Multiobjective strategies for New Product Development in the pharmaceutical industry

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Multiobjective strategies for New Product Development

in the pharmaceutical industry

José Luis Pérez-Escobedo, Catherine Azzaro-Pantel, Luc Pibouleau

To cite this version:

José Luis Pérez-Escobedo, Catherine Azzaro-Pantel, Luc Pibouleau. Multiobjective strategies for New

Product Development in the pharmaceutical industry. Computers & Chemical Engineering, Elsevier,

2012, vol. 37, pp. 278-296. �10.1016/j.compchemeng.2011.10.004�. �hal-00878406�

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Catherine and Pibouleau, Luc Multiobjective strategies for New

Product Development in the pharmaceutical industry. (2012)

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Multiobjective

strategies

for

New

Product

Development

in

the

pharmaceutical

industry

José

L.

Perez-Escobedo,

Catherine

Azzaro-Pantel

_,

_Luc

_Pibouleau

UniversitédeToulouse,LaboratoiredeGénieChimique,UMR5503CNRS,ENSIACETINPT,4,AlléeEmileMonso,BP84234-31432ToulouseCedex4,France

Keywords:

NewProductDevelopment Portfoliomanagement Discreteeventsimulation Optimization

Multicriteriageneticalgorithm

a

b

s

t

r

a

c

t

NewProductDevelopment(NPD)constitutesachallengingprobleminthepharmaceuticalindustry,due tothecharacteristicsofthedevelopmentpipeline.Formally,theNPDproblemcanbestatedasfollows: selectasetofR&Dprojectsfromapoolofcandidateprojectsinordertosatisfyseveralcriteria(economic profitability,timetomarket)whilecopingwiththeuncertainnatureoftheprojects.Moreprecisely,the recurrentkeyissuesaretodeterminetheprojectstodeveloponcetargetmoleculeshavebeenidentified, theirorderandthelevelofresourcestoassign.Inthiscontext,theproposedapproachcombinesdiscrete eventstochasticsimulation(MonteCarloapproach)withmultiobjectivegeneticalgorithms(NSGAIItype, Non-SortedGeneticAlgorithmII)tooptimizethehighlycombinatorialportfoliomanagementproblem.In thatcontext,GeneticAlgorithms(GAs)areparticularlyattractivefortreatingthiskindofproblem,dueto theirabilitytodirectlyleadtotheso-calledParetofrontandtoaccountforthecombinatorialaspect.This workisillustratedwithastudycaseinvolvingnineinterdependentnewproductcandidatestargeting threediseases.Ananalysisisperformedforthistestbenchonthedifferentpairsofcriteriabothforthe bi-andtricriteriaoptimization:largeportfolioscauseresourcequeuesanddelaystimetolaunchandare eliminatedbythebi-andtricriteriaoptimizationstrategy.Theoptimizationstrategyisthusinteresting todetectthesequencecandidates.TimeisanimportantcriteriontoconsidersimultaneouslywithNPV andriskcriteria.Theorderinwhichdrugsarereleasedinthepipelineisofgreatimportanceaswith schedulingproblems.

1. Introduction

Traditionally,ProcessSystemsEngineering(PSE)isconcerned withtheunderstandinganddevelopmentofsystematicprocedures forthedesignandoperationofchemicalprocesssystems,ranging frommicrosystemstoindustrialscalecontinuousandbatch pro-cesses.ThistraditionaldefinitionofPSEhasbeenbroadenedbythe conceptofthe“chemicalsupplychain”.ProcessSystems Engineer-ingisnowconcernedwiththeimprovementofdecisionmaking processesforthedesignandoperationofthechemicalsupplychain. Moreprecisely,itdealswiththediscovery,design,manufactureand distributionofchemicalproductsinthecontextofmanyconflicting goals.TheareaofR&DandProcessOperationshasemergedamong themajorchallengesinthePSEarea:thistopics,whichhasashorter historythanprocessdesignandcontrol,expandsupstreamtoR&D anddownstreamtologisticsandproductdistributionactivities.

Inthatcontext,optimalplanningandschedulingforNew Prod-uctDevelopment (NPD) need increasedattentionto coordinate betterproductdiscovery,processdevelopmentandplantdesignin

∗ Correspondingauthor.

E-mailaddress:[email protected](C.Azzaro-Pantel).

theagrochemicalandpharmaceuticalindustries.Fordownstream applications,areasthatreceiveincreasedattentionatthebusiness level include planningof process networks, supply chain opti-mization,realtimescheduling,andinventorycontrol.Duetothe pressure forreducing costsand inventories,in orderto remain competitive, enterprise-wideoptimization(EWO)thatmight be consideredasanequivalenttermfordescribingthechemical sup-plychain(Shapiro,2001)hasthusbecomeacornerstoneinprocess industries.

Enterprise-wideoptimizationisanareathatliesattheinterface ofProcessSystemsEngineeringandOperationsResearch.As out-linedinGrossmann(2005),anewgenerationofmethodsandtools thatallowthefullintegrationandlarge-scalesolutionofthe opti-mizationmodels,aswellastheincorporationofaccuratemodels forthemanufacturingfacilitiesisneeded.Giventhestrong tradi-tionthatchemicalengineershaveinprocesssystemsengineering andintheoptimizationarea(seeBiegler&Grossmann,2004fora review),theyareideallypositionedtomakesignificant contribu-tionsinEWO.

Thedevelopmentofdecisionsupportstrategiesandsystemsfor managingnewproductportfoliosmustbeabletoprovideinsights tomanagersonhowtominimizeriskwhileoptimizingan objec-tiveorasetofobjectives(e.g.maximizationofexpectednetpresent

value,minimizationoftimetomarket,etc.)inthepresenceof con-straints.Moreover,thesimultaneousconsiderationofallcandidate projectsisthekeyaspectinmanagingaNPDpipeline.This com-plexityhasledtothecommonuseofdecompositionbasedineither strategicoroperationalstrategies.Eachofthetwobranchescanbe furthersubdividedaccordingtothecharacteristicsofthemodel usedtosupportthedecisionmakingprocess.

Aninterestingcontribution(Zapata,Varma,&Reklaitis,2007)

proposesarecentstate-of-theartoftheconcernedproblem.The mainguidelinesoftheiranalysisarebrieflyrecalledtoposition ourwork.First,aprojectcanbeanalyzedinisolation(e.g.thenet presentvalue(NPV)oftheproject),orasperformanceassessment attheportfoliolevel(e.g.NPVoftheportfolio),includingallthe interactionsbetweenprojects.Thetimedimensiondistinguishes dynamic and staticapproaches. A dynamic model providesthe specificstateofthesystemsalong each pointof thetime hori-zon(e.g.numberofprojectswaitingforagivenresourceatagiven time),whileastaticoneusesaveragevaluestorepresentthe sys-tem(e.g.averagenumberofprojectswaitingforagivenresource atanytime).Itisthenpossibletochoosebetweendeterministic andstochasticmodels.However,dynamicstochasticmodelscan beviewedaseitheropenlooporclosedlooporiented.Openloop modelsonlycapturetheresponseofthesystemtoinputsfrom deci-sionmakers,whileclosedloopmodelsalsocapturetheresponseof thedecisionmakerstotheoutcomesfromthesystem.

Amongtheinvestigationsdedicatedtostrategicdecision sup-port systems,the differenttechniques availabledepend onthe typeofdataused,namely,qualitativeandquantitative.Ontheone hand,themethodologiesrelativetostaticstrategies are numer-ousinthisarea:itmustbeemphasizedthatamajordrawbackof suchapproachesisthattheydonottakeintoaccountproject inter-actions.Theyincludescoringmethods(Coldrick,Longhurst,Ivey, &Hannis,2005;Cooper,Edgett,&Kleinschmidt,1999),analytical hierarchyapproaches(Calantone,Benedetto,&Schmidt,1999;Poh, Ang,&Bai,2001)andfuzzylogicbasedapproaches(Buyukozkan &Feyzioglu,2004; Lin &Hsieh,2004;Lin, Tan,&Hsieh,2005). Ontheotherhand,themethodologiesthatarebasedon quanti-tativeinformation strivetoprovidearealistic simulationofthe behaviourofeachindividualprojectalongthetimehorizon con-sidered,inordertodeterminewhatthepossibleoutcomesarein termsofrewardsandrisk.Thisgroupincludesdynamic determin-isticstrategiessuchasclassicalfinancialmodels(e.g.NPV,internal rateofreturn,etc.)(Cooperetal.,1999),aswellasdynamic stochas-ticstrategies,bothclosedloopsuchasrealoptions(Copeland& Antikarov,2001;Jacob&Kwak,2003;Loch&Bode-Greuel,2001; Newton,Paxson,&Widdicks,2004;Santiago&Bifano,2005),and openloopsuchasdiscrete eventsimulation(Chapman&Ward, 2002),andneuralnetworks(Thieme,Song,&Calantone,2000).

Mostoftheapproaches thatcaptureprojectinteractionscan beclassifiedasdynamicstochasticopenloopmethodologies.An importantcontribution is theworkof Blau, Pekny,Varma, and Bunch(2004)whichproposestheuseofstochasticoptimization: theportfolioismodelledusingadiscreteeventsimulationandthe optimizationisimplementedbyageneticalgorithm;Rogers,Gupta, andMaranas(2002)formulatearealoptionsdecisiontreethat cap-turestechnicalandmarketuncertaintyasastochasticMILP(Mixed IntegerLinearProgramming)thatrelatesprojectsthroughabudget constraint.Rajapakse,Titchener-Hooker,andFarid(2005)present adecisionsupporttoolthatusessensitivityandscenario analy-sisonadiscreteeventmodelofthedevelopmentpipeline.Finally,

DingandEliashberg(2002)approachtheproblemofdetermining howmanyprojects,thatareassignedtodevelopthesameproduct, havetobeincludedinthepipelinetomaximizethetotalexpected profit.Allofthetechniquesinthisgrouparemainlyfocusedon timeindependentdecisions(excludingtheworkreportedinRogers etal.,2002)andthereforedonotrequireclosedloopmodels.Some

workhasbeendonetoaccommodatethehigherlevelof complex-ityrequiredbytimedependentstrategicdecisionssuchascapacity expansion/contraction(Wan,Pekny,&Reklaitis,2006).Itmustbe yetpointedoutthatthenon-Markoviannatureoftheassociated decisionproblemhasyetlimitedthesizeofthetreatedproblemas expressedinprojectnumber.

At the operational level, decisions are time dependent and mostlyMarkovianinnature.Thishasmotivatedthedevelopmentof operationaldecisionsupportsystemsexclusivelybasedon quanti-tativeinformationandwithadynamiccharacter(Honkomp,1998; Jain&Grossmann,1999; Subramanian,Pekny,Reklaitis,&Blau, 2001;Varma,2005).

Thisliteraturereviewrevealsthatitisdifficulttoembedallthe peculiaritiesoftheprobleminagenericformulationandto recon-cilealllevelsattheinvolvedscales.Thecomplexityoftheproblem is attributed toseveral combinedissues such as thestochastic behaviour of the system, the combinatorial aspect and conse-quently thesize of industrial problems as well as theinduced multilevel approach. Some recent works (Colvin & Maravelias, 2008,2009)aretryingtoreconcilebothstrategicandoperational levels,namely, theschedulingofclinicaltrialsandtheplanning of theresources necessary tocarry these trials out.A stochas-tic programming framework that addresses the two problems simultaneouslyisproposedinColvinandMaravelias(2009).The underlyingphilosophyimpliesthreelevels:first,thestructureof theproblemisstudiedinordertoreducethenumberofpairsof sce-narios;second,afinite-horizonapproximationisdevelopedsothat problemscanbeformulatedusingfewerstageswithout compro-misingthequalityofthesolution;third,thesequentialnatureofthe testingprocessisconsideredandmodelledwithamixed-integer programming(MIP)formulation;arelaxationofthisformulationis thenusedtoobtainfeasibleandmostoftenoptimalsolutionsover thestages of interest.Finally, a rolling-horizon-based approach isimplemented,wherethedecisionsoftherelaxedproblemare usedoverfewearlyperiodsandanewproblemisformulatedand solvedas time evolves.Thisframework wasrecently improved including:(i)theselectionandschedulingofR&Dtaskswith gen-eralprecedenceconstraintsunderpass/fail uncertainty, and (ii) resourceplanningdecisions(expansion/contractionand outsourc-ing).Furthermore,interdependenciesbetweentasksin termsof probabilityofsuccess,resourceusageandmarketimpactare con-sidered withrisk management approaches, taking intoaccount conditionalvalueatrisk(Colvin&Maravelias,2011),thatwasnever consideredinpreviousworks.Itmustbealsoemphasizedthatall thereportedapproachesarebasedonamonoobjective optimiza-tionformulationeveniftheproblemismultiobjectivebynature.

Thisworkisdevotedtothedevelopmentofadynamicstochastic open loop methodology and involves a bi-and tricriteria opti-mizationformulationoftheNPDproblem.Itinvolvesmulti-stage decisionsunderuncertainty.Therecurrentkeyissuesarecanbe statedas follows:what aretheprojects todevelop oncetarget moleculeshavebeenidentified?Inwhatorder?Whichisthelevel ofresourcestoassign?Theproposedmodellingapproachisbased onadiscreteeventsimulatorwhichisparticularlyusefulfor deci-sioncriteriaevaluation,suchaseconomic andriskmetrics.This workcanbeviewedasanextensionoftheinvestigations previ-ouslydedicatedtobatchplantdesignandschedulingwhichareof majorimportanceforsuchindustriesandwhichcanbeconsidered aspart and parcelof themore generaltopics of NPD manage-ment.Thiskindofinvolvesseveralcriteria,theNetPresentValue ofasequence,itsassociatedrisk(measuredbyanattractiveness ratioorbytheso-calledpositivityprobability)andthemakespan thatmustbeoptimizedsimultaneously.Section2isfirstdevoted tothekeyissuesinvolvedinNewProductDevelopment.Section 3presentstheprinciplesofthediscreteeventsimulationmodel developedandimplementedfordescribingthepipelinebehaviour.

Fig.1. Processfordrugdevelopment.

Section4describestheformulationofthemultiobjective optimiza-tionproblem.Thedifferentoptimizationmethodsthatmaybeused arebrieflyrecalledwithaspecialemphasistoGeneticAlgorithms (GAs),thatareparticularlyattractivefortreatingthiskindof prob-lem,duetotheirability todirectly leadtotheso-calledPareto front.AmongthevariousGAs,adiscussionisthenperformedto selectthemostappropriatevariant.TheselectedNSGAIIalgorithm isthenappliedtothetreatedcaseinSection5.Sections6and7 ana-lyzeanddiscussthetestbenchexamplesandprovidewithsome guidelinesforthetreatmentofnewcasesfromthebiandtricriteria viewpoints.Finally,Section8summarizesthemajorresultsofthe paper.

2. KeyissuesinNewProductDevelopment 2.1. Lifecycleofapharmaceuticalproduct

Basically, several stages are involved in the life cycle of a pharmaceutical product as it can be summarized in Fig. 1. In theDiscoverystage, thousandsofmolecules areappliedto tar-getsdevelopedtosimulatevariousdiseasegroups.Onceanactive molecule,i.e.amoleculethatisidentifiedtohaveacurativeeffect onthetarget,isdiscovered,variouspermutationsofthestructure ofthemoleculearetestedtoseeiftheactivitycanbeenhanced.The mostactivemoleculefromthesestructure–activityrelationshipsis testedfortoxicologicalresultsonratsormice.Ifnoparticular wor-risometoxicendpointsareobserved,themoleculeispromotedto thestatusof“lead”moleculeandbecomesacandidatefor devel-opment.IntheDevelopmentstage,enormoussumsofmoneyand resourcesarecommittedtotheleadmoleculetofirst,observeits

behaviourinhealthyvolunteers,secondly,inpatientssmittenwith thediseaseandfinally,inlargescaleclinicalstudiesconductedin concertwiththeFoodandDrugAdministration(FDA).Coincident withthesestudies,processresearchandformulationworkis con-ductedtobothsupplythedrugfortestingpurposesaswellasto designandconstructacommercialplantiftheproductislaunched. Otherparallelstudiesinvolveextensivelong-term(i.e.twoyears) chronicstudiesinanimalstoidentifyanyindicationof oncogenic-ityatdifferentdosagelevels.Ifthedrugiseffectiveintheclinical studies,hasnounacceptablesideeffectsandisblessedbytheFDA, itmovestotheCommercialStage.Targetmarketsareidentifiedfor astagedlaunchor“ramp-up”ofthenewcompound.Afterafew years,amaturesaleslevelisusuallyreachedandmaintaineduntil patentcoverageonthemoleculeexpiresand/orcompetitionfrom genericsisrealized.Oncegenericsareavailable,anattemptis usu-allymadetogetapprovalofthedrugforalternativemarketsand perhapsindifferentdosageforms.Regardless,salesarediminished afterexpirationofthepatent.

Somedependenciesareconsideredforrepresenting relation-shipbetweendrugsforthesamedisease:financialdependency; technicaldependency;manufacturingcostdependency;resources dependency.ThesystemparametersaresummarizedinFig.2.It mustbepointedoutthatarelationshipbetweenactivitytimesand costsforspecificdrugcandidatesistypicallyconsideredasinBlau etal.(2004).Thisrelationshipiscapturedwithasimpleparameter calledthedegreeofdifficulty(DoD).SubjectiveestimatesofDoD canbeobtainedfromthevariousprincipalinvestigators,although thevalues maybedifferentbetweenworkprocesses. However, sincethefocusisonprojectselectionandsequencingratherthan resourceplanning,theanalysiscanbesimplifiedbyusingasingle

Fig.2. Systemparameters.

valueofDoDrangingfrom1(veryeasy)to10(verydifficult).The casestudyusedinthisworkisderivedfromtheworkofBlauetal. (2004).Itmustbehighlightedthatwehaveseveralfruitful discus-sionswithaFrenchpharmaceuticalcompanytoassessthevalidity oftheexamplethatistackledhereandthatservesasaguideline ofthemethodologicalframework.Sometypicalproblemdataare displayedinTable1.Theinterestedreadercanhaveallthe comple-mentaryinformationinPerez-Escobedo(2010).Bylackofplace,all thedatawillnotbeextensivelypresentedhere.Itmustbesaidat thatlevelthatsomeslightdifferencesexistbetweentheexample treatedbyBlauetal.(2004)andtheoneadoptedhere.They con-cernthetypeofdependenciesbetweenproducts.Actually,product developmentisgenerallyinfluenced bytheotherproducts con-sideredinthepipelineandbycompetitorproducts.Itisassumed thatalldependencies(exceptfortechnicaldependency)willoccur. Whenconsidered,this kindofdependencymodifiesthesuccess probability.Ontheonehand,ifthefirstdruginthesequenceof drugstargetedforDiseaseIfails,theprobabilityoftechnical suc-cessforallsucceedingdrugsdecreasesbyagivenpercentage.On

theotherhand,ifthefirstinthesequencefortestingDiseaseI suc-ceeds,theprobabilityoftechnicalsuccessforallsucceedingdrugs forDiseaseIincreasesbyagivenpercentage.Itmustemphasized thatthistechnicaldependencyisnotquitecommoninthe pharma-ceuticalindustryandisevenacontroversialissuefromthefruitful discussionswithFrenchpharmaceuticalmanagers.Thisexplains whyithasnotbeentakenintoaccountformodelling.

2.2. Classicalapproachesofresolution

A fundamental challenge in managing a pharmaceutical or biotechnologycompany is identifyingthe optimalallocation of finiteresourcesacrosstheinfiniteconstellationofavailable invest-mentopportunities.Inthatcontext,theoptimalmanagementofthe newproductpipelinehasemergedattheforefrontofallstrategic planninginitiativesofacompany.Thisissueistraditionally iden-tifiedasacomplexonesinceitintegratesvariousareassuchas productdevelopment,manufacturing,accountingandmarketing. Thecomplexityoftheproblemismainlyattributedtothegreat

Table1

Dataforthe9-drugproblemwithresourcelimitations.

Activity Duration(days) Cost(M$) Totalavailable

resources(M$)

Min ML Max Min ML Max

FHDP 300 400 500 72 80 88 275 Sampleprep 300 400 500 1.8 2 2.2 10 PhaseI 225 300 375 70 80 90 350 PhaseII 375 500 625 75 80 85 175 PhaseIII 575 775 975 150 200 250 250 ProcessdevelopI 600 800 1000 7 10 13 16 ProcessdevelopII 600 800 1000 7 10 13 16 Designplant 550 750 950 8 10 12 12 FSA 275 375 475 18 20 22 100 Prelaunch 75 100 125 45 50 55 550 Buildplant 600 750 900 52 62 72 120 RampupI 250 350 450 9 12 15 25 RampupII 250 350 450 19 22 25 50 RampupIII 250 350 450 35 40 45 100 Maturesales 250 350 450 46 53 60 150

variety of parameters and decision-making levels involved. A strategicinvestmentplanshouldsimultaneouslyaddressand eval-uateinapropermannerthefollowingfourmainissues:product management,clinicaltrialsuncertainty,capacitymanagementand trading structure. It is also generally viewed as a multistage stochasticportfoliooptimizationproblem.Themainchallengeisto configureaproductportfolioinordertoobtainthehighestpossible profit,includinganycapacityinvestments,inarapidandreliable way.Thesedecisionshavetobetakeninthefaceofconsiderable uncertaintyasdemands,salespricesandoutcomesofclinicaltests thatmaynotturnoutasexpected.

Thiskindofproblemhasrecentlyreceivedattentionfromthe processsystemsengineeringcommunityutilizingpreviousworks fromtheprocessplanningandschedulingarea.VariousMILP opti-mizationmodelsareproposedinSchmidtandGrossmann(1996)

fortheschedulingoftestingtaskswithnoresourceconstraintswith adiscretizationschemeinordertoinducelinearityinthecostof testing.ThesemodelsareextendedinJainandGrossmann(1999)to accountforresourceconstraints.Asimulation-optimization frame-work (Subramanian, Pekny, Reklaitis, & Blau, 2003) takes into accountuncertaintyinduration,costandresourcerequirements as wellas risk. AnMILP model is proposedin Maravelias and Grossmann(2001)thatintegratestheschedulingoftestswiththe designandproductionplanningdecisions.Aliteraturereviewof optimizationapproaches in thesupply chain ofpharmaceutical industriescanbefoundinShah(2004).TheworkofBlauetal.(2004)

isbasedonamonoobjectiveGeneticAlgorithmtooptimizeproduct sequenceevaluatedbyacommercialdiscrete-eventsimulator.

This work lies in the perspective of implementing efficient optimizationtools:theunderlyingideaistouseamultiobjective frameworkasalreadyinitiatedbyAguilar-Lasserre,Azzaro-Pantel, Pibouleau,and Domenech(2007)tomodel boththe conflicting natureofthecriteria(i.e.riskminimizationandprofitability maxi-mization)andtheimprecisenatureofsomeparameters(demand, operatingtimes,...).Inthatcontext,thisworkaimsatthe develop-mentofanarchitecturethatcombinesanoptimizationprocedure andasimulationmodeltorepresentthedynamicbehaviourofthe pipelinewithitsinherentuncertaintyandtohelpdecision-making. Thegeneralobjectiveis thustoproposea generalmethodology frameworktosupportdecisionsandmanagementof pharmaceuti-calproductsinvolvedintheirlifecycle,fromearlystagestomature sales.

3. DiscreteeventsimulationforNPDpipelinemodelling The main motivation of this work was to propose an opti-mizationframeworktoselectasetofR&Dprojectsfromapool ofcandidateprojectsinordertomaximizetheexpectedbenefits whilecopingwiththeuncertainnatureoftheprojects.Thisisa challengingproblemduetothecharacteristicsofthedevelopment pipeline,namely,thepresenceofuncertainty,theinterdependency betweenprojects,thelimitedavailabilityofresources,the over-whelmingnumberofdecisionsduetothelengthofthetimehorizon andthecombinatorialnatureofaportfolio.

Inthatcontext,discreteeventsimulationisacommontoolused tounderstandhowasystemworksandhowthedifferentitems interacteach other.It mustbesaid thatdiscrete event simula-tionhasbeenmostlyconfinedtoproductionsystems(batchplant scheduling for production debottlenecking, batch plant design, etc.),butthetrendinmanyindustriesofmovingtowardsan inte-gratedapproachforsupplychainmanagementhasexpandedthe areasinwhichthistechnologycanbeused.Theanalysisalso high-lightsthatalltheprocessesinvolvedinNewProductDevelopment arecharacterizedbyuncertaintyatvariouslevelsofthepipeline: impreciseparameters for activitycost anddurations aswellas

Table2

TerminologyinBPSandNPDprojectproblems. Batchplantscheduling(BPS) NPDproject

Product#i Projectrelatedtoaproduct#i(PRP#i) Equipmentitem#j Resourceofagivenstep#j

Recipe#k Successionofactivities#k(alsocalledrecipe) Unitoperationofarecipe#l Activity#l

successprobabilitiesatPhasesI,IIandIIIofthepipeline.Inour researchgroup,thedevelopmentofdiscreteeventsimulatorsfor batchplantdesignandschedulinghasbeenaconstantfocusforthe pastdecade(Baudet,Azzaro-Pantel,Domenech,&Pibouleau,1999; Bérardetal.,1999;Dietz,Azzaro-Pantel,Pibouleau,&Domenech, 2005).Moreover,ontheimplementationside,itisnotsoeasyto usecommercialsimulatorscapableofinteractingwith optimiza-tionpackagesoruserwrittencode.Amajorincentivetousediscrete eventsimulationisthatprocessescharacterizedbyuncertaintyand suitableforprobabilisticmodellingcanbeeasilyanalyzedand syn-thesizedusingdiscreteeventsimulationbyuseofaMonteCarlo approach. Anobject-orientedmodelstructure previously devel-oped forbatchplantscheduling and designwasthen extended toembedthecaseofproductmanagement,whichisparticularly adequateforreuseofbothstructureandlogic.Itsdetailed presen-tationisnotthepurposeofthispaperwhichhasbeenpresentedin

Perez-Escobedo(2010).

AfourlayerframeworkwasproposedinBérardetal.(1999)

basedonthefollowingitemsengine,event,object,supervisor,the aimbeingthedevelopmentofastandardlibraryforthe simula-torclassesthataregeneraltoanycase,thusminimizingthetask oftreatingdifferentstudycasesorthevariantsofagivenone(i.e. designorschedulingobjectives).Inthis approach,atthelowest level,thecommonenginecanbefound.Initially,theeventsinthe nextlevelaregenericeventscommontoallbatchplant simula-tions:inthiscase,thedefinitionmustbeadaptedsincewehave toconsiderthewholelifecycleofaprojectrelatedtoaproduct.In thesameway,theobjectstakenintoaccountpresentsome simi-laritiesbutdifferintheirappreciation:forinstance,inbatchplant schedulingproblems(BPS),materialresourcesareconstitutedby equipmentwhereasinNPDproblems,resourcesmaybeviewed moreglobally.Infact,themaindifferencesatthisstepoccurfrom aterminologypointofviewandthiscanbeeasilytransposedin theNPDformulation.ThecoreofthesimulatoristheEngine,which hastwofunctions:theformeristoordertheEventsinitsCalendar bytheiroccurrencedatewhereasthelatteristoactivatethemif thenecessaryresourcesareavailable;ifnot,itreportstheEventto anextdate.AnEventrepresentsachangeoftherealsystemata giventime.TheclassEventisabasisclassfromwhichthedifferent eventsmustbedefined.Ifresourcesforthisactivityareavailable, theEventisactivated;conversely,ifresourcesarenot available, theactivitywillbescheduledlater.AnEventischaracterizedbyits occurrencedate,itsactionoverthesystemandatypethatenables togiveprioritieswhentwoormoreEventshavethesame occur-rencedate.Asageneralrule,Eventswhichreleaseresourceshave priorityovertheothers,andwhenEventshavethesametype,the classicalFIFOrule(FirstInFirstOut)isapplied.Thiswillbeuseful whendifferentprojectscompeteforthesameresources.TheEvent ClasspreviouslydevelopedwasgenericenoughtoembedtheNPD formulation(seeTable2).

Wefocusedondevelopingasimulationdecisionsupporttool thatusesprobabilisticdataintheformofdurationsofactivities, resourcerequirements(modelledascapitalandoperatingcosts), clinicalsuccessprobabilitiesandproductsales,and computesa schedule.Theresultingschedulesand resourceallocation levels canbeusedtoinferefficientprojectprioritizationandresource allocationpoliciesunderuncertainty.

Fig.3.Relativefrequenciesforproduct-project3.

Beforeconsideringaportfolioofproducts,itisinterestingto examinethebehaviourofeach individualdrugcandidate.Using netpresentvalue(NPV)withaninternalrateofreturnof15%as theeconomiccriterion,thebehaviourofeachdrugcanbe simu-latedbyusingthediscreteeventsimulator.Atypicalexampleis presentedinFig.3.Totakeintoaccounttheimprecisenatureof someparameters,NPVdistributionwasobtainedfromasufficiently largenumberofMonteCarlotrials.Atwo-peakeddistributionis observedwhichistypicalofanewdrugcandidateinthe pharma-ceuticalindustry.Thefirstpeakcorrespondstothelossofmoney inthoseinstanceswhenthedrugfailstopassalltheclinical tri-als.Theseconddistributioncorrespondstothereturnsfollowing asuccessfulproductlaunch.Duetothisbimodaldistribution,the economiccriterionmustbeclearlydefinedandnecessarily asso-ciatedwithariskcriteriontoevaluatethequalityofsequence:it mustbepointedoutthattheExpectedNetPresentValue(ENPV) thatiscommonlyusedforprojectevaluationmustbeconsidered carefully.Itcorrespondstoameanvaluebetweenthepositiveand negativepartsofthedistribution.Ifconsideredatthe optimiza-tionstep,itrepresentsapessimisticvalueoftheNPVbutevolves inthesamedirectionoftheENPV.Theriskisappreciatedbythe computationofthepositivevaluesofNPVoverthetotalnumberof samplesthathavebeenevaluated.Moreover,themakespanthatis thetimetomarketisalsoacriterionthatneedstobeconsideredat portfolioselection.Thisanalysisallowstodefinethemost impor-tantcriteriathatmustbetakenintoaccounttodefinethebestdrug portfolio.

Aspecialemphasishasthenbeendevotedtouncertainty mod-ellinginNPD.Theuncertaintyconsideredinthisworkistwofold: itisassociatedtoprojectsuccess,whichisanendogenous uncer-taintythatcanberepresentedbyadiscretenumberofrealizations (i.e.,clinical trial failed or approved) but also withthe uncer-taintyassociatedwithtimeandcostparameters.Traditionally,two classes of methods of imprecision representation have become important: probabilitytheoryand non-probabilistic uncertainty modelling.Theformerclassattemptstomodeluncertain param-etersas randomvariables:imprecise parametersare associated withaprobabilitydistributionwithinaMonteCarloframework. TheconceptofDegreeofDifficultywasusedtoreflectthemoreor

lessdifficultytocarryoutaprocesstask.Thelatterclassincludes intervalcomputationandfuzzysettheory.Twoapproacheswere implemented, a classical probability approach and an interval-basedone.Bothapproacheshavebeenillustratedbyanumerical examplewhich hasshownthat thetendenciesobtainedbythe interval-basedapproachmaybedifficulttointerpretforthe deci-sionmaker, due tothegrowinguncertainty along thepipeline. Besides,therisk,which istakenintoaccountvia failure proba-bilityofsomestagesandwhichisstronglyinvolved intheNPD processmustbepartandparcelofthemodellingapproach.Atthis level,itwasdifficulttomodelthisparameterbyanintervalandthe repetitiveuseofsimulationwithrepresentativesamplingwasthe adoptedproceduretoaddressthisissue.Allthesereasonsexplain whythereisnoneedtodevelopaproperinterval-based frame-workforNPDproblemwithuncertainty.Amoreaccurateanalysis ofaninterval-basedoptimizationmethodasanouterloopofthe discrete-eventsimulationmodelforNPDhasthusnotbeen devel-oped.

However,evenifitis particularlyusefulfordecision criteria evaluation,suchaseconomicandriskmetrics,theuseofdiscrete eventsimulationasastand-alonetechnologyconsiderablylimits thenumberofsystemconfigurationsthatcanbeconsidered.This hasmotivatedtheuseofahybridsimulation-optimization strat-egythatnotonlyaccuratelycapturesthedynamicsofthesystem butalsoprovidesastructuredwaytosearchfortheoptimal config-urationsaccordingtoseveralobjectivefunctionsinaconstrained space.Theuseofthediscreteeventsimulatorisparticularly use-fulforcriteriaevaluationandwillnowbeembeddedinanouter optimizationloopassummarizedinFig.4.

4. Multiobjectiveoptimizationproblemformulation

Realengineeringdesignproblemsaregenerallycharacterized bythepresenceofmanyoftenconflictingobjectives.Thisraisesthe issueabouthowdifferentobjectivesshouldbecombinedtoyielda finalsolutionandtosearchforoptimalsolutionstotheconsidered problem.

Fig.4. SED-optimizationframework.

4.1. Generalmultiobjectiveoptimizationproblemformulation Ageneralmultiobjectivedesignproblemcanbeexpressedas follows:

f(x)=(f1(x),f2(x),...,fk(x))T

s.t. x∈S

x=(x1,x2,...,xn)T

wheref1(x),f2(x),...,fk(x)arethekobjectivefunctions,(x1,x2,...,

xn)arethenoptimizationparametersandS⊂Rq×Nr:q+r=nisthe

solutionorparameterspace.Thesub-spaceSmightbedefinedby linear/nonlinearconstraintslinkingbothcontinuousanddiscrete variables.

Onepropertycommonlyconsideredasnecessaryforany can-didatesolutiontothemultiobjectiveproblemisthatthesolution isnotdominated.TheParetosetconsistsofsolutionsthatarenot dominatedbyanyothersolutions.Asolutionxissaidtodominate yifxisbetterorequaltoyinallattributes,andstrictlybetterinat leastoneattribute.Consideringaminimizationproblemandtwo solutionvectorsx,y∈S,xissaidtodominatey,denotedx<y,if:

∀

∃

Thespacein(Rv

×Nw_:

_v

ofParetooptimalsolutionsisknownastheParetooptimal fron-tier,P:anyfinaldesignsolutionshouldpreferablybeamember oftheParetooptimalset.Paretooptimalsolutionsarealsoknown efficientsolutionswhenscalarizationmethodsareused.

IfthefinalsolutionisselectedfromthesetofParetooptimal solutions,therewouldnotexist anysolutionsthatarebetterin allattributes.Inpractice,thedecisionmakerhastoselectasingle solutionbysearchingamongthewholeParetofront,anditmay bedifficulttopickone“best”solutionoutofalargesetof alterna-tives.Branke,Deb,Dierolf,andOsswald(2004)andTaboadaand Coit(2006)suggesttopickthekneesintheParetofront,thatisto say,solutionswhereasmallimprovementinoneobjective func-tionwouldleadtoalargedeteriorationinatleastoneoftheother objectives.

4.2. Generaloptimizationmethods

A great varietyof applications,drawnfrom a widerange of investigationareas,can beformulatedascomplex optimization problems.Thislargenumberofoptimizationproblemsarisesfrom modelsthathavetoenable,forindustrialrequirements,atruly real-isticrepresentationofthesystemtheyaccountfor.Consequently, thesemodels tendtoshow anincreasingsophistication degree thatderivesintoahighercomplexityand,thus,solution difficul-ties.Thecomplexityoftheformulatedmodelsisbasicallydueto thenatureofthefunctionsand ofthevariables involvedin the optimizationproblem.Theformeronesmaybenotonly nonlin-ear,butmoreover,theyalsooftenprovetobenonconvex,which isastronglypenalizingcharacteristicinthetypicalminimization case. Then, fora constrained problem, determiningthe feasible spaceturnsouttobeareallydifficulttask. Withregardto vari-ablenature,mostengineeringproblemsconsiderbothcontinuous anddiscretevariables,introducingdiscontinuitiesintheobjective functionandinthesearchspace:thosearecalledmixed-integer problems.Furthermore,thediscrete variablesinducean impor-tantcombinatorialeffect:thispointisemphasizedwithNP-hard problems,forwhichnoalgorithmleadingtopolynomialsolution times is known. In order to face these problems, a significant investigationefforthasbeencarriedouttodevelopefficientand robustoptimizationmethods.Atthebeginning,thisaimwas pur-suedspeciallyintheoperationalresearchandartificialintelligence areas.But,this trendwassubsequently followedbytheprocess systemengineering community,since this oneprovidesa wide numberofapplicationsformulatedascomplexoptimization prob-lems.Atypicalreferenceisconstitutedbydesignproblems:heat ormassexchangernetworks(Zamora&Grossmann,1998), sup-plychaindesign(Guillen,Badell,Espuna,&Puigjaner,2006),and multiproduct(Ravemark&Rippin,1998)ormultipurpose(Dedieu, Pibouleau,Azzaro-Pantel,&Domenech,2003)batchplantdesign orretrofitting(Montagna&Vecchietti,2003).Asaconsequence, a great diversity of optimizationmethods wasimplemented to meettheindustrialstakesandprovidecompetitiveresults.But,if theyprovetobewell-fittedtotheparticularcasetheypursue,the

performanceofthesetechniquescannotbeconstantwhateverthe treated problemis. Actually,a methodefficiency for a particu-larexampleishardlypredictable,andtheonlycertaintywehave isexpressedbytheNoFreeLunchtheory(Wolpert&Macready, 1997): there is nomethod that outdoes all theother ones for anyconsideredproblem.Thisfeaturegeneratesacommonlackof explanationconcerningtheuseofamethodforthesolutionofa par-ticularexample,andusually,norelevantjustificationforitschoice isgivenapriori.

Optimizationmethodscouldbedividedintoderivativeand non-derivativemethods.Thederivativeorscalarizationproceduresaim attransformingthe multiobjectiveoptimizationprobleminto a nonobjectiveoneandsolvingitwithclassicalNLPorMINLPtools. Non-derivative methods are particularlyinteresting for general engineeringdesign problems.Onereasonis thatnon-derivative methodsdonotrequireanyderivativesoftheobjectivefunction inordertocalculatetheoptimum.Therefore,theyarealsoknown asblackboxmethodswherenumericalvaluesofvarious objec-tivesand/orconstraintsaccordingtoagivenentrancevectorx,are returnedbycomputercodes.Anotheradvantageofthesemethods isthattheyaremorelikelytofindaglobaloptimum,andnotbe trappedonlocaloptimaasgradientmethodsmightdoinsofaras somedegradationsinobjectivefunctionscanbeadmittedduring thesearch.

Fora generaldesign problem, itis hardtoexpressobjective functionsintermsofthedesignvariablesdirectly,whichis par-ticularlythecaseinourproblem,sincetheperformancefunctions areevaluatedfromadiscreteeventsimulator.Therefore,thereisno straightforwardwayofcalculatingthederivativesofthedifferent objectivefunctions.Anotherincentivetousenon-derivative meth-odsparticularlyGeneticAlgorithmsisthattheyarewell-suitedto tacklehighlycombinatorialproblems.

4.3. Geneticalgorithms

4.3.1. Geneticalgorithmroadmap

Geneticalgorithms(GAs)andthecloselyrelatedevolutionary strategiesareaclassofnon-gradientmethodswhichhasgrown inpopularityeversinceHolland(1975)firstpublishedtheirinthe early1970s.ThebasicideaofGAsisthemechanicsofnatural selec-tion.Eachoptimizationparameter,(xi),iscodedintoageneasfor

examplearealnumberorstringofbits.Thecorrespondinggenes forallparameters,x1,...,xn,formachromosome,whichdescribes

eachindividual.Achromosomecouldbeanarrayofrealnumbers, abinarystring,alistofcomponentsinadatabase,alldepending onthespecificproblem.Eachindividualrepresentsapossible solu-tion,andasetofindividualsformapopulation.Inapopulation,the fittestareselectedformating.Matingisperformedbycombining genesfromdifferentparentstoproducechildren,calledacrossover. Finallythechildrenareinsertedintothepopulationwheresome mutationsarerandomlyperformed,andtheprocedurestartsover again,thusrepresentinganartificialDarwinianenvironment.The optimizationcontinuesuntilthepopulationhasconverged(non evolutionofstatisticalparameterslikemeans,standarddeviations, ordominationranks)oruntilamaximumnumberofgenerations predeterminedhasbeenreached.

Thepopularityofgeneticalgorithmshasgrowntremendously underrecent yearsandtheyhavebeenappliedtoawide range ofengineeringproblems(Altiparmak,Gen,Lin,&Karaoglan,2009; Deb&Srinivasan,2005;Dietzetal.,2005;Yoshikawa&Terai,2005). ThereisalsoalargevarietyofgeneticalgorithmssuchassimpleGA, steadystateGA,GAwithmultiplepopulations,GAwithcrowding andsharingtechniques(seeZitzler,Deb,&Thiele,2000fora com-pletesetofreferences).ThedifferentGAsallhavedifferentfeatures inordertosolvevarioustypesofproblems.Therearealsoanumber ofmultiobjectivegeneticalgorithmswhichaimatconvergingthe

populationontheParetooptimalfrontinsteadofonjustonesingle optimalpoint.

Multiobjectivegeneticalgorithmsaregenerallydividedin non-ParetoandParetobasedapproaches:

1. Non-Paretobasedapproaches:Thefirstmultiobjectivegenetic algorithm was VEGA (Vector Evaluating Genetic Algorithm) developedbySchaffer(1985).VEGAusestheselection mecha-nismoftheGAtoproducenon-dominatedindividuals.Fourman (1985)presentsageneticalgorithmusingbinarytournaments, randomlychoosingoneobjectivetodecideeach tournament.

Kurasawe (1991)further developed this schemeby allowing theobjectiveselectiontoberandom,fixedbytheuser,orto evolvewiththeoptimizationprocess.Allofthesenon-Pareto techniquestendtoconvergetoasubsetofthePareto-optimal frontier,leavingalargepartoftheunexploredParetoset. 2.Paretobasedapproaches:Anon-dominated sortingtoranka

searchpopulationaccordingtoParetooptimalityisintroduced inGoldberg(1989).First,non-dominatedindividualsinthe pop-ulationareidentified.Theyaregiventherank1andareremoved fromthepopulation.Thenthenon-dominatedindividualsinthe reducedpopulationareidentified,giventherank2,andthen theyarealsoremovedfromthepopulation.Thisprocedureof identifyingnon-dominatedsetsofindividualsisrepeateduntil thewholepopulationhasbeenranked.Goldbergalsodiscusses usingnichingmethodsandspeciationtopromotediversityso thattheentireParetofrontieriscovered.

In themultiobjective GA(MOGA) presentedby Fonseca and Fleming (1995, 1998) each individual is ranked according to a degreeofdominance.Therankingsarethenscaledtoscore indi-viduals in the population. In MOGA both sharing and mating restrictionsareusedinordertomaintainpopulationdiversity.

ThenichedParetoGA(NPGA)byHornand Nafpliotis(1993)

isPareto-basedbutdoesnotuserankingmethods.Rather,Pareto dominationbinarytournamentsareusedtoselectindividualsfor thenextgeneration.ZitzlerandThiele(1999)developeda multiob-jectivegeneticalgorithmcalledthestrengthenParetoevolutionary algorithm(SPEA)whichusestwopopulations.

The non-dominated sorting GA (NSGA) of Srinivas and Deb (1995)implementsGoldberg’sconceptsabouttheapplicationof nichingmethods.InNSGA,non-dominatedindividualsinthe pop-ulationareidentified,givenahighinitialindividualscoreandare then removed from the population. These individuals are con-sideredtobeofthesamerank.Thescoreisthenreducedusing sharingtechniquesbetweenindividualswiththesameranking. Overtheyears, themain criticismsof theNSGAapproach have beenasfollows:highcomputationalcomplexityofnondominated sorting;lackof elitism;need forspecifying thesharing param-eter. Allof these issues have beenaddressed in the improved versionofNSGA,calledNSGA-II.Fromthesimulationresultsona numberofdifficulttestproblems,ithasbeenfoundthatthat NSGA-IIoutperformstwoothercontemporaryMOEAs:Pareto-archived evolutionstrategy (PAES)(Connor &Tilley, 1998)and strength-ParetoEA(SPEA)(Goldberg, 1989)intermsof findinga diverse setofsolutions andinconvergingnearthetruePareto-optimal set.Thewayconstraintsaretreatedisbrieflyrecalledinwhat fol-lows.

4.3.2. Constrainthandling

Constrainedmultiobjectiveoptimizationisthemostcommon kindofprobleminengineeringapplications.Ingeneral,threekinds

ofconstraintsareconsidered:simpleinequality(≤),strict inequal-ity(<),andequality:

g(x)≤c1 r(x)<c2 h(x)=c3

)

(

where(g,r,h)arereal-valuedfunctionsofadecisionvariablex=(x1,

...,xn)onann-dimensiondecisionalsearchspaceU,and(c1,c2,c3)

areconstantvalues.Inthemoregeneralcase,theseconstraintsare writtenasvectorsofthetype:

constr1(x)=(constr1(x)1,...,constr1(x)n1)≥0

constr2(x)=(constr2(x)₁,... ,constr2(x)n2)>0

constr3(x)=(constr3(x)1,...,constr3(x)n3)=0

wheren1,n2,andn3 arerespectively,thenumberorinequality,

strictinequalityandequalityconstraints.Thisconstraint formu-lationimpliesthateach constraintvaluewillbenegativeifand onlyifthisconstraintisviolated.Inpractice,duetoround-offerror onrealnumbers, theequalityconstraint constr3 isreplaced by contr3(x)+





exam-ple).Thisapproximationisnotnecessarywhenequalityconstraint involvesonlyintegerorbinaryvariables.

Theconstraintsatisfactionimpliesthemaximizationofviolated constraintsin vectorsconstr1,constr2,andconstr3.Accordingto

FonsecaandFleming(1998),thesatisfactionofanumberofviolated inequalityconstraintsisamultiobjectivemaximizationproblem.A moresimplesolutionconsistsincomparingthesumofvaluesof violatedconstraintsonly,asinNSGAIIalgorithmofDeb,Pratap, Agarwal,andMeyarivan(2002),whichimpliestherearenopriority rulesbetweenconstraints.Thisstepisperformedfirst,beforethe secondone,whichconcernsthecomparisonoftheobjective func-tionvectors.Onfourproblemschosenfromtheliterature(Debetal., 2002),NSGA-IIhasbeencomparedwithanotherrecentlysuggested constraint-handlingstrategyandprovedtobemoreefficient.These resultsleadustoapplyNSGA-IItotheNPDproblem.

4.4. CombinatorialaspectsoftheNPDproblemandsearchspace definitions

Asabovementioned,evolutionaryprocedures,andparticularly GAs,arewell-suitedforhandlinghighlycombinatorialproblems. Oneoftheobjectivesof theNPDoptimizationbeingthe deter-minationofthebestsequenceofproducts,this itemintroduces averyhighcombinatorialaspectintheproblem.Forexample,asit isshownbelow,forasimpleprobleminvolvingthreediseases,two drugsfordiseaseI,twofordiseaseIIandonefordiseaseIII,itexists 240possiblesequences,andthisnumbergrowsupto951,744for theproblemunderconsideration.

GivenaprobleminvolvingMDdiseases.Foreachdiseasedii(i=

1,MD),nditherapeuticaxisinvolvingndidrugscanbeconsidered.

Asequenceisthusconstitutedbytheunionofsub-sequencesof drugs,eachdevotedtoadisease.Adrugrelatedtoadiseasedii(i=1,

MD)isdenotedwithp=1,...,ndi.Anintegervaluepisallocated

toeachdruginapartialsequencerangingfrom1tondi.Thedrugs

canbearrangedasfollows: [1,...,nd1]

|

{z

}

|

{z

}

|

{z

}

LetusconsiderthesetSofallthepossiblesequencesinwhichthe numberofproductscanvarybetweenMD(atleastonedrugper

disease)andn_d

1+nd2+ndMD andinwhich allthepermutations

canbeconsidered: NTOT=nd₁+nd₂+···+nd_MD Card(S)=NTOT!+ NTOT−MD

X





X





Thiscanbeappliedtotheexamplewhichservesasatestbench involvingthreediseases(fourdrugsford1;fourdrugsford2;one

drugford3).

Card(S)=(9)!+(8)!(2C₄3C₄4C₁1)+(7)!(2C₄4C₄2C₁1+2C₄3C₄3C₁1) +(6)!(2C₄3C2₄C₁1)+(5)!(2C2₄C₄2C₁1)+(4)!(2C₄1C₄2C₁1)+(3)!(2C₄1C₄1C1₁) Thetotalnumberofpossibilitiesforthisexampleis951,744.This meansthat951,744 possibleportfoliodrugs canbeconsidered, takingintoaccountthatportfolioswithlessthan3drugsarenot possibleduetotheconstraintsdefinedforthemodel,atleastone drugperdisease.

5. ImplementationoftheNSGAIIkeyproceduresforNPD modelling

ThemethodologyusedforsolvingtheNPDproblemsinvolves atwo-stepapproach:atthelowerlevel,thepreviouslydeveloped discreteeventsimulatorisusedtoevaluatetheproduct develop-mentsequences,accordingtodifferentcriteria:NetPresentValue, riskmetricsandmakespan;attheupperlevel,amultiobjective pro-cedurebasedonNSGAIIprinciplesisusedtodetermineboththe numberofdrugproductsinthesequenceandtheorderinwhich thedrugsarereleasedinthepipeline.

5.1. Coding,crossoverandmutation

Asequenceismodelledbyuseoftwotypesofchromosomes withanidenticalnumberofgenes,equaltothenumberofproducts toconsiderintheglobalportfolio.ToeachproductP1corresponds

anindexiwhichisthechromosomepositioni.Thefirst chromo-someChrom1isrelatedtotheproductorderinasequence.Genes

areintegervariables,rangingfrom1tothetotalnumberof prod-uctsinasequence.Thevalueofeachgenemayoccuronlyoncein thechromosome.Forapositioniofagene,itsvaluecorresponds totheproductpositioniinthesequence.Thesecondchromosome Chrom2isonlyconstitutedbybinaryvariables,theunityvalueof

ageneinpositioni(respectively0)correspondingtothepresence (respectivelyabsence)ofaproduct.

The chromosome corresponding to each sequence is then obtainedbymultiplyingeachgeneofChrom1withthe

correspond-ingoneofChrom2,locusbylocus.Itmustbehighlightedthatthis

codingisnotuniquewhichmayintroducesomebiasinthesearch. Yet,amoreattractivealternativewouldbetodirectlycodethe chro-mosomesrepresentingasequencewithvariablelengthinfunction oftheproductnumberinthesequence.Yet,thisapproach may leadtounfeasibleindividualsinthecrossoverphase,withalarger sizethantheonecorrespondingtotheeffectivenumberof prod-uctsinthesequence.Theefficiencyoftheformerprocedurehas beentestedsuccessfullythroughseveralexamplesandhasthus beenselectedinthiswork.Fig.5illustratestheusedcoding rep-resentingasolutionthatisthenevaluatedbythesimulatorfor9 products.Crossoverandmutationhavebeencarriedoutbyspecific proceduresforeachtypeofchromosome.

Chromosomesdedicatedtoproductorderarehaploid,yet,all theintegergenesmustbedifferent,rangingfrom1tothetotal num-berofproductsinasequence.Forthispurpose,acrossoveroperator

Fig.5. Codingforgeneratingasequence.

withrespecttogenotypeconstraintswithoutclonegenerationin theoffspringgeneticcodehasbeencarriedout,theso-callMPX operator(MaximalPreservativeX)(Andersson,1999).This under-lyingideaistoinsertasegmentofaparentchromosomeinthe chromosomeoftheotherparentsothattheresultingcrossoveris closertohisparents.Itisatwo-pointcrossoverandthetwosonsare obtainedinasymmetricalmanner.Concerningmutation,aclassical mutationoperatorthatrandomlypermutestwogenesofa chro-mosomeisused.Thisoperatorisappliedtotheindividualsderived fromcrossoverwithanadaptedrate(preferably0.5).Then,thenew offspringisplacedinanewpopulation.

5.2. Optimizationparameters

TheoptimizationparametersarepresentedinTable3.

5.3. Optimizationcriteria

Theoptimizationcriteriaareevaluatedbyuseofthepreviously developeddiscreteeventsimulator.TheyinvolvetheglobalNet PresentValueofasequence,classicallycomputedfromtheaverage valueofnetpresentvaluesofthesamples.Anactualizationrateof 15%hasbeenchosen:

f1: n

X

"

X

#

Anothercriterionistherisk,correspondingtothenumberoftimes anegativevalueofNPV isobservedamongthetotal numberof samples.Notethatriskf2isthecomplementaryriskrelativetothe

alreadymentionedpositivityprobability:

f2: n

X

"

X

#

Numberofindividualspergeneration 80 Numberofsimulationsperindividual 300

Then,finally,themakespanofasequenceiscomputedfromthe averagemakespanofthesamples:

f3: n

X

whereW→numberofdrugsinthesequence;n→numberofruns bysequence;dur→makespanforasequence.

5.4. Constraints

It must be emphasized that the resource constraints have alreadybeentakenintoaccountinthecapacityrequirementsof eachtaskinthepipelinewithinthediscreteeventsimulator.The constraintsthatareconsideredherearerelatedtothepresenceofat leastonedrugtargetingatherapeuticaxis.Theycanbeformulated asfollows:

X

mp_d

i≥1 with p=1,...,ndi

Letusrecallthatfortheexample,3diseasesand9moleculeshave beenconsidered:Disease 1:ProductsP2,P3, P6,P7;Disease 2: ProductsP4,P5,P8,P9;DiseaseM3:ProductP1.Theseconstraints involveatleastonegenevalueequalto1forchromosomeChrom2

inthelocicorrespondingtothegenesoftheproductsofthegiven diseasegi,thatis:

g2+g3+g6+g7≥1

g4+g5+g8+g9≥1

g1>1

6. Resultpresentationanddiscussion

The case study results are now discussed in the following sections, focusing on analyzing thePareto front generated and identifyingtrendsconcerningportfoliocomposition.Inallthe opti-mizationruns,unlessexplicitlymentioned,theinitialpopulation wasgeneratedrandomly.

6.1. Introduction

Optimizationrunswerefirstcarriedoutinabicriteriawayand thenanalyzedfromatricriteriaviewpoint.Totakeintoaccountthe stochasticnatureoftheGeneticAlgorithm,eachoptimizationrunis repeated5times(atleast).TheCPUtimeofeachoptimizationrunis

Table4

NetPresentValueandRiskfor4-drugsequences(9drug-portfoliooptimization). Solutions NPV(M$) Risk(%) Makespanby

simulation (days) Releaseorder 1,3,4 1546–1730 14–20 3721 P2 P7 P5 P1 2 1683 17 4055 P7 P5 P2 P1 5 1507 13 3604 P5 P7 P2 P1 6 1472 12 3721 P5 P2 P7 P1

difficulttoevaluate,duetocombinedeffects:first,itdependsonthe numberofproductsinthesequence,second,thestochasticaspect oftheMonte-Carloapproachusedthroughsimulationmayleadto prematurestopoftheevaluationofacandidate.Anoptimization runtakesaround36hforthisstudywith9drugsfor3diseases. 6.2. BicriteriaoptimizationNetPresentValue-Risk

6.2.1. OptimizationNPV-Riskwithrandominitialization

AfirststudyconcernstheNetPresentValueandRiskto opti-mizesimultaneously.Asmightbeexpected,randominitialization ofthepopulationhasresultedinadispersionofthevarious solu-tions.Theoptimizationprocedurewasconsideredtobeconverged whengeneralprogressionoftheParetofrontwasinsignificant.It canbeobservedthattheriskvariationliesbetween10and40%.No solutionexistsforriskvaluesgreaterthan40%.Aninterestingresult concernsthenumberofdrugsintheportfolio.Forriskvalues com-prisedbetween12and20%,thenumberofdrugsinthesequences isequalto4andthedrugsthatcanbesystematicallyfoundare [P1∧P2∧P5∧P7]withanNPVvaluecomprisedbetween1400and

1700M$.Forriskvaluesbetween27%and39%thenumberofdrugs intheportfolioisequalto6andtheproductsthatcanbe system-aticallyobservedare[P1∧P2∧P3∧P5∧P6∧P7]withanNPVvalue

comprisedbetween1800and2000M$.Theresultsarealso pre-sentedinTables4(4drugs)and5(6drugs).Asitcanbeseenin

Fig.6,afirstcommentconcernstheParetofrontsolution.Forallof them,thehighertherisk,thehighertheNetPresentValue.Second, itcanbehighlightedthatseveralsolutions(solutions1,3,4)are foundseveraltimes(P2∧P7∧P5∧P1)and(P2∧P7∧P5∧P1∧P1).

FurthermoreinthecodingoftheGA,asequenceisnotrepresented byauniquechromosome.Althoughsomesignificantdifferencesare observedaboveallfortheriskcriterion,thesesolutionscanbe con-sideredasparticularlyattractivesincetheprocedurehasidentified themseveraltimesasParetocandidates.

TheunionoftheParetofrontsobtainedfromtheoptimization runs,canbevisualizedinFig.6duetoboththestochasticnatureof theNPDmodel(asequenceisevaluated300times)andtotheGA. Thisfiguredisplaysthenon-dominatedindividualsobtainedfrom 5optimizationsandarerelativetosequencesof4drugs(P1,P2,P5, P7)and(P1,P3,P5,P9).

Forthesetwodistinctbehaviours,thesolutionscanbe distin-guishedbytheproductorderinthesequence.Acloserlookatthese solutionsindicatethattheoptimizationstrategytendstoeliminate longsequences,reproducingtheso-calledattritionphenomenon occurringinNPDproblems.Thisisduetothecomplexitythatis inherentin themodel ofthepharmaceuticaldrugdevelopment pathwayandoftheinteractionsbetweenthevariousdrugs.Itis difficulttopredictthis behaviourwithouttheuseof a numeri-caltool.Forthe12sequencesevaluatedbysimulation,thesame behaviourasinthepresentsectionforsequenceswithfourandsix drugs,wasalreadyobserved.TheNPVvariedfrom80to685M$ andtheriskwasintherange(0.77–0.38).However,theefficiency ofthegeneticalgorithmimprovesconsiderablytheperformances. Anotherimportantobservationisthatstrategieswithdifferences ineitherdrugselection,timing,cancompetewithsimilarreward versusriskprofiles.Henceitisusefulforthedecisionmakerto identifyandcloselyexaminethedifferentoptionsthatcanyield thedesiredreturnandacceptablerisk.

6.2.2. Optimizationwith9-drugsequencesinthefirstgeneration Anoptimizationrunwasthenperformedinordertostudythe influenceofthenumberofproductsinthefirstgeneration.Theidea istoinitializetheAGwithsequencescontainingexactly9drugs,in ordertoexaminehowthenumberofproductsevolvesnaturally alongthegenerations.TheresultsarepresentedinTable6for gen-erations1,40,80,120,160and200.Itcanbeclearlyobservedthat thesequenceswithalownumberofproductsarefavouredinthe optimizationprocess.

Fig.7displaystheParetofrontinwhichsequenceswith4,6 and7productscanbefound.Itmustbeemphasizedthat7-drug sequenceswerenotfoundinthepreviousoptimizationruns:this maybeduetothefactthatthenumberofgenerationsneedstobe increased.Table7presentsthenumericalvaluesofNPVandrisk aswellasthereleaseorder.Finally,themainresulthereisthat thenaturalevolutionofthealgorithmistowardstheelimination oflongsequences.

Table5

NetPresentValueandRiskfor6-drugsequences(9drug-portfoliooptimization). Solutions NPV(M$) Risk(%) Makespanby

simulation (days) Releaseorder 1 2206 39 5155 P6 P1 P3 P2 P7 P5 2 2203 37 4878 P6 P2 P1 P7 P5 P3 3 2097 36 4914 P2 P6 P1 P7 P5 P3 5 1900 28 4928 P7 P6 P3 P2 P5 P1 4,6 1887–2077 27–29 4869 P3 P6 P7 P2 P5 P1 Table6

Evolutionofthenumberofdrugsbysequence.

Generation Numberofdrugsinthesequence

1 2 3 4 5 6 7 8 9 1 0 0 0 0 0 0 0 0 80 40 0 0 0 0 0 11 69 0 0 80 0 0 0 0 0 30 50 0 0 120 0 0 0 0 0 45 35 0 0 160 0 0 0 24 0 31 25 0 0 200 0 0 0 21 0 35 24 0 0

Fig.6.ParetoFront.NetPresentValue/Risk.

Table7

ResultsofNetPresentValueandRiskfor9-druginitializedportfolio(Paretofront solutions).

Solutions NPV(M$) Risk(%) Releaseorder

1 798 22 P7 P1 P3 P5 2 783 21 P7 P3 P5 P1 3 690 18 P7 P1 P5 P3 4 782 19.5 P7 P3 P5 P1 5 700 19 P7 P3 P5 P1 6 992 23 P7 P1 P3 P5 7 1857 30 P7 P3 P6 P5 P2 P1 8 1592 26.5 P7 P3 P6 P5 P2 P1 9 1722 28.5 P1 P7 P6 P5 P2 P3 10 1696 28 P7 P1 P6 P5 P2 P3 11 1355 24.5 P6 P7 P1 P5 P9 P2 P3 12 1379 26 P7 P1 P6 P5 P9 P2 P3 13 1343 24 P7 P1 P6 P5 P9 P2 P3

6.2.3. Optimizationwith9-drugsequencesintheoptimized portfolio

Toconfirm oncemore that longsequences arenot interest-ing,anoptimizationisperformedundertheconstraintofa9-drug portfolioalongthealgorithmevolution.Here,thefirstpopulation wasgenerated randomly(withouttaking intoaccountthe con-straint).

ThisisconfirmedbytheParetofront(Fig.8)onlyconstituted of3sequenceswithanNPVbetween1171and1313M$andarisk valuebetween35and44%.ThenumericalvaluesofNPVandRisk foreachnon-dominatedindividualaswellasthereleaseorderare presentedinTable8.Asaconclusion,theresultsindicatethatlong sequencesarenotrepresentativeofattractivevaluesbothforNPV andrisk.Thisexplainswhy9-drugindividualsareeliminatedfrom theParetofrontinthepreviousbicriteriaoptimization.

Fig.8. ParetofrontfortheoptimizationNetPresentValue/Riskfor9-drugoptimizedportfolio.

Fig.9.ParetoFront.NetPresentValue/Makespan.

6.3. BicriteriaoptimizationNetPresentValue-Makespan

ThesecondoptimizationstudyisbasedonNetPresent Value-Makespan (expressed in days). The resultsrelative tothis pair of criteria are presented in Fig.9. Solutions withnegative val-uesfornetpresentvalues arefound, whichcorrespondtovery low values of the time horizon, that obviously will not be

Table8

ResultsofNetPresentValueandRiskfor9-drugoptimizedportfolio(Paretofront solutions).

Solutions NPV(M$) Risk(%) Releaseorder

1 1314 44 P9 P2 P3 P6 P4 P7 P5 P1 P8 2 1305 37 P9 P2 P3 P6 P4 P7 P5 P1 P8 3 1172 35 P9 P2 P3 P6 P4 P7 P5 P1 P8

consideredbythedecision maker.It canbehighlightedthatan importantnumberofsequenceswith3or4productsarefound again.Amongthe3-product sequence,thecorrespondingdrugs are [P1∧(P2∨P6)∧(P5∨P8)]. The 4-product portfolio involves thedrugs[P1∧P2∧P5∧P7],thathavebeenalreadyidentifiedas potentialcandidatesfornetpresentvalue-riskoptimization(see

Tables9and10).

6.4. Bicriteriaoptimizationmakespan-risk

The resultsof the bicriteria optimization makespan-riskare presented in Fig. 10. From these results, it can be seen that a decrease in risk has a strong impact in the pipeline dura-tion, that can be quantified. Risk ranges from 13% to around 70% when the duration decreases from 11.5 to 7.7 years. The

Fig.10.ParetoFrontMakespan/Risk.

Fig.11.TricriteriasolutionsprojectionforNPVRisk.

sequences for which risk lies between 13 and 18% are exclu-sively composed of 3 products [P1∧(P2∨P7)∧(P5∨P8)]. The sequences for which risk is comprised between 21 and 73% involve4 products [P1∧P2∧P5∧P7]. Thereis a small overlap-ping zone from 18 to 21% with mixed sequences of 3 and

Table9

NetPresentValueandmakespanfor3-drugsequences. Solutions NPV(M$) Makespan(days) Riskby

simulation Releaseorder 1 1312 3739 54 P1 P5 P2 2,3,5 57–713 2850–3402 33 P5 P2 P1 4 136 3043 50 P8 P2 P1 6 11 2848 69 P1 P8 P2 6 2 2846 55 P5 P1 P2 7 -35 2827 73 P2 P1 P8

4 products (the same ones as those previously found). Once more,thehigher number ofsolutions presented relativetothe combinatorial aspectof theproblemis due to slightvariations for clones, due to the stochastic aspects of the problems (see

Tables11and12).

Table10

NetPresentValueandmakespanfor4-drugsequences. Solutions NPV(M$) Makespan (days) Riskby simulation Releaseorder 1 1752 4017 39 P5 P7 P2 P1 2 1357 3845 22 P2 P5 P7 P1 3,7 413–1253 3241–3691 39 P5 P1 P2 P7 4,5 856–1423 3587–3878 47 P7 P5 P2 P1 6 152 3107 45 P2 P1 P5 P7

Fig.12.TricriteriasolutionsprojectionforNPVDuration.

Fig.13.TricriteriasolutionsprojectionforRiskDuration.

Asapartialconclusionofthisbicriteriastudyinvolvingtimeas acriterion,itcanbesaidthatdecisionsontimingareanimportant constituentoftheportfoliodevelopmentstrategyastheyareused tofavourablyorganizecash flows.Thisisparticularlyimportant whenhavingtoconsidertheprobabilitythataprojectwillsucceed, i.e.thefinancialimpactoffailedprojects.

Table11

Riskanddurationfor3-drugsequences. Solutions Risk(%) Makespan

(days) NPVby simulation Releaseorder 1 40 3389 628 P7 P5 P1 2 35 3458 603 P7 P1 P5 3 30 3608 461 P1 P5 P2 4 25 3757 532 P2 P1 P5 5 21 3906 554 P5 P2 P1 6 18 3973 474 P1 P2 P5

7. TricriteriaoptimizationNPV-Duration-Risk 7.1. Studypresentation

In this study, thetricriteriaoptimization isperformed NPV-Duration-Risk. To make the interpretation easier, the results

Table12

Riskanddurationfor4-drugsequences. Solutions Risk(%) Makespan

(days) NPVby simulation Releaseorder 1,2,3 17–21 3889–4090 1375 P2 P5 P7 P1 4,5 16–17 4101–4105 1379 P7 P5 P2 P1 6,7,10,12,13 14–19 4038–4204 769 P5 P7 P2 P1 8 16–20 3922–4110 953 P5 P2 P7 P1 11 15 4150.48 584 P7 P2 P5 P1

Fig.15.Frequencyandbehaviourfornon-dominatedtricriteriaoptimization.

relativetoa givenpairof criteriaarepresented asaprojection ona2D-axis(see Figs.11–13).Globally, itcan besaidthatthe sametrendsasthebicriteria approachesareobserved,thatis a smallnumberofproductsintheportfoliofavoursthebest compro-misebetweenthecriteria.Acloserexaminationattheevolutionof NPVvs.riskneedssomeadditionalcomments.Fig.11showsthat forriskvaluescorrespondingtoa15–35%range,itseemsthatan increaseinriskleadstoanincreaseinNPV.Thistrendisnomore observedwhenexploringriskvaluesbetween35and70%,where thehighertherisk,thelowertheNPV.Thisphenomenoncanbe nowattributedtoastrongdecreaseinmakespanwhichis opti-mizedsimultaneouslyinthiscase.Sincethebicriteriastudyhas shownthatthesolutionsetisdifferentaccordingtothepairof cri-teriaconsidered,thetricriteriaanalysisseemsmoreconsistentto findthemostinterestingsolutions.

Fromthesolutionsobtainedfromthe3-criteriaParetofront,the decisionmakercanselectasequence,fromarisklevelthatseems acceptableforhim.TheresultspresentedinTable13allconstitute potentialcandidates.

Acloser examinationof thesolutions presented inTable 13

ispresentedinFigs.14and15whererelativefrequencyis plot-ted vs. net present value. This kind of representation is more

meaningful since the analysis of simulation results of some sequenceshasshownthattheyexhibitabimodalbehaviour.The meannetpresentvalueisinterestingfromanoptimization view-pointsincetheobjectiveistoshifttowardspositivevaluesforNPV: theinterpretationisconsistentheresincemeanNPViscombined withariskcriterionasmeasuredbytheratioofthenumberof pos-itivevaluesforNPVtothetotalnumberofNPVevaluations.This two-peakedphenomenonisstillobservedforthese5sequences, withmoredispersedvaluesforthe3-drugportfolio.

Solutions1and2areequivalentfromthemakespancriteria, butdiffersignificantlyfromtheriskandNPV criterion.Solution

Table13

Someinterestingsolutionsfromtricriteriaoptimization. Solutions Risk(%) NetPresent

Value(M$)

Duration(days) Releaseorder

1 16 1316.29 4141.52 P7 P2 P5 P1 2 20 1620.7 4110.4 P2 P5 P7 P1 3 25 1280.99 3952.73 P5 P2 P7 P1 4 30 721.11 3709.74 P5 P2 P1 5 35 850.35 3645.91 P5 P7 P2 P1

2exhibitssomepeakshigherthan9000M$.Thedecisionmaker hastodecideifthehigherriskinducedbySolution2isjustified. Solution3mustbeinvestigatedifthetimecriterionisimportant toconsideratthatlevel,evenifriskandNPVarelowerthanfor solutions1and2.Solutions4and5showthesameorderof magni-tudeformeanNPVanddurations(riskishigherforsolution4).Yet, Solution3concerningthe3-drugsequencehaspoorperformances (nopeakhigherthan3600M$)andmaybefinallydiscardedbythe decisionmaker.Atthislevelofdiscussion,itisdifficulttosaymore sincetheexamplehasjustadidacticvalue.

7.2. Conclusionofthebi-andtricriteriastudy

Asa conclusionof this bicriteria analysis performed onthe different pairs of criteria and of the tricriteria optimization, it canbehighlightedthatamongtheconstellationofpotential can-didates,theoptimization strategy seemsefficient todetectthe sequenceswhichcanbeconsideredbythedecisionmakers.Onlya fewsequencesaredetected.Amongthesessequences,large port-folioscauseresourcequeuesanddelays timetolaunchandare eliminatedbythebicriteriaoptimizationstrategy.Smallportfolio reducesqueuingandtimetolaunchappearasgoodcandidates. Theoptimizationstrategy,basedonNSGAII,thatisparticularly eli-tist,isinterestingtodetectthesequencecandidates.Timeisan importantcriteriontoconsidersimultaneouslywithNPVandrisk criteria.Theorderinwhichdrugsarereleasedinthepipelineisof greatimportanceaswithschedulingproblems.Theuseofa deci-sionanalysismethod(Pirdashti,Ghadi,Mohammadi,&Shojatalab, 2009)asTOPSISwillallowtoselectasequencesaccordingtothe decisionmakerpreferences.Thebasicconceptofthismethodisthat theselectedalternativeshouldhavetheshortestdistancefromthe negativeidealsolutioningeometricalsense(Pirdashtietal.,2009). AthoroughanalysisisproposedinMorales-Mendoza.etal.(2011).

8. Conclusions

ThedevelopmentofamultiobjectiveGeneticAlgorithm opti-mizationframeworkcoupledwithadiscreteeventsimulatorhas beenpresentedthataddressestwokeydecisionssimultaneously: portfoliomanagement andschedulingofdrugdevelopmentand manufacturing.Twocasestudieswereusedtoillustratethe capa-bilitiesof theframework andalsohighlightedthat thescopeof decisionsthatadrugdevelopermaybeconfrontedwithcanbe vastandcomplex.

Our analysis on both case studies suggests that optimizing projectprioritiestakingintoaccountresourceallocationsyieldsa significantlyimprovedportfolioperformance,ratherthanasimple useofabubblechartthatcannottakeintoaccountthe interdepen-denciesbetweenprojects.Duetothecomplexityofthisproblem,a contributionofthisworkisindemonstratingaformulationbased ontechniquesfromevolutionarycomputationemployedforan effi-cientsearchofthedecisionspaceandoftheobjectivespace.All theresultstendtohighlightthatpharmaceuticalproduct devel-opmentstrategiesintherealworldmaybebetteranalyzedwhen consideringtheimpactofdecisionsholisticallyratherthanonly individually.

Oneimportantissueinthiskindofproblemsis thedecision flexibilitythatmanagementcanexerciseduringthecourseofthe developmentpath.Giventhehighfailurerateandlargepotential investmentloss,aperspectiveofthisworkcouldbetotakeinto accountforthevaluationofproductportfoliosnotonlythe uncer-taintiesandrisksbutalsothedecisionflexibility.Thesequenceof projects,asproposedinthiswork,isfixedduringthewhole hori-zon.Itwouldbeinterestingtoconsiderthepossibilityofadynamic

evolutionofthesequencealongthesimulationpathembeddedin theoptimizationloop.

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