifextraactivitieshadbeenadded(scopecreepoccurred),thentheearnedvaluewouldapproachahigherlevelthantheplannedvalue.Sincethenumberofactivitieshasnotchanged,thetotalearnedvaluemustbethesameasthetotalplannedvalue.ThisisconfirmedintheWarburtonmodelbythefollowingrelations,whichshowthePlannedandEarnedValuesast→∞: CPI(t)andSPI(t) TheCostPerformanceIndex(CPI)isdefinedastheratioofEarnedValuetoActualCost,whiletheSchedulePerformanceIndex(SPI)isdefinedastheratioofcumulativeEarnedValuetocumulativePlannedValue(PMI,2000).BothCPIandSPIaretraditionallydefinedintermsofthecumulativevalues.However,fromequations(3)–(5),onecanimmediatelyseethatthesequantitiesareafunctionoftime,andso: ThebehaviorofCPI(t)andSPI(t)areshowninExhibit4.Theperformanceovertimeofthetwoindexesisdifferentandquiteinteresting.CPI(t)immediatelyfallsto1–αbecauseinthefirsttimeinterval,someactivitiesarerejected.Astherejectedactivitiesarecompletedaftertime,t=τ,creditisearned,andtheCPIrisesslightly,butitremainslowovertheentirelifeoftheproject.Attheendoftheproject,CPI(t)doesnotapproach1.0,itapproachesthevalueshowninequation(10),whichdependsdirectlyontherejectionrate. Exhibit4–CumulativeCPI(t)andSPI(t)asFunctionsofTime However,thebehaviorofSPI(t)isquitedifferent.Italsofallsimmediatelyto1–α,butiteventuallyclimbsbackto1.0attheendoftheproject,asitshould.However,SPI(t)reachesthevalue1.0aftertheprojectedscheduledcompletion.Theactualvaluesattheendoftheprojectare: AcriticismoftheuseofSPIisthatsinceitapproaches1.0attheendoftheproject,itisnotusefuloverthelastthirdoftheproject(Corovic,2007;Fleming&Koppelman,2003;Lipke,2003,2004;Henderson2004;Vandevoorde&Vanhoucke,2006).However,Exhibit4showspreciselyhowandwhenSPI(t)approaches1.0.Knowingtheprecisetime-dependentbehaviorofSPI(t)somewhatbluntsthiscriticism,becausewiththeabovemodelonecancomparetheactualperformanceofSPI(t)totheprojectedperformance.ThecriticismthatthevalueofSPI(t)isinmonetaryunitsisstillvalid,andwewilladdressthislater. SoftwareLaborCurves Putnam(1978)pioneeredtheuseoftheNorden-Rayleighcurvetodescribethenumberofpeopleworkingoncomplexsoftwareprojects.ThePutnam-Norden-Rayleighcurve,nowknownasPNR,appearstoapplytomanytypesofsoftwareprojects,particularlyembeddedsoftwaresystems(Warburton,1983).Thenumberofpeopleworkingonaprojectasafunctionoftime,m(t),isgivenby: Tisaconstantthatdenotesthetimeatwhichthenumberofpeopleisatthemaximum—thelaborpeak.Kisaconstantthatcanbedeterminedbytheconditionthatthetotalcostoftheproject,i.e.,thetotalnumberofman-yearsis,N.AcomparisonofthePNRandtrapezoidallaborcurvesisshowninExhibit5.TheinstantaneouslaborratesareshowninExhibit5a,whilethecumulativevaluesareshowninExhibit5b,bothofwhichshowthetypical“S”shape.TheWarburton(2008)modelcanbeappliedtothePNRcurveexactlyaswasdoneforthetrapezoidallaborcurve.Theplannedvalue,actualcostandearnedvaluearecomputedusingintegralsasbefore,justreplacingthec(t)ofthetrapezoidalcurvewithm(t)fromequation(11). Exhibit5–ComparisonofTrapezoidalandPNRLaborCurvesa)InstantaneousValues                     b)CumulativeValues Exhibit6comparesthecurvesthatresultforCPI(t)andSPI(t)fromthetwomodels.Theparametershavebeenselectedsothatbothmodelshavethesametotalcostandthesamelaborpeak.Despitetheapparentlyquitedifferentlaborcurves,thebehaviorofCPI(t)andCPI(t)overtimeisremarkablysimilarinthetwocases—seeExhibit6. Exhibit6–ComparisonofCPI(t)andSPI(t)forTrapezoidalandPNRLaborCurvesa)TrapezoidalLabor                                       b)PNRLabor ChristianandKallouris(1991)establishedthatformostprojectsthetypicalcumulativelaborprofileisan“S”curve.ThissuggestsanintriguingandpotentiallysignificantpropertyoftheWarburtonmodel: ThebehavioroftheCPI(t)andSPI(t)curvesovertimeformostprojectsshouldbesimilartothoseshowninExhibit6. Therefore,aslongasaproject'slaborcurveistheusual“S”shape,theconclusionisthattheshapeoftheCPI(t)andSPI(t)curveswillfollowthatshowninExhibit6. EarlyEstimationofCostandScheduleOver-runs Wenowturntoestimatingintheearlystagesofaproject,thefinalvaluesoftheproject'scostandscheduleover-runs.Intheearlystagesofaproject,thelaborcurveislinear,andgivenbythefirstpartofequation(3).Inthisregion,theplannedvalue,actualcost,andearnedvalueareeasilycomputed,andareallfunctionsoftime: CalculationoftheCostVariance,CV(t) UsingthestandarddefinitionofCV(t),wehavefromequation(14)and(13): Rearrangingequation(15)givesthevalueforthecostover-runparameter,α,whichisaconstant: Tocalculateα,weneedthevaluesforPandTe.ThisratiocanbefoundbynotingthattheslopeofthePlannedValuelaborcurveis2P/Te.Therefore,intheearlystagesofaprojectwecanestimatetheslopeoftheplannedvaluecurve,whichwedenoteasSPVinequation(16).Sowecancalculateαfromthetimedependentcostvariance,CV(t),accordingtoequation(16).Noticethatthedelayparameter,τ,doesnotoccurhere,andsowecanuseequation(16)tocalculateα.Weemphasizethatαisaconstant,andisaglobalpropertyoftheentireproject. CalculationoftheScheduleVariance,SV(t) UsingthestandarddefinitionofSV(t),wehavefromequations(14)and(12): Forconvenience,wehavedefinedtheconstant,Q,whichdependsonα,whichweknowfromequation(16).Thequadraticinequation(18)hasastraightforwardsolution: Therefore,wecancalculatethescheduleover-run,τ,fromSV(t).Weemphasizeagainthatτisaconstant,andisthepredictionofthetotalscheduledelayfortheproject;thatis,itisaglobalpropertyoftheentireprojectanddoesnotdependontime. Intherealworld,therewillofcoursebenoiseinthedata.However,ifwecanobtainreasonablevaluesforCVI(t)andSV(t),thenwecanestimatevaluesforαandτ.Thenwecanestimatethecostandscheduleover-runsfortheproject.Bothoftheseestimateswillremainconstantovertime,andsoaregenuinelyusefulpredictionsofthecostandscheduleover-runs.Further,thescheduleover-runparameterisintimeunits. AnExampleoftheEstimationofCPI(t)andSPI(t) Wenowshowhowthemodelisusedinpractice.Intheearlystagesofaproject,theplannedvaluecurveisknown.Astheprojectproceeds,anddeliverablesareaccumulated,onecandeterminetheactualcostandassessthecorrespondingearnedvalue.Exhibit7shows3samplesofsuchaprocess,attimes5,7,and9. Exhibit7–ComparisonofPlannedValue(SolidLine)WithSamplesofEarnedValues(Crosses)andActualCosts(Circles)a)InstantaneousValues                                                           b)CumulativeValues InExhibit7a,wehaveusedinstantaneousvalues,whileinExhibit7bwehaveplottedthesamedatausingcumulativevalues.OneofthecharacteristicsthatisimmediatelyobviousfromExhibit7isthatthecalculationismucheasiertoaccomplishusingtheinstantaneousvalues.Anumberoffeaturesoftheinstantaneousrepresentationmakeitmorepracticaltouse:thelinearityofthecurvesmakethedataeasiertoanalyze—itisalwayshardertoestimatequantitiesfromcurves;andthegreaterdeviationsoftheactualcostsandearnedvaluesonthelinearchartmakethemeasiertorecognizeandcompute. Oncewehavethevaluesfortheplannedcosts,actualcosts,andearnedvalues,itisstraightforwardtocomputethevaluesforCV(t)andSV(t)ateachofthepointsinExhibit7.Then,usingtheslopeoftheplannedvaluecurve,SPV,weuseequation(16)tocalculateα,andequation(19)tocalculateτ.Atthispoint,wehaveestimatesfortheeventualtotalcostover-runandthetotalscheduledelay. EstimatingCPI(t)andSPI(t) Exhibit8showstheestimatesforthevaluesofCPI(t)andSPI(t)correspondingtothesamepointsasinExhibit7.ThedottedlinesshowhowCPI(t)andSPI(t)changeovertime,evenwhentheover-runinbothcostandscheduleareinfactconstant.Onecanseethatbasedonthethreeestimatedvalues,itmightbehardtodecidewhattheoverallCPIandSPIfortheprojectmightbe.Theinherentcurvatureofthelineswouldmakeitdifficultpredictthefuturevaluesofthesequantities.OnlythroughtheknowledgeofthemodelpresentedheredoweunderstandthatthecurvesinCPI(t)andSPI(t)doinfactrepresentconstantvaluesfortheover-runinbothcostandschedule. LookingatExhibit8,oneseesthatthethreeestimatesforCPI(t)appeartoberising.Aninexperiencedprojectmanagermightinterpretthistomeanthatthingsareimproving.Infact,nothinghaschanged,andtheprojecthasanover-runinbothcostandschedule.The“apparent”slightimprovementintime3–9isduetothefactthatsomeactivitieswerenotcompletedduringthetime0–3.Thesehavenowbeencompletedandcreditisbeingearnedforthem.However,aconstantfractionofactivitiesareinfactdelayedineachtimeinterval,andCPI(t)willsoonleveloffasseeninExhibit6. Exhibit8–EstimationofCPI(t)andSPI(t) Ontheotherhand,SPI(t)risessteadily,eventuallyreaching1.0.However,asseeninExhibit6,SPI(t)willonlyreach1.0(denotingthecompletionofallactivities)afterthescheduledcompletiontime.Theprojectisrunninglate. Conclusions EVMprovidesprojectmanagerswithtriggersorearlywarningsignalsofprojecttrouble.Suchindicatorshavebeenfoundtobereliableasearlyas15%intoaproject.Betterplanningandresourceallocationassociatedwiththeearlyperiodsofaprojectmightbethecauseofthisreliability(Fleming&Koppelman,2000). However,thesewarningsignsaretime-dependent,andmustbeinterpretedwithgreatcare.Untilnow,therewasnowaytodeterminethistimedependence.TheextensionoftheWarburtonmodelprovidedhereestablishesthatthebehaviorofCPI(t)andSPI(t)overtimecanbecalculated.Laborcurvesintwodiverseindustries(constructionandsoftware)produceverysimilarcurvesforCPI(t)andSPI(t),leadingtothetentativeconclusionthatCPI(t)andSPI(t)arerelativelyindependentofthepreciseformoftheproject'slaborcurve.ThisagreeswithChristianandKallouris's(1991)observationthatmostprojectcurvesfollow“S”shapes.Thissuggeststhatthemodelhaswideapplicability. Exhibit8showsthatestimatesofCPI(t)earlyoninaprojectareinherentlyafunctionoftime.Aninexperiencedprojectmanagermightinterpretthistomeanthatwhiletheprojectdidnotstartwell,theriseinCPI(t)indicatesthatimprovementshavebeenmade.Infact,nothinghaschanged,andtheprojecthasanover-runinbothcostandschedule.Exhibit8alsoshowsthatSPI(t)isrisingsteadily,whichagainmightbeinterpretedasanimprovementintheproject'sprogress.Thisisnotso,astheprojectisrunninglate.Exhibit8showsthatinterpretingmeasuredvaluesforCPI(t)andSPI(t)istrickyunlessoneappreciatestheirinherenttime-dependentbehavior. TheWarburtonmodelrequirestwoconstants:thecostover-runrateandthetimedelaytorepairtherejectedactivities.Becausenotallactivitieswillbecompletedontime,aconstantfractionofthemareassumedtoberejectedandrequireextrawork.Byplottingtheinstantaneouslaborcurve,projectmanagersshouldbeabletoestimatetherequiredparametersearlyonintheproject,asthefirstfewactivitiesarecompleted. Morecomplexissuesneedtobeincludedinthemodelandaretopicsforfutureresearch.Themostinterestingissueisthatof“scopecreep,”whichincreasesthenumberofactivitiesandsoincreasesboththeactualcostandtheearnedvalue,whichinturnaffectsbothCPI(t)andSPI(t).Also,themodelassumesthatactivitiesareindependent,whichisclearlynottrue,asthecriticalpathdependsontheconnectionbetweenactivities.Thismightbeaddressedbyassumingthatthedelayinoneactivityresultsinthedelayofotheractivitiesfurtherdownthepath.Thisshouldmagnifytheeffectofthescheduledelayandwillpresumablymakethemodelmoresensitivetothescheduledelayparameter. Aprojectmanager'stoolkitshouldincludeamethodfordetermininghowparticularvaluesforCPI(t)andSPI(t),measuredatsomedefinitepointintime,canbeusedtoestimatetheeventualcostover-runandthescheduledelay.DespitethesimplicityoftheWarburtonmodel,anumberofinterestingfeaturesemerged.Whilemuchworkremainstobedone,itappearsthatthemodelisusefulandshowssomepromise. 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Wideman,M.(2001).Whatcanbelearnedofpracticalvalue?RetrievedNovember23,2007fromhttp://www.maxwideman.com/papers/resource/learned.htm Thismaterialhasbeenreproducedwiththepermissionofthecopyrightowner.Unauthorizedreproductionofthismaterialisstrictlyprohibited.Forpermissiontoreproducethismaterial,pleasecontactPMIoranylistedauthor. ©RogerD.H.Warburton,PhD,PMP&VijayKanabar,PhD,PMPOriginallypublishedaspartofPMIGlobalCongressProceedings2008-DenverColorado Advertisement Advertisement RelatedContent Article Innovation,Scheduling,Strategy,CostManagement,OrganizationalProjectManagement,Construction 1April2019 ProjectManagementJournal DeterminingContingenciesintheManagementofConstructionProjects ByOrtiz,JoséI.|Pellicer,Eugenio|Molenaar,KeithR. 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