modeling of technological performance trends design...

Modelingoftechnologicalperformancetrendsusingdesigntheory

SubarnaBasnet

MassachusettsInstituteofTechnology,DepartmentofMechanicalEngineering,77MassachusettsAve,Cambridge,Massachusetts02139

ChristopherL.Magee

MassachusettsInstituteofTechnology,InstituteforData,Systems,andSociety,77MassachusettsAve,Cambridge,Massachusetts02139

Abstract

Functionaltechnicalperformanceusuallyfollowsanexponentialdependenceontimebuttherateofchange(theexponent)variesgreatlyamongtechnologicaldomains.Thispaperpresentsasimplemodelthatprovidesanexplanatoryfoundationforthesephenomenabasedupontheinventivedesignprocess.Themodelassumesthatinvention‐novelandusefuldesign‐arisesthroughprobabilisticanalogicaltransfersthatcombineexistingknowledgebycombiningexistingindividualoperationalideastoarriveatnewindividualoperatingideas.Thecontinuingproductionofindividualoperatingideasreliesuponinjectionofnewbasicindividualoperatingideasthatoccursthroughcouplingofscienceandtechnologysimulations.Theindividualoperationalideasthatresultfromthisprocessarethenmodeledasbeingassimilatedincomponentsofartifactscharacteristicofatechnologicaldomain.Accordingtothemodel,twoeffects(differencesininteractionsamongcomponentsfordifferentdomainsanddifferencesinscalinglawsfordifferentdomains)accountforthedifferencesfoundinimprovementratesamongdomainswhereastheanalogicaltransferprocessisthesourceoftheexponentialbehavior.Themodelissupportedbyanumberofknownempiricalfacts:furtherempiricalresearchissuggestedtoindependentlyassessfurtherpredictionsmadebythemodel.Keywords:Modeling,design,combinatorialInvention,technologicalperformance

Nomenclature and terminology QJ=intensiveperformanceofartifactswithinatechnologicaldomain,Jt=timeIOI=individualoperatingideasPIOI=probabilityofcombinationofanytwoIOIIOI0=basicIOI‐IOIthatfirstintroduceanaturalphenomenonintheOperationsregimeIOIC=cumulativenumberofIOIintheOperationsregimeIOIL=maximumnumberofpossibleIOIinOperationsregimeattimetIOISC=IOICsuccessfullyintegratedintoadomainartifactK=annualrateofincreaseinIOIcintheOperationsregimeKJ=annualrate(whentimeisinyears)ofperformanceimprovementmeasuredbytheslopeofaplotoflnQJvs.timefi=fitnessinUnderstandingregimeforascientificfieldiFU=cumulativefitnessofUnderstandingregimedJ=interactionparameteroftechnologicaldomainJdefinedasinteractiveout‐linksfromatypicalcomponenttoothercomponentsinartifactsindomainJsJ=designparameteraffectingtheperformanceofanartifactindomainJAJ=exponentofdesignparameterinpowerlawfordomainJ,relatingperformanceandthedesignparameter

1. Introduction Inventionsaretheoutputsofthedesignprocesswhentheyreachsufficientnoveltyandutilitytoratethatterm:theyareabasicbuildingblockoftechnologicalprogressandthefundamentalunitofthispaper.Inourformulation,technologicaldomainsconsistofdesignedartifactsthatutilizeaspecifiedbodyofknowledgetoachieveaspecificgenericfunction(Mageeet.al.2014).Thus,technologicaldomainsinvolvealargenumberofinter‐relatedinventionsasevensingleartifactscanembodymultipleinventions.Arthur(2006)usedtheterm“technologies”todescribesomethingthatbridgesinventionsandtechnologicaldomains;accordingtoArthur,theseuse“effects”toachievesome“purpose”.Thus,onecanalsosaythateachartifactisamaterialrealizationofitsdesignthatintentionallyembodiestheeffects.Thispaperbringstogetherthreebodiesofresearchthatdonotusuallyinteract.Thefirstisthedesignresearchfield,particularlyitscognitivescientificinsightsonthedesignprocess.Thesecondisthetechnologicalchangefieldwheremostresearchershavebeeneconomistsorbusinessscholars.Thethirdareaisquantitativemodelingofperformanceofartifacts.Theobjectiveoftheworkreportedhereistouseunderstandingofthedesignandinventionprocesstomodelperformance‐howwellaspecificdesignedartifactachievesitsintendedfunctionorpurpose.Inparticular,weexamineperformancetrends‐thetimedependenceofperformanceasrealizedinaseriesofimproveddesignsofartifactsthatarise

overtime.Wedosoinanattempttodevelopanexplanatoryandquantitativepredictivemodelforwhyperformanceimprovesexponentiallyovermultipledesignswithwidelyvaryingratesamongtechnologicaldomains,rangingfrom3to65%annuallyfordomainscharacterizedsofar.Ourresearchquestioniswhetheraquantitativepredictivemodelbaseduponfoundationsandinsightsaboutthedesignprocessleadstoresultsconsistentwiththisexponentialbehaviorandwhethersuchamodelhelpsexplainandpossiblypredictthevariationintherateofimprovement.Wefirstdiscusssomerelevantliteratureineachofthethreeintersectingfields.

2 Background

2.1 Design, invention and cognitive psychology literature Whatconnectionsbetweentechnologicalchangeanddesignresearchcanbeinferredfromtheexistingliterature?Businessscholarsandeconomistsoftenviewtechnicalchangeasoccurringinsideablackbox,andhaveusuallyavoidedexaminingdesignactivitiesthatarethesourceoftechnologicalchange.AnimportantrecentpublicationthatbeginstobuildabridgebetweenaspectsofdesignresearchandtheeconomicsoftechnologicalchangeisthepaperbyBaldwinandClark(2006).Theseauthors(andLuoetal.2014)pointspecificallytoacentralrolefordesigninachievingeconomicvalue.Inadditiontoeconomicperspectives,anotherviewthatsomewhatignoresdesignisthelinearmodelaccreditedtoVannevarBush(Bush,1945),whichconsiderstechnologicalchangeoccurringthroughapplicationofscience.Asacounterview,inhisseminalbook,TheSciencesoftheArtificial,HerbertSimon(1969,1996)wasthefirsttohighlightthatdesignisanactivitystandingonitsownright,likenaturalsciences,andhasitsownsetoflogic,concepts,andprinciples.Whiletheprimarygoalofnaturalscienceistoproducepredictiveexplanationsofnaturalphenomena,theprimarygoalofdesignistocreateartifacts.Thedesignactivityiscentraltocreationandimprovementofartifactsinalltechnologicaldomainsandinvolvescognitiveactivitiessuchastheuseofknowledge,reasoning,andunderstanding.Theseindisputablecognitiveactivitieshavebeennotedbymanyscholarswhohavestudiedinventionanddesign(Simon1969,Dasgupta1996,GeroandKannengiesser(2004),HatchuelandWeil2009).Inthecontextofrealizinghigherperformancefromsubsequentgenerationsofartifacts,theroleofinvention,asoneoutcomeofthedesignprocess,isacriticalonesinceimprovementinperformanceofartifactsmuststronglyreflecttheinventions.AsVincenti(1990,pg.230)putsit,inventiveactivityisasourceofnewoperationalprinciples,andnormalconfigurationsthatunderliefuturenormalorradicaldesigns.Theoperationalprinciples(Polyani1962,Vincenti1990)ofanartifactdescribehowitscomponentsfulfilltheirspecialfunctionsincombiningtoanoveralloperationtoachievethefunctionoftheartifact.ModelsfoundusefulindescribingthecreativedesignprocessincludetheGeneploremodel(Finke,WardandSmith1996),topologicalstructures(BrahaandReich2003),FBStheory(GeroandKannengiesser2004),CKtheory(HatchuelandWeil2009),infuseddesign(Shaietal.2009),analyticalproductdesign(Frischknechtetal.2009),andothermodelingapproaches.Althoughalloftheseframeworksinclude–tosomedegree‐thekeyideaof

combiningexistingideas(forexample,intheformofconceptualsynthesis,andblendingofmentalmodelsdescribedindiscussionoftheGeneploremodel),theframeworkfoundmosthelpfulinourmodelingofperformancechangesresultingfromacumulativedesignprocessisanalogicaltransfer.AlthoughthisideacanbetracedasbeginningwithPolya(1945)orearlier,theframeworkremainsanactiveareaindesignresearch(Clementetal.1994,HolyoakandThagard1995,Goel1997,GentnerandMarkman1997,LeclerqandHeylighen2002,DahlandMoreau2002,ChristensenandSchunn2007,Linseyetal.2008,Tsengetal.2008,Linseyetal.2012,Fuetal.2013).Scholarsofanalogicaltransfer(GentnerandMarkman1997,Holyoaketal.1995andWeisberg2006)explainanalogicaltransferasinvolvingtheuseofconceptualknowledgefromafamiliardomain(base)andapplyingittocreateknowledgeinadomainwithsimilarstructure(target):analogicaltransferexploitspastknowledgeinboththebaseandtargetdomains.Theanalogiesutilizedcanbelocal,regionalorremote,dependingonsurfaceandstructuralsimilaritiesbetweenobjectsinvolvedinthebaseandtargetdomains.WeisbergdiscussestheexampleoftheWrightbrothersusingseveralanalogicaltransferstofirstrecognizeandsolvetheproblemofflightcontrol.First,theyviewedflyingasbeingsimilartobikinginwhichtheriderhastobeactivelyinvolvedincontrollingthebike,anapplicationofregionalanalogy.Interestingly,manyothersattemptingtodesignartifactsforflyingdidnotaccessthisregionalanalogyandthusdidnotevenidentifythekeycontrolproblem.Second,theWrightbrothersstudiedbirdstoseehowtheycontrolledthemselvesduringflight,andlearnedthattheyadjustedtheirpositionabouttherollingaxisusingtheirwingtips.Fromthisinsight,theyhadtheideaofusingsimilarmovingsurfaces,anotherinstanceofusingregionalanalogy.Lastly,theydevelopedtheideaofwarpingthewings,demonstratedbyusingatwistedcardboardbox,toactlikevanesofwindmillstomaketheairplaneroll.Theuseofthreeanalogicaltransfersincombinationtoseeandsolvetheflightcontrolproblemisaclearcaseofanalogicaltransferbutthereisalsoevidence(citedearlierinthisparagraph)ofmuchwiderapplicability.Therearemoreabstractversionsofcombinatorialanalogicaltransferthathavebeenproposedinthewiderliterature.BasedonanextensivehistoricalstudyofmechanicalinventionsanddrawinginsightsfromGestaltpsychology,Usher(1954)proposedacumulativesynthesisapproachforcreationofinventions.Thenotionofbisociation(Koestler1964,Dasgupta1996)developsthecumulativesynthesisapproachfurtherandsaysthatanewinventiveideaisideatedcombiningdisparateideas.Morerecently,Fleming(2001)andArthur(2006)haverespectivelyusedthesamecombinatorialnotionsofinventioninstudyingtechnologicalchange.Otherresearchinthetechnologicalchangeliteraturealsodiscussesarelatedconceptthatisusuallycalled“spillover”.Rosenberg(1982)showedthatsuchtechnologicalspillovergreatlyimpactedthequantityandqualityoftechnologicalchangeintheUnitedStatesinthe20thcentury–aresultsupportedbyNelsonandWinter(1982)andRuttan(2001).Indeed,arecentpaperbyNemetandJohnson(2012)statesthat“oneofthemostfundamentalconceptsininnovationtheoryisthatimportantinventionsinvolvethetransferofknowledgefromonetechnicalareatoanother”.Wenotethatthesedescriptionsdonotalwaysmakeacleardistinctionregardingwhetherthetransferisoccurringattheidealevelorattheartifactlevel.Theyaresilentregardinghowandfromwheredesignersorinventorsgettheirdisparateideastocombineandregardingdetailsaboutthecomplexitiesoftransferandcombination.

Analogicaltransferofideasasabroadmechanismandexpertiseasthefoundationofideas(Weisberg,2006)providesadequatespecificityformodelingscienceandinventioninthispaper.Weisbergcontendsthatanalogicaltransferisutilizedingenerationofbothscientificandtechnologicalknowledge.Existingknowledgeprovidesthefoundationalbasisforanalogicaltransfertooccur.Asimilarargumenthasbeenappliedtothemoreabstractnotionofcombinations.Usherdescribesacumulativesynthesisapproach‐afourstepsocialprocess(perceptionoftheproblem,settingthestage,theactofinsight,criticalrevision)‐whichbringstogetherinventivestructurestocreatenewinventions.Ruttan(1959),hasarguedthatUsher’sformulationprovidesa“theoryofthesocialprocessesbywhich‘newthings’comeintoexistencethatisbroadenoughtoencompassthewholerangeofactivitiescharacterizedbythetermsscience,invention,andinnovation”.ModelsofbothUnderstandingandOperationsregimeinourpaper(definedinthenextparagraph)utilizetheabstractionthatknowledgeiscreatedbyprobabilistically1combiningexistingknowledgemadeavailablebyanalogicaltransfer.Vincenti(1990),andMokyr(2002)taketheviewthatscientificandtechnologicalknowledgecanbeclassifiedintodescriptive(Understanding)andprescriptive(Operations)knowledge2regimes.TheUnderstandingregimecanbeseenasabodyof‘what’knowledgeandincludesscientificprinciplesandexplanations,naturalregularities,materialsproperties,andphysicalconstants.Aunitofunderstandingisfalsifiable(Popper1959)andenablesexplanationandpredictionaboutspecificphenomena,includingbehaviorofartifacts.TheOperationsregime,ontheotherhand,canbeviewedasabodyof‘designknowledge’,whichsuggestshowtoleveragenatural‘effects’(Arthur,2006,Vincenti,1990))toachieveatechnologicaladvantageorpurpose.Itincludes,operatingprinciples,designmethods,experimentalmethods,andtools(Dasgupta1996,Vincenti1990).Basedonthisdistinction,understandingenablesgenerationofoperationalknowledge,whichultimatelycontributestowardsdesignofsomeartifact.However,operationsisnotentirelybaseduponexistingunderstandingandinfactinnovationsinknow‐howcanandoftendooccurbeforeanyunderstandingofrelatednaturaleffectsisavailable.AnimportantaspectofdesignandinventionisthecooperativeinteractionbetweenUnderstandingandOperationsregimes(Musson,1972,MussonandRobinson1989).Usingahistoricalperspective,Mokyr(2002)hascarefullyobservedthatasynergisticexchangebetweenthetwohasbeenoccurring,whereeachenablestheother.ThecontributionofUnderstandingtoOperationsiswellknown:itprovidesprinciples,andregularitiesofnaturaleffects,includingnewones,intheformofpredictiveequations,anddescriptive1Atapointintime,notallpossiblecombinationsofexistingknowledgeleadtonewknowledge.2Weusetheterms“Understanding”and“Operations”,sinceeachonebringsmoreclaritytothenatureofunderlyingactivity.Understandingreferstoconceptualinsightthatisgeneratedaboutanobjectorenvironment,whereasOperationsreferstotheideaofactingonanobjectorenvironmenttogetsomedesiredeffect,aswellasexperimentalmethodsincludedintheterm‘science’.

facts,suchasmaterialproperties.FlemingandSorenson(2004)provideevidencethatunderstandinghelpsinventorsbyprovidingarichermaptosearchforoperatingideas,whichcanbecombinedtogether.Understandingalsoprovidesinsightaboutwherenewtechnologicalopportunitiesmaybefound(Klevoricetal.1995).Beyondthesecontributions,thereisthemoregeneralview,discussedintheinitialparagraphofthissection,thatnewoperationalideascanbederivedfromnewunderstanding.Whatislessdiscussedisthemulti‐facetedcontributionsofOperationstotheUnderstandingregime.Inhispaper,Sealingwaxandstring,deSollaPrice(1983),aphysicist,andhistorianofscience,highlightedthatinstruments(anoutputoftheOperationsregime)wereadominantforceinenablingscientificrevolutions.Hestates:“changesinparadigmthataccompanygreatandrevolutionarychanges(inscience)werecausedmoreoftenbyapplicationoftechnologytoscience,ratherthanchangesfrom‘puttingonanewthinkingcap’“.Operationsprovidetoolsandinstrumentstomakemeasurements,andtomakenewdiscoveries.Inhisbook,TheScientist:AHistoryofScienceToldThroughtheLivesofitsGreatestInventors,Gribbin(2002),aBritishastrophysicist,andsciencewriter,hasdescribedhowtheabilitytogrindeyeglasslensesmadeitpossibletomakebettertelescopes,andhencepavedthewayforastronomerstomakenewdiscoveries.Neworimprovedobservationaltechniquesarestillamajordriverofprogressinscience.Gribbinhasaptlysummarizedtheenablingexchangebetweenthetworegimes:“newscientificideasleading…toimprovedtechnologyandnewtechnologyprovidingscientistswiththemeanstotestnewideastogreaterandgreateraccuracy”.Additionally,theOperationsregimeprovidesnewproblemsfortheUnderstandingregimetostudy,andhasledtobirthofnewfieldsinUnderstanding(Hunt2010).Basedupontheseinsightsandwithourfocusonexplainingperformanceimprovementarisingfromcontinuingstreamsofinventions,ourmodeltreatsmutualexchangebetweenUnderstandingandOperations.Indesignofartifacts,Simon(1962)introducedthenotionofinteractionsinhisessayonthecomplexityofartifacts.Whenadesignofanartifactischangedfromonestatetoanother(withdifferencesbetweenthetwostatesasdefinedbymultipleattributes,sayD1,D2,andD3)bytakingsomeactions(say,A1,A2,andA3),inmanycases,anyspecificactiontakenmayaffectmorethanoneattribute,thuspotentiallymanifestingasinteractionsoftheattributes.Thesamenotionofinteraction/conflictsiscapturedbytheconceptofcouplingoffunctionalrequirements(Suh2001),ordependenciesbetweencharacteristics(Weber2003),whichcanoccurwhentwoormorefunctionalrequirementsareinfluencedbyadesignparameter.Theoreticallyitseemsidealtohaveonedesignparametercontrollingonefunctionalrequirementtoachieveafullydecomposable(modular)design(Suh2001,BaldwinandClark2000).However,Whitney(1996,2004)hasarguedthat,inreality,howdecomposableadesignofanartifactcanbedependsonthephysicsinvolvedoradditionalconstraints,suchaspermissiblemass.Thesearereflectedascomponent‐to‐component,andcomponent‐to‐systeminteractions,orasaneedtohavemulti‐functionalcomponents.Consequently,Whitneyargues,complexelectro‐mechanical‐optical(CEMO)systems,primarilydesignedtocarrypower,cannotbemadeasdecomposableasVLSIsystemsprimarilydesignedtotransmitandtransforminformation.Forexample,inenergyapplications,theimpedanceoftransmittingandreceivingelementshastobematchedformaximumpowertransfer,thusmakingthetwoelementscoupled.Further,CEMOsystemstypicallyneedtohavemulti‐functionalcomponentsinordertokeeptheartifactsize

reasonable,creatingcouplingofattributesatthecomponentlevel.AnothertypeofinteractionWhitneyhasidentifiedarethesideeffects,suchaswasteheatincomputers,andcorrosionofelectrodesinbatteries‐thatoccurinartifacts,whichinsomeelectro‐mechanicalsystemscanconsumesignificantportionofthedesigneffortfortheirmitigation.Thepresence,andthustheresolution,ofthesedifferentinteractionscausesignificantdelay,consumesignificantengineeringresourcesandpotentiallystopapplicationsofsomeconcepts,thusmakingthelevelofinteractionsofatechnologicaldomainapotentiallystrongfactorinfluencingitsrateofimprovement.BaseduponWhitney’swork,theeffectofinteractionsonratesofimprovementwassuggestedqualitativelybyKohandMagee(2008)andaquantitativemodeloftheeffectwasdevelopedbyMcNerneyetal.(2011)–seesection2.3.Theinfluenceofdesignparametersonartifactperformanceisanessentialpartofdesignknowledge.Manytechnologicaldomainshavecomplexmathematicalequationsrelatingsomeaspectsofperformancewithdesignparameters.Indeed,theso‐calledengineeringscienceliteraturehassuchequationsformanyaspectsaffectingthedesignofartifactsofperhapsalltechnologicaldomains.Simplerrelationshipsconcerningthegeometricalscaleofartifactsarealsoavailableandgenerallygiveperformancemetricsasafunctionofadesignvariableraisedtoapower.Useofpower‐lawrelationshipscanbefoundin:1)Sahal(1985)whostudiedscalinginthreedifferentsetsofartifacts‐airplanes,tractors,andcomputers;and2)BelaGold(1974)whodemonstratedthatdoublingthesizeofablastfurnacereducestheircostbyabout40%.Theconstantpercentchangeperdoublinginsizeresultsfromthepowerlaw(assumedbyGold)betweenperformance/costandgeometricalvariablessuchasvolume.

2.2 Technological change literature Whatdescriptivemodelsandtheorieshelpusunderstandwhytechnologiesimproveandhowtheimprovementpatternsarestructured?Schumpeter(1934)introducedtheideathatentrepreneurs,whoseprimaryroleistoprovideimprovedproductsandservicesthroughinnovation,driveeconomicprogress.Theseinnovations,whichSchumpeterdescribesasindustrialmutations,displacecompetingproductsandservicesfromtheeconomy.However,they,too,aredisplacedbyhigherperforminginnovationsthatfollow,thusperpetuatingthecycleofcreativedestruction.BuildinguponSchumpeter’snotion,Solow(1956)recognizedandincorporatedtechnologicalchangeasthekeyelementinhisquantitativeexplanatorytheoryofeconomicgrowth.Thebasicconclusionthattechnologicalchangeisthefoundationofsustainedeconomicgrowthhasstoodthetestoftime.Latertheoristsofeconomicgrowth(Arrow1962,Romer1990,Acemoglou2002)haveattemptedtodealwiththemorecomplexproblemofembeddingtechnologicalchangewithintheeconomy(endogenoustodifferentdegrees).Althoughthelatertheoriesareimportant,theissuesareoutsidethescopeofthispaperandwillnotbecoveredhere.Arelatedquestionofdemand‐pullandtechnology‐pushdoeshavemorerelevance. Whatdrivestechnologicalinnovation?Someearlyexplanationsemphasizedpuredemandpush(CarterandWilliams(1957,1959),Bakeretal.1967,MyersandMarquis1969,Langrishetal.1972,Utterback1974)wheretheneedsoftheeconomyatagiventime

dictatetechnologicaldirection.MoweryandRosenberg(1979)reanalyzedthedataandmethodologyinthisearlyworkandarrivedatastrongroleforscience/technologypush(thediscoveriesofscientistsandinventorsprimarilydeterminetechnologicaldirection).Takingabalancedview,Dosi(1982)arguedthatbothmarket‐pull(customerneedsandpotentialforprofitability)andtechnology‐push(intheformofpromisingnewtechnology,andtheunderpinningprocedures)areequallyimportantforbeingsourcesofinnovation.TushmanandAnderson(1986)discussdiscontinuitiesashavinglargesocio‐technicaleffectsandnotethatsuchdiscontinuitiesareanessentialelementoftechnologicalchange.Inanotherhighlyreferencedpaper,HendersonandClark(1990)emphasizetheimportanceofarchitecturalchangeofartifacts‐asopposedtocomponentchange‐havinglargeeffectsonthefirm‐levelimpactofchange.Christensen(1996),ontheotherhand,viewstechnologicalchangeoccurringasaseriesofdisruptiveproductinnovationsthatstartinanichemarketcateringtodifferentfunctionalrequirements,butthenrapidlyimprovetowardstherequirementsofmainstreamperformance.Thedisruptivetechnologysurpassesthematuremarketleaders(byachievingthenecessaryperformanceinsmaller,cheaperartifacts),anddisplacesthem.Alloftheconceptsoftechnologicalchangedescribedintheprecedingparagraphs‐atleastimplicitly‐dependuponrelativeratesofchangeofperformance.Thisisthefocusofourmodelingeffortsowewillnowbrieflyreviewconceptsrelatedtotrendsinperformanceofdesignedartifacts,andwhatpatternstheyhavefollowed.Wefirstreviewtwoestablishedframeworks–generalizationsofWright’searlyresearch,andMoore’sLaw‐fordescribingtrendsintechnologicalperformance.In1936TheodorePaulWright(1936)inhisseminalpaper“FactorsaffectingtheCostofAirplanes”forthefirsttimeintroducedtheideaofmeasuringtechnologicalprogressofartifacts.Fromhisempiricalstudyofairplanemanufacturing,hedemonstratedthatlaborcostortotalcostofspecificairplanedesignsdecreasedasapowerlawagainsttheircumulativeproduction.Thisrelationshipisexpressedas: C=C0P‐w (1) WhereC0,andCareunitcostofthefirst,andsubsequentairplanesrespectively,andwherePandwarecumulativeproductionanditsexponentthatrelatesittounitcost.Wrightexplainsthatlaborcostreductionsarerealizedasshopfloorpersonnelgainexperiencewiththemanufacturingprocesses,andmaterialusageandhaveaccesstobetterproductiontools.SinceWright’swork,thisapproachhasbeenusedtostudyproductionofairplanesandshipsduringWorldWarII,andextendedtoprivateenterprises(Yelle,2007).ItshouldbenotedthatWrightdidnotlookatimprovementduetonewdesigns,insteadheonlyconsideredimprovedmanufacturingofafixeddesign.GordonMoore(1965)presentedthesecondapproach‐usingtimeastheindependentvariableandinvestigatingaseriesofnewlydesignedartifacts‐inhisseminalpaperthatdescribesimprovementofintegratedcircuits.Heobservedthatthenumberoftransistorsonadiewasdoublingroughlyevery18months(modifiedto2yearsin1975).This

exponentialrelationshipbetweenthenumberoftransistorsonadieandtime,famouslyknown3asMoore’sLaw,canbemathematicallyexpressedas: QJ(t)=QJ(t0)exp{KJ(t‐t0)} (2)

WhereQJ(t0)andQJ(t)arethenumberoftransistorsperdie(ameasureofperformance)attimet0andtimet,andKJistherateofimprovement(annualiftimeisinyears).Forintegratedcircuits,theexponentialrelationshiphasheldbroadlytrueforfivedecades.Others(Girifalco1991,Nordhaus1996,KohandMagee(2006,2008)andLeinhard2008)utilizedthistemporalapproachtostudyperformanceofdifferenttechnologies,andhavedemonstratedthatmanytechnologiesexhibitexponentialbehaviorwithtime.Morerecently,Mageeetal.(2014)extendedthestudyto73differentperformancemetricsin28differenttechnologydomains.Theperformancecurveshavecontinuedtodemonstrateexponentialbehavior,althoughannualratesvarywidelyacrossdomains.WenotethatMooreandallotherswhousedhisframeworkbasicallycomparedtheperformanceofdifferentdesignsovertimedifferentiatingtheWrightandMooreframeworks.However,itisalsopossibletousetheWrightframeworkfordifferentdesignsbutonlyiftheamountproducedincreasesexponentiallywithtime(Sahal,1979,Nagyetal.2013,Mageeetal2014).Inordertoclarifyforreadersthenatureofempiricalperformancedata,wepresentperformancedatafortwosampledomains,magneticresonanceimaging(MRI)andelectricmotors(Fig.1a),andasummaryofimprovementratesfor28domains(Fig.1bfromMageeetal.2014).Theexponentialtrendforeachdomaincanbedescribedbyequation(2),whereQJ(t)andQJ(t0)aretheintensiveperformanceofanartifactindomainJattimetandt0,andKJistheannualrateofimprovementofthedomaininquestion.

3ThisdesignationwasgiventotherelationshipbyCalTechprofessorCarverMead.

Fig. 1a: Exponential growth of performance in sample domains – Electric motor and Magnetic resonance imaging (MRI). Adapted from Magee et al. 2014 with permission.

Fig. 1b: Annual rate of performance improvement, KJ, for 28 domains. Adapted from Magee et al. 2014with permission.

Elec. MotorRate = 3.1 %R² = 0.9657

Rate = 21.3 %R² = 0.8561

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1.E+01

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1880 1910 1940 1970 2000 2030 2060

MRI (resolution/tim

Electric motor (W

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Electric Motor MRI

Arecentpaper(BensonandMagee,2015a)hasempiricallyinvestigatedthevariationoftheimprovementratesinthese28domains.Theworkhasimportantrelationshipstothecurrentworksowedescribeittonotetherelationshipsbuttoalsoclarifythefundamentaldifferences.BensonandMageefoundstrongcorrelationsbetweenspecificmeta‐characteristicsofthepatentsinthe28domains4andtheimprovementrateinthedomains.Theseauthorsfoundthatpatentmeta‐characteristicsreflectingtheimportance(citationsperpatentbyotherpatents),recency(ageofpatentsinadomain)andimmediacy(theaverageovertimeoftheusageofcurrentnewknowledgeinthedomain)areallcorrelatedwiththeimprovementrate.Theyfoundaparticularlystrongcorrelation(r=0.76,p=2.1x10‐6)withametricthatcombinesimmediacyandimportance(theaveragenumberofcitationsthatpatentsinthedomainreceiveintheirfirstthreeyears).Thefindings(andassociatedmultipleregressions)arerobustovertimeandwithdomainselectionandareofpracticalimportanceinpredictingtechnologicalprogressindomainswhereperformancedataisnotavailable(BensonandMagee,2015).Nonetheless,theconceptualbasisforthefindingsisobservedattributesoftheinventiveoutputfromatechnologicalfield(importance,recencyandimmediacyofapatentset)andnottheprocessofinvention,designknowledgeorothertechnicalaspectsofdesignedartifactsinthedomain.Theaimoftheworkreportedinthepresentpaperistodevelopamodelthatyieldsinsightsaboutthepaceofchangewithoutrecoursetoconceptsbaseduponobservationoftheoutputovertime.Iffullysuccessful,wewouldbeabletojudgethepotentialforchangebasedonlyuponthenatureofthedesignknowledgeandwemightevenbeabletofindnewapproachesthatmightachievetechnologicalgoalsatmorerapidimprovementrates.

2.3 Literature on quantitative modeling of technological change Whatresearchhasattemptedtomodelthetechnologicalperformancetrendsthatwejustdiscussed?Muth(1986)andAuerswaldetal.(2000)havedevelopedmodelstoexplainWright’sresultsbyintroducingthenotionofsearchfortechnologicalpossibilities.Eachpaperassumesthatrandomsearch,akeyelementoftechnologicalproblemsolving,forabettertechniqueismadewithinafixedpopulationofpossibilities.Consideringacaseofasinglemanufacturingprocess,Muth(1986)developedamodeltocapturetheideaofsubstitutingmanufacturingsequenceswithbetterones.Hearguesthatshoppersonnelimprovetheprocessbylearningthroughexperienceandmakingrandomsearchfornewtechniques,whichenableimprovementofprocessesleadingtocostreductions.Muthdemonstratedthatthenotionoffixedpossibilitieseasilyleadstofewerandfewerimprovementsthatcanberealizedandhearguesthatthedata(forfixeddesigns)showsalevelingoffandeventualstoppageasthemodelsuggests.BuildingonMuth’sideaofrandomsearchwithinasetoffixeddesignpossibilities,Auerswaldetal.modeledamulti‐processsystem,inwhichdifferentprocessescanbecombinedtocreatediverserecipes,andforthefirsttimeintroducedthenotionofinteractionsbyallowingadjoiningprocessestoaffecteachother’scost.

4Thepatentsarefoundbyanewtechnique‐BensonandMagee2015b

FollowingsimilarreasoningasMuthandAuerswaldetal.,McNerneyetal.(2011)havedevelopedastochasticmodeltoexplainhowthecostreductionofamulti‐componentsystemisinfluencedbycomponentinteractions,whichtheyrefertoasconnectivitybetweencomponents.McNerneyetal.operationalizedthenotionofinteractionsasout‐linksrepresentinginfluenceofacomponentonothercomponents.Whenaspecificcomponentinadomainartifactchangesbyintroducinganewoperationalidea,thechangeaffectsthedesignofallthecomponentsitinfluences.Iftheperformanceoftheartifact(influencingandinfluencedcomponents)asawholeimproves,thenMcNerneyetal.considertheinteractionstoberesolvedandtheoperatingideaisconsideredsuccessful.TheMcNerneyetal.paperdemonstratesthatartifactswithmoreinteractionsimprovemoreslowlythanartifactswithlessinteractions.Usingagent‐basedmodeling,Axtelletal.(2013)havedevelopedacompetitivemicro‐economicmodeloftechnologicalinnovationutilizingthenotionoftechnologicalfitness.AlthoughtheydonotdiscussorciteMoore’slaworhiswork,theyhavedemonstratedthatcumulativetechnologicalfitnessofallagentsincreasesexponentiallyovertime.ThisisdifferentfromotherresearcherswhohavepredominantlybeenfocusedonWright’sframework.Consistently,Axtelletal.considernewdesignsandnotjustprocessoptimization.Usingasimulationapproach,ArthurandPolak(2006)havemodeledhownewgenerationsofartifactsarisebycombiningcurrentlyavailableartifacts.Theartifactsconsideredareelectroniclogicgates.Newdesigns(combinations)aremorecomplexlogicgatesthatcanthenalsobecombinedintoevenmorecomplexlogicgates.Intheirmodel,ArthurandPolakspecifyseveraldesigngoalstowardstowhichthelogicgatesevolve.Theyhavedemonstratedthatdesignswithhigherlevelsofcomplexitycannotbeattainedwithoutrealizingdesignconfigurationswithintermediatelevelsofcomplexity,andnewdesignswithhigherfunctionalitysubstituteforcurrentdesignswithinferiorfunctionality.Thismodelismuchricherthanothermodelsinrepresentingtheartifactpartofthedesignprocess;however,itdoesnotconsiderperformanceimprovement,asdotheothermodels.Itisalsolimitedtodevelopingpre‐specifiedartifactsandisthusaspecificprocess;consequently,itisnotopen‐endedorgeneralwhicharecharacteristicsnecessaryformodelingperformancetrendsforgeneraltechnologicaldomains.Althoughsomearemoreexplicitthanothers,onefeaturecommontoallthesemodelsisthatallutilizethenotionofbuildingupontheperformance(intheformofcost)ordesignsofthepast,akeyfeatureofcumulativeprocessesincludedinthemodelpresentedhere.Ontheotherhand,theydonotconsidertwoaspectswebelieveusefulinansweringourresearchquestion.First,noneofthemdiscussesorincludestheinfluentialroleplayedbyexchangebetweenscienceandtechnology.Inthispaper,wetreatthedesignprocessandtheexchangebetweenscienceandtechnologyasimportantelementsforunderstandingthechangeinperformanceovertimethatinturnisessentialtounderstandingtechnologicalchange.Second,noneconsiderthedesignprocessoroperatingprinciplesandinsteadlookatcombinationsattheartifactlevelinsteadofcombinationofideas.

3. Overview of the model

3.1 Conceptual basis of model Thedesiredoutputfromtheconstructedmodelareperformanceimprovementrates.Toagreewithknownempiricalresults,performanceshouldincreaseexponentiallywithtime.Weutilizetwosetsofmechanismsfromdesigntoconstructtheoverallmodel.Thefirstset,whichgivesrisetoexponentialtrends,includesgrowthofknowledge‐understandingandoperations‐usingcombinatorialanalogicaltransferaidedwithmutualexchangebetweenthetwo.Thesecondset,whichgivesrisetovariationinimprovementrates,includescomponentinteractionsandscalingofdesignvariables.Sincethegoalofthemodelistodevelopanexplanatoryandquantitativepredictivemodel,whilemodelingthesemechanismswehave,wherenecessary,simplified(removeddetails)andutilizedabstractiontokeepthemodeltractable.TheoverallarchitectureofthemodelisshowninFigure2.BasedontheworkofVincenti(1990)andMokyr(2002)thatwediscussedearlier,weclassifyscientificandtechnicalknowledgeintoUnderstandingandOperationsregimes.WefurthersplittheOperationsregimeintoideaandartifactsub‐regimeswherenon‐physicalrepresentationofartifactsareintheideasub‐regime.Theideasub‐regime,representedasanideaspool,consistsofindividualoperatingideas(IOI).TheIOI(individualoperatingidea)conceptisanabstractionandgeneralizestheideaofoperatingprincipleintroducedbyPolyani(1962)andincludesanyideas,includingoperatingprinciples,inventionclaims,designstructures,componentintegrationtricks,tradesecretsandotherdesignknowledgethatleadtoperformanceimprovementofartifacts.AnIOIisdifferentthanaunitofunderstanding(UOU)whichincludesscientificprinciples,andfactualinformation.Anexampleofaunitofunderstanding(UOU)istheprincipleoftotalinternalreflection,whichdescribeshowabeamoflightundergoesreflectioninsideadensemedium,whentheangleofincidenceisaboveacriticalvalue(seeFig.3).Thisprincipleaccuratelydescribesanaturaleffect,butitdoesnotprescribehowwecanuseittotransmitinformation.Ontheotherhand,apairofparallelsurfaces(orafiber)enclosingadensemediumandutilizingtheprincipleoftotalinternalreflectionprovidesamechanism–anoperatingprinciple‐tomakearayoflighttraveldownthelengthofthemedium(seeFig.3).SuchamechanismisanexampleofanIOI.Unlikeartifacts,whichbelongtoaspecifictechnologicaldomain,wemodelIOIintheideas(IOI)poolasbeingnon‐domainspecificandavailabletoalltechnologicaldomains.Forinstance,theoperatingprincipleoftotalinternalreflectionisutilizedinfiberoptictelecommunications,fluorescentmicroscopy,andfingerprinting,verydistincttechnologicaldomains.Intheideasub‐regime,designers/inventorssourceexistingideas(IOI)usinganalogicaltransferandcombinethemprobabilisticallytocreatenewideas(IOI).OncenewIOIaresuccessfullycreatedthroughprobabilisticcombination,theybecomepartoftheIOIpool,thusenlargingthenumberofideas(IOI)inthepoolforcombination.Itisimportanttoclarifythatmodelconsiderscombinationsattheideaslevelrathercombinationofcomponents,withtheformerbeingfundamentalandallowingcombinationofideasfromdifferentfieldsusinganalogicaltransfer.

Fig.2:ModelofexchangebetweenUnderstandingandOperationsregimesandmodulationofIOIassimilationbyinteraction(dJ)andscaling(AJ)parametersofdomainJ.Exampleofunitofunderstanding(UOU)

Exampleofincrementaloperatingidea(IOI)

Principleoftotalinternalreflection

Totalinternalreflectionbetweenparallelsurfacesenclosingdensemedium:mechanismtomakelighttravellongitudinally(fiberoptics)

Fig.3Examplesofunitofunderstanding(UOU)andincrementaloperatingidea(IOI)

Understanding

regime

IOIpoolIOI C,K

Operationsregime

DomainJ

Interactions

Perf.ScalingAJ

Interactions

Artifact

WemodelgrowthintheexplanatoryreachoftheUnderstandingregimebysimulatingasimilarcombinatorialanalogicaltransferprocess.TheUnderstandingregimeisconceptualizedtoconsistofunitsofunderstanding(UOU).Theunitsofunderstanding(UOU)fromdifferentfieldswithintheunderstandingregimeparticipatetocreateanewunitofunderstanding(UOU)thatpotentially(probabilistically)hasagreaterlevelofexplanatoryandpredictivepower.FollowingthetreatmentinAxtelletal.(2013),wemodeltheexplanatoryandpredictivepowerofafieldofUnderstandingasafitnessparameter,fi.IfthenewUOUhasagreaterfitnessvalue,itreplacestheUOUwiththesmallestfitnessvalue.Sinceourprimaryfocusisonperformance‐theoutputoftheOperationsregime,wesimulatetheUnderstandingregimeonlyatthishigherabstractionlevel.Althoughbothregimes–UnderstandingandOperations–evolveindependently,theycannotdosoindefinitely.WemodelthedeSollaPriceandGribbininsightsbyhavingeachregimeactasa“barrier‐breaker”fortheotherregime.Wheneachregimehitsabarrier,theothercaneventuallyaidinbreakingthebarrier:infusionofunderstandingenablescreationofimportantIOIintheOperationsregime;andinfusionofnewoperationaltoolsenablenewdiscoveriesintheUnderstandingregime.Theperformancesoftheartifactsintechnologicaldomainsareimprovedbyaseriesofdesigns/inventions(IOI)overtime.IOIenabledesignerstochangespecificcomponentsinthedomainartifactleadingtoapotentialimprovement.FollowingMcNerneyetal.’streatment,theIOIinquestionisassimilatedonlyiftheperformanceoftheartifactoverallimproves.Another,andfinal,factorthatwemodelisscaling,apropertyinherentinthephysicsofthedesignoftheartifact.5,6ThesuccessfullyassimilatedIOI,whichwerefertoasIOIS,effectimprovementofthedomainartifactbyenablingfavorablechangeofarelevantdesignparameter.Thedesignparameterisincreasedordecreasedsuchthatitleadstoimprovedperformance7.Scalingreferstohowchangeinadesignparameterrelatestorelativechangeintheperformanceofanartifact.Theformulationweuseinthemodelisthatrelativeperformancechangeisrelatedtodesignparametersraisedtosomepower,inotherwordsscaled.Ascoveredinsection2.1,thisisthemostwidelyusedfunctionalrelationshipwithdecentempiricalsupportandtheoreticaljustificationinsomecases(Barenblatt1996).

5Recallthattheperformanceweconsiderinthispaperisintensive,e.g.,energydensity,w/cm3.6Inrelationstoartifactssuchassoftware,physicsreferstothemathematicsbehindthesoftware.7Taguchi(1992)notedthatsomephenomenatendtoworkbetterwhencarriedoutatasmallerscale(“smallerisbetter”),whileotherarebetteratlargerscale(“largerisbetter”).Integratedcircuits,forexample,performbetterasdimensionsarereduced,sincesmallerdimensionsleadtoshorterdelays,andhigherdensityoftransistors,bothofwhichcontributetowardsimprovedcomputationpervolumeorcost.

3.2 Mathematical summary Aperformance(intensive)metricofadomain,labeledQJ,isafunctionofasetofdesignparameters(s1,s2,s3)ofadomainartifactandtimebutforsimplicityhereweconsideronlyasingleparameter(s).ThedesignparameterischangedbyIOIs(successfullyassimilatedIOIintodomainartifacts),whichinturnareassimilatedfromIOIC(numberofaccumulatedoperatingideasintheIOIpoolshowninFigure2).IOICisafunctionoftime.Equationsdescribingthesenestedvariablesinlogarithmicformare: lnQJ=f1(lns);lns=f2(lnIOISC);lnIOISC=f3(lnIOIC);lnIOIC=f4(t) (3) Assumingthatthefunctionsarecontinuousandalldependenceisthroughthenamedvariables,thechainruleisappliedandyields dlnQJ/dt=dlnQJ/dlns∙dlns/dlnIOISC∙dlnIOISC/dlnIOIC∙dlnIOIC/dt (4) Thefirsttermontherighthandsiderepresentsrelativeimpactofdesignvariablechangeonperformancechange,whichwillbeshowninsection4.5tobeequaltothescalingparameter(AJ)whenQJfollowsapowerlawins:dlnQJ/dlns=AJ.Thesecondtermisthe‘smaller‐is‐better/larger‐is‐better’factor,andcapturesthenotionwhetheradesignvariablehastobeincreasedordecreasedinordertoimproveperformance.Wecapturethisdependenceusinganabstractionandequatedlns/dlnIOIsc=+/‐1. Thus,equation(4)becomes

dlnQJ/dt=AJ∙(±1)∙dlnIOISC/dlnIOIC∙dlnIOIC/dt

Thethirdtermontherightofequation(5)represents‘difficultyofimplementingideas’inspecificdomains,andthusrelatesthedomainspecificsuccessfulIOISCtotheIOICinthepool:wewillshowinsection4.4‐followingMcNerneyetal.‐thatdlnIOISC/dlnIOIC=1/dJ,wheredJistheinteractionparameterintroducedbyMcNerneyetal.Finally,thefourthtermrepresentstherateofideaproduction.K=dlnIOIC/dtisarrivedatbyasimulationofcombinatorialanalogicaltransferwhichispresentedinthefirst(following)sectionoftheresults.

4. Results

4.1 Overall IOI simulation Asnotedinsection3.1,wemodeltheIOIasresultingfromcombiningknowledgefrompriorIOIbyprobabilisticanalogicaltransfer.Fig4aschematicallyrepresentscombinationofIOI,inwhichspecificIOIaandbcombinetocreateIOIdwithaprobability,PIOI.Ifthiscombinationattemptsucceeds,thenewlycreatedIOIdthenisaddedtothepoolofIOI(Fig4b).Insubsequenttimesteps,IOIdcanattempttocombinewithanotherspecificIOIinthepool,suchasIOIc,toprobabilisticallycreateamoreadvancedIOIe.Ascombinationadvances,thecumulativenumberofindividualoperatingideas,IOICgrows.WefurthermakethedistinctionbetweenderivedIOIandbasicIOI,whichwelabelasIOI0.IOI0are

fundamentalIOI,whichfirstintroduceanaturaleffectintoanoperationalprincipletoachievesomepurpose.Theexample(describedinsection3.1)ofapairofcloseparallelsurfaces(orafiber)enclosingadensemediumandutilizingprincipleoftotalinternalreflectiontotransmitabeamoflightlongitudinallycanbeviewedasanexampleofanIOI0.Incontrast,derivedIOI,justasthetermsuggests,areobtainedthroughcombinationoftwoIOI0,orbetweenanIOI0andaderivedIOIorbetweentwoderivedIOI.Inthissense,IOIa,b,andcinthefigurerepresentIOI0andIOIdande,derivedIOI.

Inonerunofthesimulation,westartwiththeinitialnumberofbasicindividualoperatingideas,IOI0.Ateachtimestep,themaximumnumberofcombinationsweallowtobecreatedisequaltohalfthenumberoftotalIOIavailable.Theintentionistoalloweachoperatingideatocombinewithanotheroperatingideaoncepertimesteponaverage.Figure5showsresultsfromasimulationrunstartingwith10basicIOIandaprobabilityofcombination,PIOI,equalto0.25.Figures5aand5bwithtimestepsontheX‐axisandthecumulativenumberofoperatingideas,IOIContheY‐axisshowthatthecumulativenumberofoperatingideas,IOIC,growsexponentiallywithtimeatanimprovementrate(K)of0.116.

Forthissimplifiedcase,therateofgrowthofIOI,K,canbemathematicallyshowntobeequaltoln(1+PIOI/2),=0.118whichcanbeeasilyderivedasfollows:

Atinatimestept,numberofIOInewlycreated=PIOI∙IOIC(t)/2 (6)

Fig.4:Combinationofindividualoperatingideasa)basicandderivedIOIb)accumulationofIOIthroughfeedback

IOIpool BasicIOI:a,b,cDerivedIOI:d,e

IOIC IOIC

Fig.5:GrowthofIOICovertime:initialIOI0=10,probabilityofcombination,PIOI=0.25:(a)linearY‐axis(b)logarithmicY‐axis.

IOIC(t+1)=IOIC(t)+PIOI∙IOIC(t)/2=IOIC(t)∙(1+PIOI/2) (7)

RatioofIOICbetweenconsecutivetimesteps,r=IOIC(t+1)/IOIC(t)=(1+PIOI/2) (8)

Then,ingeneral,IOIc(t)canbewrittenintermsofaninitialIOI0andratio,randtimestep,t;theexpressioncanbestatedinanexponentialform.

IOIC(t)=IOI0rt=IOI0exp{lnr∙t}=IOI0∙exp{ln(1+PIOI/2)∙t}=IOI0∙exp{k∙t} (9)

Where,therateofgrowthofIOIC(t),

K=ln(1+PIOI/2) (10)

ForverysmallvaluesofPIOI,

K≈PIOI/2 (11)

Thesimulationresultstothispointassumethatindefinitelylargenumbersofoperatingideas,IOI,canbecreatedoutoffewbasicIOI.ThisisbecausethemodelassumesthatthesameoperatingideascanberepeatedlyusedtocreatenewIOIwithoutlimit.(Forexample,recombining(a,b)witha,thenwithbwouldgivenewoperatingIOI(((a,b),a),b)andeventuallyanarbitrarilylargenumberofa,bpairs.IndefinitemultipleusesofthesamebasicideatocreateinnumerableIOIdoesnotappeartoberealistic.Inordertobetterreflectthisintuition,weintroduceaconstraintthatanyderivedIOIcanutilizeanIOI0onlyonce.TheconstraintoperationalizesthenotionthatcountingrepetitioususeofbasicIOIasnewdesignsthatpotentiallyimproveperformanceisunrealistic.Accordingtothis

0 20 40

IOI0 = 10

1.E+00

1.E+01

1.E+02

1.E+03

0 20 40

IOI0 = 10

constraint,derivedIOI((a,b),c)inFigure4wouldbeallowed,butnot((a,b),b).Employingthisconstraint,thesimulationyieldstheresultsinFig.6a,asemi‐loggraph,showingthecumulativenumberofIOIinitiallygrowingexponentiallywithtime.However,lateronthecurvebendsoverandhitsalimit,demonstratingthatallcombinationpossibilitieshavebeenusedup,andthepoolofoperatingideasstagnateswhichisalsoshownonthelinearplot(Figure6b)resemblingawell‐known“Scurve”.

Fig.6:GrowthofcumulativeIOIC(t)afterimplementingtheconstraintthatIOI0canbeusedonlyoncebyanyspecificderivedIOIs;a)semi‐logplotandb)linearplot.

Themaximumnumberofcombinationpossibilities,whichisafunctionofIOI0inthepool,definesthelimit.Thislimit,ormaximumnumberofcombinationpossibilities,isgivenbyasimplecombinatoricsequation(Cameron1995):

2 1 (12)

Equation12entailsthatthelimitincreasesrapidlyasIOI0increases,duetoitsgeometricdependenceonIOI0.Forexample,forIOI0equalto5,10,15,and20thecorrespondinglimitsare31,1023(Figure6),32767,and1,048575combinationpossibilities.

AnaturalquestionthatarisesfromthisresultiswhatmightdeterminetheIOI0overtime?WepostulatearoleforUnderstandinginthisregardandwefirstbrieflylookathowUnderstandingevolvesovertime.

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0 20 40 60 80

IOI0 = 10

0 20 40 60 80

IOI0 = 10

4.2Combinatoric simulations for Understanding regime JustliketheOperationsregime,wemodeltheUnderstandingregimetoalsogrowthroughaprobabilisticanalogicaltransferprocess,inwhichunitsofunderstandingcombinetocreatenewunitsofunderstanding.Inthismodel,weenvisionthattheUnderstandingregimeiscomposedofmanyfields,witheachfieldhavinganexplanatoryreach.UsingatreatmentsimilartotheoneusedbyAxtelletal.(2013),theexplanatoryreachofafieldmaybeviewedasafitnessvalueofthetheoreticalunderstandingofthatfield,whichwedenotewithfi.FollowingAxtelletal.,whenunitsfromtwofieldswithfitnessvalues,f1andf2,combine,thefitnessoftheresultingunitisrandomlychosenfromatriangulardistributionwiththebaseorX‐axisdenotingthefitnessvaluesrangingfrom0tof1+f2,andtheapexrepresentingthemaximumvalueoftheprobabilitydistributionfunction,givenby2/(f1+f2).SeeFig7a.Iftheresultingfitnessofthenewunderstandingunitishigherthanthefitnessofeitherofthetwocombiningunits,thenewunderstandingunitreplacestheunitwhosefitnessisthesmallestamongthethree.WeassumethecumulativefitnessoftheUnderstandingregime(FU)asawholetobeequaltothesumoftheindividualfitnessvalueofeachfield.Oursimulationassumes10fieldswithstartingfitnessvaluesrangingfrom0to1,whicharerandomlyassigned.Consequently,theaveragecumulativefitness(FU)valueisinitially5.Asthesimulationproceeds,fitnessvaluesofthe10fieldsgrowindependently,andasaresult,thecumulativefitnessoftheUnderstandingregimegrows.Fig.7bshowsresultsfromasimulationrunexhibitingroughlyexponentialgrowthofcumulativefitnessovertime.Thus,asimplemodelforgrowthoftheUnderstandingregimeisalsoexponential.However,aswiththeOperationsregime,unlimitedgrowthbysimplecombinationofscientifictheoriesisnotrealistic.TheUnderstandingregimealsocannotprogressbysimplecombinationofexistingunderstandingbutinsteadexperiencesalimitthatweenvisionasdependinguponavailabilityofoperational(technological)toolsavailablefortestingscientifichypothesesandfordiscoveringneweffects.Weexpressthisdependencethroughanequationwhichexpressesthemaximumcumulativefitnessatanytime,maxFU(t),assimplyproportionaltotheIOIexistingatthattime:maxFU(t)=ZF∙IOIC(t) (13)WhereIOICthusrepresentsanapproximationfortheeffectivenessofavailableoperationaltools,andZFisaconstantofproportionality.ThisequationcapturestheconceptfirstsuggestedbyPricethattheextent(orscope)ofexplanatoryreachoftheUnderstandingregimeisdependentuponwhatexperimentaltoolsareavailableforscientistsandresearchers.Italsorecognizesinthetermsofourmodelthatthesetoolsareessentiallyoperationalartifacts.

Fig. 7: a) Triangular distribution of possible fitness values that can be assumed by a new unit of understanding b) Growth of FU (cumulative fitness of Understanding regime) over time.

4.3 Exchanges between Understanding and Operations regimes Asdiscussedinsection3.1,priorqualitativeworkindicatesthattheinteractionofUnderstandingandOperationsisprobablybestmodeledbyassumingmutualbeneficialinteraction.Inourmodel,wecapturethisenablingexchangefromtheUnderstandingtotheOperationsregimeusingasimplemathematicalcriterion:FU(t)/FU(t_prev)≥cutoff_ratio(R) (14)Where,FU(t)andFU(t_prev)representcumulativefitnessvaluesattimesteptandthemostrecenttimestep,t_prev,atwhichaIOI0hadbeenintroduced.ThiscriterionstatesthatwhencumulativefitnessoftheUnderstandingregimegrowsbysomemultiple(R)fromthetimewhenthelastIOI0wasinvented,understandinghasimprovedenoughtogenerateanewIOI0,whichbecomesavailableforcombinationswithallexistingIOI.Thethresholdratio,R,determinesthefrequencyatwhichIOI0arecreated.

WenowshowresultsfromasimulationincludingtheexchangeandlimitsonIOI0.Inthesimulation,westudyhowsynergisticexchangefromUnderstandinginfluencestherateofgrowthofIOIintheOperationsregime,includingescapefromstagnation.Wefocusparticularlyontwovariables,namely,theinitialnumberofIOI0intheOperationsregime

andthethresholdratioRforcreationofnewIOI0.Otherpertinentvariablesaretheprobabilityofcombination,PIOI,thenumberofattemptspertimestepandthenumberoftimestepsperyearandarenotvariedinthissetofresults.Forthissimulationstudy,Table(1)presentstheparametervaluesforIOI0(column3)andthethresholdratiosofcumulativefitness(column4)thatareused.Asanexample,5B3RstartswithIOI0of5andanewIOI0iscreatedwhencumulativefitnessgrowsbyafactorof3.BoththeinitialnumberofIOI0andthethresholdratiosofcumulativefitnessaresetat3differentvalues,givingatotalsetof9parametercombinations.Forall9runs,theprobabilityforcombinationiskeptconstantat0.25,andweassumeoneattemptperyearlytimestep.

Table1:Simulationstudy:ParametervaluesofIOI0 andR (thresholdratiosofcumulativefitnessofUnderstanding)forthestudy.Results:KistheslopefittingthesimulationresultstoanexponentialwithR2forthefit(alsoshown).Otherparameters,suchasprobabilityofcombination,PIOI=0.25,arekeptconstant. Simulation

RunInitialIOI0

ThresholdratioR

Simulationavg.K(±2stddev)8

R2 K =ln(1+PIOI/2)

1 5B1.5R 5 1.5 0.123(±0.011) 0.998 0.1182 5B3R 5 3.0 0.055(±0.019) 0.959 0.1183 5B5R 5 5.0 0.039(±0.007) 0.943 0.1184 10B1.5R 10 1.5 0.122(±0.011) 0.997 0.1185 10B3R 10 3.0 0.115(±0.007) 0.998 0.1186 10B5R 10 5.0 0.117(±0.007) 0.983 0.1187 20B1.5R 20 1.5 0.116 (±0.007) 0.998 0.1188 20B3R 20 3.0 0.116(±0.009) 0.998 0.1189 20B5R 20 5.0 0.119(±0.016) 0.998 0.118

8Thestandarddeviationwasestimatedfromsevenrepetitionsforeachsimulationrun.

Fig. 8: Growth of IOIc; initial IOI0 and R (cumulative fitness ratio) for each run are shown in the legend

for each run; e.g., 10B5R represents 10 IOI0 and fitness ratio of 5.

ThesimulationresultsinFig.8showsthetemporalgrowthofIOICintheOperationsregimefortheninerunsshowninTable1.Runs5B3Rand5B5Rclearlystandout:theyhaveabumpygrowthsincetheyencounterperiodsofstagnationmultipletimes,astheyevolve.Moreover,theireffectiveratesofgrowtharemeager,standingonlyat0.055and0.04,whichismuchlowerthan0.118,therategivenbyEquation10{ln(1+PIOI/2)}.Columns5,6,and7listtheK,R2,andKcalculatedusingln(1+PIOI/2)respectively.Thesmalldeviations

0 20 40 60 80 100 120 140 160 180

5B1.5R 5B3R 5B5R

10B1.5R 10B3R 10B5R

20B1.5R 20B3R 20B5R

fromequation10foundfortheother7runsarewithinthe2‐sigmaestimatedfrommultiplesimulationrepetitionsforeachrun.Both5B3Rand5B5RstartwithlowinitialIOI0of5andhavehighercumulativefitnessthresholdratios(R)forinfusionofnewIOI0.LowinitialIOI0impliesthattheOperationsregimehasalownumberofcombinatorialpossibilitiesofIOItostartwith.Additionally,sincenewIOI0arenotcomingfastenoughtopushthefrontierofcombinatorialpossibilitiesofIOIfarenough,theOperationsregimequicklyexhauststhepossibilitiesandagainstagnates.Run5B5Rstagnatesforlongerperiodscomparedto5B3Rsinceithasahigherthresholdratio(R)forinfusionofanewIOI0andthusslowerprogress.TheOperationsregimecannotescapethestagnationuntilanotherIOI0iscreatedwithinfusionofnewunderstanding.Itisclearfromthecurvesthatthispatternrepeatsitselftimeaftertime.Othersimulationruns,exceptrun10B5Rgrowexponentiallyandsmoothlyandtheirratesareconsistentwiththetheoreticalvaluecalculatedusingln(1+PIOI/2),0.1178.ThesecurveshaveeitherhighenoughIOI0tostartwithorfastinfusionofIOI0,orboth.Run5B1.5R,forexample,startswithalownumberofIOI0buthasfastinfusionofIOI0,sincethethresholdratioRisonly1.5.Ontheotherhand,run20B5RhasslowinfusionofIOI0(highR),butstartswithhighinitialIOI0.Theserunsdonotexhibitstagnationfortworeasons.ThefirstreasonisthatthefrontierofcombinatorialpossibilitiesforsomerunsisveryfarfromthenumberofrealizedIOIatagiventimestep.Forexample,run20B5Rhasoveramillionpossibilitieswhenitstartswith20IOI0.ThesecondreasonisthatthefrontierofthecombinatorialpossibilitieskeepsonmovingfurtherawayasIOIcincreases.Run5B1.5R,forexample,startswith5IOI0,andyetitneverexperiencesstagnationduetofastinfusionofIOI0(lowR)thatpushthefrontierofcombinatorialpossibilities.ThegrowthofIOICisalsofreeofstagnationforruns(e.g.,suchasRun10B3R)withmediumnumberofinitialIOI0andmediumrateofinfusionofIOI0(mediumR).Thisistruebecausebothfactorsincombinationensurethatfrontierofcombinatorialpossibilitiesisfarenoughtostartwith,andthefrontiercontinuestomoverapidlyenoughwithtime.Run10B5Rexhibitssomewhatunusualbehavior.Althoughitgrowssmoothlyatthebeginningforquitesometime,itexperiencesstagnationlateron.Thisisbecausethefrontierofcombinatorialpossibilitiesisfarenoughawaytosustainsteadygrowthearlyon.Later,theOperationsregimeexhauststhecombinatorialpossibilitiesbeforenewIOI0arrive.However,onceanewIOI0arrives,itjumpstartsagainbutitbrieflyhaltsateachnewlimitdemonstratingthevalueoffrequentinterchangebetweenUnderstandingandOperationsinthissimulation9.9ThesimulationsarebaseduponinfusionofIOI0dependinguponaratio(R)ofgrowthincumulativeunderstanding,butsimilarresultsarefoundwithassumingamodelofdifferenceinFU.

WehaveseenthatacombinatorialprocesscombinedwithsynergisticexchangebetweenUnderstandingandOperationsleadstoanexponentiallygrowingpoolofoperatingideas,IOIC.Thisgrowthisdescribedbyanexponentialfunction:

exp (15A)

(15B)Where,K=theeffectiverateofgrowthofIOIC,IOI0(t0)=thenumberofinitialbasicIOI,t=time,t0=initialtime.Ouroverallmodel(Section3,Figure2)envisagesthatthisexponentiallygrowingpoolofoperatingideas,IOIC,providesthesourcefortheexponentialgrowthofperformanceoftechnologicaldomains.HowdoesthisexponentialgrowthofIOICresultinperformanceimprovementandwhataccountsforthevariationinratesofperformanceimprovementacrosstechnologicaldomains?

4.4 Modeling interaction differences among domains Asexplainedinsection3,twofactorspotentiallyresponsibleformodulatingtheexponentialgrowthofoperatingideasastheyareintegratedintotechnologicaldomainsarethedomaininteractionsandscalingofrelevantdesignvariables.WeconsiderdomaininteractionsfirstfollowingtheworkofMcNerneyetal.(2011)whomodeledhowinteractionsinprocessesaffectunitcost.WebuildontheirmathematicaltreatmenttoanalyzetheeffectofinteractionsbetweencomponentsuponintegratinganIOIintoanartifactinadomain,whichinturnimprovestheartifact’sperformance.Figure9ashowsasimplifiedschematicofanartifactinatechnologicaldomainthathasthreecomponents(1,2,3)withinteractionbeingdepictedbyout‐goingarrows,representinginfluence,fromacomponenttoothercomponents,includingitself.Theoutgoingarrowsarereferredtoasout‐links.Thenumberofout‐links,d,fromacomponentprovidesameasureofitsinteractionlevel,andhasvalueof1orgreaterasMcNerneyetal.assumeeachcomponentatleastaffectsitself.Forsimplicity,Figure9ashowseachcomponentwithtwoout‐links,toitselfandtoanothercomponent.Werepresentaninstanceofanattemptbeingmadetoimprovetheperformanceofcomponent2byanIOIbeinginserted.Sincecomponent2interactswithitselfandanothercomponent,theperformanceoftheinteractingcomponentisalsochangedbytheinsertionbutinafashiondescribedprobabilistically.Theperformanceimprovementattemptisaccepted,onlyiftheperformanceoftheartifactasawholeimproves.Ifthatdoesoccur,wefollowMcNerneyetal.andconsidertheinteractionsbeingsuccessfullyresolvedtoimprovetheperformance.Forasimplifiedartifactwithdnumberofout‐linksforeachcomponent(d=2inFig.9a),McNerneyet.al.’streatment(2011)forunitcostresultsinthefollowingrelationship:dC/dm=‐B∙Cd+1 (16)

Where,C=unitcostnormalizedwithrespecttoinitialcost10,m=numberofattempts,d=numberofout‐links,B=constantThisequationstatesthatthelevelofinteractioninherentinthedomainartifactinfluencestherateofunitcostreduction.Weadaptthisequationforouranalysisinthefollowingmanner.WeinterpretthenumberofattemptsasIOIcsincethenumberofIOIdeterminestheattempts(ateachattemptanIOIisbeingintroducedintoanartifacttomakeadesignchange).Secondly,costreductionisinverselyrelatedtoperformanceimprovement,suchasinatypicalmetrickWh/$.11WiththeseextensionsofMcNerneyetal.equation16canbere‐writtenas:d(Q)/dIOIc=B∙Q‐(d‐1) (17)Where,Q=performance

Fig. 9: Interactions in an artifact; a) illustration of interactions as out‐links b) sample space of probabilities for unit cost .

SinceasshowninEquations4and5,successfullyresolvedoperatingideasinadomain,IOISC,arethesourceforitsperformanceimprovement,wereplaceperformanceQofadomainwithIOISC.AnIOIisconsideredasuccessfulattemptiftheinteractionresolution

10Thenormalizedunitcostis1orlesssoincreasesindinequation16resultinlessimprovementperattempt.11Theconceptcanbefurthergeneralizedtoincludeperformancemetricswhichinvolveotherresourceconstraintssuchasvolume,mass,andtime,insteadofcost(e.g.,kWh/m3).

n=3;d=2n=#ofcomponentsd=out‐links(interactions)

C1+C2 = C1(t) + C2(t) = C

leadstonetperformanceimprovementoftheartifact,andthecountofsuccessfulIOIisdenotedbyIOISC.Themodifiedequationshownbelowstatesthattheinteractionlevel,d,hasaretardingeffectonthegrowthofIOISCinadomain.d(IOISc)/dIOIc=B∙IOISc‐(d‐1) (18)Wesolvethedifferentialequationbyseparatingthevariables(IOISContheleftandIOIContheright),andintegratingbothsidesusingdummyvariables,andexpressIOISCexplicitly.Theintegrationlimitsare:a)fortherightside,0toIOIC,b)fortheleftside,1toIOISC.Theresultis:

∙ ∙ 1 / (19)SinceBanddareclosetounity,andIOIc>>1,wecanignore1inthebrackets.Sinceourgoalistodetermine{dlnIOISC/dlnIOIC},wetakethenaturallogofbothsidesanddifferentiateitwithrespecttolnIOIC,resultinginthefollowingexpressionwhichwillbesubstitutedintoequation5insection4.6:dlnIOIsc/dlnIOIc=1/dJ (20)

4.5 Performance models ‐ scaling of design variables Ourresearchquestionisconcernedwithintensivetechnologicalperformanceofdomainartifacts.Theintensivetechnologicalperformancerepresentsaninnateperformancecharacteristicofanartifact.Weoperationalizethenotionofintensiveperformancebydividingdesirableartifactoutputswithresourceconstraints(e.g.,mass,volume,time,cost).Anintensiveperformancemetricforbatteriesisenergydensity,kWh/m3.Wenowconsiderthreeexamplesofrelationshipsbetweenintensiveperformanceanddesignvariables.

4.5.1 Selected examples Wefirstconsiderblastfurnacesusedinthemanufacturingofsteelasrepresentativeofreactionvesselsofvariouskinds.Widelyusedperformanceattributesforablastfurnacearecapacityandcost,wherecostcanbeconsideredtheresourceconstraint.So,anintensiveperformancemetriccanbedefinedascapacity(outputperhourordaytypically)perunitcost.Thecapacityofareactionvesselisproportionaltoitsvolumewhileitscostisprimarilyproportionaltosurfacearea(Lipseyetal.2005).Thefollowingdimensionalanalysisshowsthatfollowingthesesimplisticassumptions,intensiveperformanceofareactionvesselislinearlyproportionaltosize,s.QRV=capacity/costofreactionvessel=s3/s2=s1 (21)Gold(1974)hasempiricallyshownthatthecostofablastfurnacegoesupby60percentwhenthecapacityisdoubled.IntensiveperformanceQRVusingthisempiricalfindinggoesupby1.25(=2/1.6)whens3doubles,andthussgoesupby1.26(=2.333)closelyagreeingwiththesimplyderivedequation21.

Asecondexampleweconsiderisspecificpoweroutputfrominternalcombustion(andotherheat)engines.Poweroutput(kW)isproportionaltovolumeoccupiedbythecombustionchamberminustheheatlossfromtheengine,whichinturnisproportionaltotheengine’ssurfacearea.Thepower,then,is:power=As3–Bs2;B/A<1 (22)WhereAandBareconstantsforpowergenerationandheatlossrespectively.QIC=specificpowerαpower/volumeofengine;thusspecificpoweris=(As3–Bs2)/s3=A–B/s (23)Equation23indicatesthat,similartoreactionvessels,specificpoweroutputofICenginesincreaseswithsizesobothare“largerisbetter”artifacts”.ForsmallvaluesofB/As,specificpowerincreasesapproximatelylinearlywiths.Forlargervaluesofs,theincreaseislessthanlinearins.Asafinalexample,weconsiderinformationtechnologies,whoseperformanceimprovementranksamongstthehighest.Severalmoderninformationtechnologiesdependuponintegratedcircuit(IC)chips.ElectroniccomputershavebeenimprovingperformancebyreducingthefeaturesizesoftransistorsinICchipsformicroprocessors.Thenumberofcomputationspersecondperunitvolume,anintensivemeasureofperformance,dependsuponfrequencyandthenumberoftransistorsinaunitvolume.Frequencyisinverselyproportionaltothelineardimensionofafeature,s,andthenumberoftransistorsperunitareaisinverselyproportionaltoareaofthefeature.Thus,Computationpersecpercc=1/s∙1/s2=s‐3 (24)Thedimensionalanalysisindicatesthatcomputationspersecondincreasesrapidlyforadecreaseinalineardimensionofafeature.Thisisduetothecubic(orhigher)12dependenceofcomputationspersecondonfeaturesize.Thenegativesigncapturesthefactthatreductionofthedesignvariableincreasesperformance–smallerisbetterforthisartifact.

4.5.2 Generalization of scaling of design variables Thethreeexampleswehavepresentedillustratethenotionthatintensiveperformanceimprovedbydifferentdegreesdependinghowthedesignvariablesarescaled.Inthefirsttwocases,a10percentincreaseinadesignvariablewillimproveperformanceby10percentorless.However,inthecaseofcomputations,forthesame10percentchangeindesignvariable(featuresize),theperformancewouldimprovebyover33percent.Thisdependenceismodeledasapower‐law13:

12Iftheverticaldimensionalsodecreasesovertimeasthefeaturesizedecreases,ahigherpower‐perhapsapproaching4‐wouldapply.13Theengineexampledemonstratesthatthisisanapproximationinmanycases.

(25) lnQJ=AJlns (26)

dlnQJ/dlns=AJ (27)

Where,AJisthescalingfactorfordomainJ,sisthedesignvariable.

4.6 Bringing all elements together WenowbringtheresultsforrateofIOISCgrowthandinfluenceofinteractionandscalingtogether.Forthereader’sconvenience,wereproduceequation4here,andsubstitutetheresultsforthefourfactors: dlnQJ/dt=dlnQJ/dlns∙dlns/dlnIOISC∙dlnIOISC/dlnIOIC∙dlnIOIC/dt (4)Substitutingtheresultsfromequations27,20,and15Bforthefirst,thirdandfourthterms,±1forthesecondterm,andthenrearranging,weget:

∓1 1

Equation28representstheoverallmodeloftheannualrateofimprovementfordomainJ.Accordingtothisequation,KJ,theannualrateofimprovementofdomainJdependsuponK,theexponentialrateatwhichtheIOICpoolincreasesinsize.Kisthenmodulatedbydomainspecificparameters,dJ(interaction)inverselyandAJ(scaling)proportionallytoresultinadomainspecificrateof improvementKJ.TheminussignisconvertedintopositiveonebynegativesignofAJ(forthosecaseswheresmallerisbetter).OneobservationtonoteisthatAJand dJ are constants for a given domain, thus resulting in a time invariant rate (or asimpleexponential)foradomain.

5. Discussion

Thegoalofthispaperwastodevelopamathematicalmodelthatutilizesmechanismsinthedesign/inventionprocesstoexaminethenatureoftechnologicalperformanceimprovementtrends.TheexplorationhasutilizedsimulationtogaininsightintoacombinatorialprocessbaseduponanalogicaltransferandUnderstanding/Operationsexchangeandquantitativelymodeledinteractionsandscaling.Inthissection,wefirstbrieflyreviewtheconsistenciesofthemodelwithempiricalresults(andwhatisknownabouttechnologicalchange).Allempiricalresultsweareawareofarefoundtosupportthemodel.Wethenconsidertheasyetuntestedpredictionsfromthemodelaswellastheassumptionsmadeinthemodel.Accordingtothemodel,theexponentialnatureofperformanceimprovementforalltechnologicaldomainsarisesintheidearealmoftheoperationalknowledgeregime,where

newinventiveideasarecreatedusingcombinatorialanalogicaltransferofexistingideas,which,inturn,becomethebuildingblocksforfutureinventiveideas.Weemphasizethatthecombinationsmodeledareoccurringattheidealevel,althoughcombinationscanalsotakeplacebetweencomponents.Asnotedinsection3.1,wemakethisdistinctionastheformerismuchmorepervasiveandallowscombinationofideasfromdifferentfields;however,itislikelythatsomeideascannotbecombinedandthisistreatedprobabilisticallysincemanycombinationattemptsfail.ThemodeldemonstratesthisincessantcumulativecombinatorialaspectofknowledgeinboththeUnderstandingandtheOperationsregimesmanifestsasexponentialtrends.ThecombinatorialmodelissimplebutitleadsnaturallytotheexponentialbehaviorwithtimethathasonlybeenobtainedpreviouslybyAxtelletal.inamodelthatwentbeyondperformancetodiffusionoverasetofagents.Sincesuchexponentialbehaviorwithtimeisoneofthemostwidelynotedbehaviorsoftechnicalperformance(Moore1965,KohandMagee2006,2008,Nagyetal.2013,Mageeet.a.2014),thecombinatoricmodelenactinganalogicaltransferthatwasdevelopedinthecurrentpaperisclearlysupportedbywhatisknownempiricallyaboutperformancetrendswithtime.TheOperationsandtheUnderstandingregimescanimproveindependentlyinthemodelbutnotindefinitely.HowlongtheOperationsregimecanimprovedependsinthemodeluponthesizeofthetechnologicalpossibilityspace,whichaccordingtothemodelisdependentonthenumberofbasicIOI,fundamentaloperationalprinciples,existing.TheUnderstandingregimecanalsoexperiencestagnation,butthishappenswhentheoperationaltoolsthatscientistsandresearchersusefordiscoveryandtestinghypothesesarenotadequate.TheOperationsregimecomestoitsrescuebyprovidingtheseoperationaltoolsinformofempiricalmethods,toolsandinstruments(increasednumbersofindividualoperatingideas),whichgreatlyenhancesthescientistsabilitytodiscoverandtest,andthusfurtherpushthelimitsofunderstandinginthemannersuggestedbyPrice(1983),Gribbin(2002)andinthefollowingquotefromToynbee(1962).

Physical Science and Industrialism may be conceived as a pair of dancers both of whom know their steps and have an ear for the rhythm of the music. If the partner who has been leading chooses to change parts and to follow instead there is perhaps no reason to expect that he will dance less correctly than before.

Inthissense,theOperationsregimeandtheUnderstandingregimeareliketwoindependentneighborswhointeractformutualbenefit.Inthemodel,theirfrequencyofinteractionhoweverinfluencestheireffectiverateofgrowth.Ourmodelisaspecificrealizationthatachievesthismutualinteractionthathaspreviouslybeenwidelynotedfromdeepqualitativeresearch.TheresultsinFigure8aresummarizedasasurfaceplotinFigure10.K,theeffectiverateofgrowthofIOICwasdeterminedbytheinitialIOI0,andthefrequencyofinteraction(α1/lnR).Theformerdeterminedtheenvelopeoftechnologicalpossibilityspace.WhenIOI0arehigh,theeffectiverateofgrowthKisclosetothetheoreticalcombinatorialratedeterminedbyEquation10{ln(1+PIOI/2)},irrespectiveofwhethertherewasfrequentexchange.However,whentheIOI0arelow,thelimitishitrepeatedly,translatinginto

haltingandareducedeffectiverateofgrowth.ThevalueofKinthiscasewasdeterminedbythefrequencyofenablingexchangefromtheUnderstandingregime,withhigherfrequency(lowR)leadingtohighereffectiverate.Withsufficientlyhighfrequency,evenwithlowinitialIOI0,theeffectiverateKeventuallyapproachesthetheoreticalrate.

Fig.10:VariationofKasafunctionofinitialIOI0andR.LowerRreferstohigherfrequencyofinteractionwiththeUnderstandingregime.Detailedhistoricalstudiesoftechnologicalchange(Mokyr2002)notecenturiesofslow,haltingprogressthateventuallybecomesmuchmorerapidandsustainedstartinginthelate18thcenturyintheUK.Aninterestingconsistencyoftheseobservationswithourmodelisseensinceourmodelattributesthetransitiontosustainedhigherimprovementratetothecombinatorialgrowthofindividualideasthatareabletoreinforceoneanotherbytheanalogicaltransfermechanism.ThatourmodelpartiallyaccomplishesthisthroughthesynergisticexchangebetweenUnderstandingandOperationsisalsoconsistentwiththedetailedhistoricalstudiesasinterpretedbymanyobservers(Schofield1963,Musson1972,RosenbergandBirdzell1986,MussonandRobinson1989,Mokyr2002,Lipseyetal.2005).TheKJvaluesfoundempiricallyvarybyapproximatelyafactorof22(from0.03to0.65accordingtoMageeetal.(2014).Equation28statesthatannualimprovementrateforadomainisdeterminedbytheproductofKtimesthescalingparameter,AJ,andthereciprocaloftheinteractionparameter,dJ.Accordingtothisresult,thelasttwoparametersproducethevariationofimprovementratesacrossdomains.Duringtheembodimentprocess,interactionsprevalentinthedomainartifactsinfluencehowmanyinventiveideascanbeabsorbed.ThepercentincreaseinsuccessfullyabsorbedideasbyadomainartifactisinverselyproportionaltotheaverageinteractionparameterofthedomaindJ.By

0.12‐0.16

0.08‐0.12

0.04‐0.08

0‐0.04

definition,theminimumvalueofdis1andthemaximummightbehigherbutavalueof6appearsreasonable.Theotherfactorthatispredictedtodifferentiatedomainsisperformancescaling.Inventiveideasaffectartifactperformancebymodifyingthedesignparametersindomainartifacts.ThemodelindicatesthattherelativeimprovementofperformanceforagivennumberofabsorbednewoperatingideasisgovernedbythescalingparameterAJ.Theexamplespresentedinsection4.5illustratedthatthevalueofAJcanvaryacrossdomains.Inparticular,fortheICdomain(wheresmallerisbetter),AJisapparently3to4timeslargerthanfortypicallarger‐is‐betterdomainssuchascombustionengines.Thus,therangeofKJempiricallyobservedispotentiallyexplainablebychangesindJandAJ,butmuchmoreempiricalworkisneededtofullysupportthesequantitativeimplicationsofEquation28aswillbediscussedfurtherbelow.TheempiricalfindingsofBensonandMagee(2015a)alsosupportthemodel.Inparticular,theyfoundnocorrelationofratesindomainswitheffortinadomain(measuredbynumberofpatentsorpatentingrate)orwiththeamountofoutsideknowledgeusedbyadomain(thisisverylargeforalldomains).Theyinterpretedtheirfindingsbya“risingseametaphor”thatrepresentsallinventionsandscientificoutputbeingequallyavailabletoalldomainsbutthatfundamentalsinthedomainsdeterminetherateofperformanceimprovement.OveralleffortinUnderstanding(science)andinventionincreasetheratesinalldomainsbutthedifferencesamongratesofimprovementareduetodifferencesinfundamentalcharacteristicsamongthedomains.Themodelinthispaperidentifiesinteractionsandscalingastwosuchfundamentalsandequation28isspecificaboutthevariationexpectedduetothesetwofundamentalcharacteristics.Thus,ourmodelissupportedbywhatisknownempiricallyincludingexponentialdependenceofperformanceontime;slow,haltingprogressintheearlystagesoftechnologicaldevelopment;aroleforscienceinenablingtechnologicalperformanceimprovement;therangeofvariationinperformanceimprovementacrossdomains;andtheimportanceofdomainfundamentalstovariationinperformance.However,towhatextentdoesitachievetheideallevelofunderstandingmentionedinsection2whendiscussingtherelatedBensonandMageeresearch?Itis‐asdesired‐baseduponwhatisknownaboutthedesign/inventiveprocessanddoesnotrelyuponcharacteristicsonlydeterminedbyobservationofoutputinadomain.Moreover,itprovidesexplanationsofexistingempiricalresultsnotmadebypriormodels.However,doesitmakeanynewpredictions;doitsassumptionsappearreasonable;andwhatnewavenuesofdesignresearch,ifany,doesitopenupforfurtherexploration?Weconsidertheseissuesintheremainderofthediscussion.TherearethreenewpredictionsmadebythemodelasinstantiatedinEquation28.Theseare:1)thatthenoiseinestimatingKJshouldvarywithKJlinearlyratherthanforexamplebeindependentofKJ;2)thatperformanceimprovementcomparisonsacrossdomainsvaryas1/dJwheredistheinteractionparameter;and3)thatperformanceimprovementacrossdomainsvaryasAJ.ThefirstpredictionfollowsfromthefactthatthemodelascribesallvariationintheprocesstotheprobabilisticanalogicaltransferprocessthatcreatesIOIandthusanynoisegeneratedintheprocessisamplifiedbythesamefactorsthatdetermineKJ(namely1/dJandAJ.).Veryrecentworkappearstoconfirmthefirstprediction.Inacareful

studyoftheobservednoiseinawidevarietyofdomains,FarmerandLafondhavefindthatthevariationinKJisproportionaltoKJofferingempiricalsupporttotheformofEquation28.ThisispotentiallyanimportantconfirmationofapredictionofthemodelbutthecarefulworkbyFarmerandLafondhaspotentialdatalimitations(detailedintheirpaper)andfurtherworkofthiskindishighlydesirable.Prediction2isthatcomponentinteractions(dJ),whichcharacterizethedomains,influenceimprovementratebymodulatingtheimplementationofIOIinthedomainartifacts.ThispredictioncanbetestedbystudyoftheperformanceimprovementratesoveravarietyofdomainswhereanindependentassessmentofdJismade.Theauthorshaveperformedsuchatestusingpatentdata(BasnetandMagee2016)andtheresults,whichdemonstratepositivecorrelationbetweenimprovementrates(KJ)andinteractionparameter(dJ),offersupportfortheanalysisofMcNerneyetal.thatweuseinourmodel.Prediction3isthatrelativeimprovementamongdomainsvariesproportionallytothescalingparameterforthedomaindesignparameters,aconsequenceofperformancefollowingapowerlawwiththedesignparameters.Ifscalinglawswerefound(orderived)foravarietyofdomainswhoserateofprogressisknown,prediction3canalsobetested.Inthispaper,weshowedthatthefactorAisatleast3timeslargerforIntegratedcircuitsthanforcombustionengines.WhilethisprovidespreliminarysupportforthemodelsinceIntegratedcircuitsimproveabout7timesfasterthancombustionengines(Mageeetal,2014),twopointsdonotachievearigoroustest.Onewouldneedtohavereliablescalingfactorsforatleast10domainswithvaryingKJtodeterminewhetherthispartofthemodelisempiricallysupported.Afundamentalaspectoftheoverallmodelisthatitdifferentiatesbetweentheidea/knowledgeandartifactaspectsofdesignandinvention.Suchdecompositionisanessentialstepinarrivingatourkeyresult(equation28throughequation5).Itisnotclearthatthisassumptionistestablesoitmustremainanunverifiedassumptionordefinitionbutwedonotethatitappearstoaccordwithrealityinthatinventors/designersspendsignificantamountoftimeworkingwithideasandrepresentationsofartifacts,forexampleintheformofsketchesanddrawings,wellbeforetheybuildartifacts.Othershavenotedthehigherleverageofanalogicaltransferbetweenideasasopposedtodesignedartifacts(Weisberg2006).Apotentiallyimportantandnon‐obviousassumptionmadeinthemodelisthatinventiveeffortincreasesasthecumulativenumberofindividualoperatingideas‐IOIC‐increases.ThisassumptionisintroducedwhenweassumethateveryexistingIOIundergoesacombinationattemptineachtimestep.AsIOICincreases,thismeansthatmoreinventionsareattemptedineachsuccessivetimestep.ThisassumptioniscriticaltoobtainingtheexponentialtimedependenceforIOICandthusforQbecausethegrowthofIOICwouldbechokedoffifinventiveattemptsdidnotincreaseovertime.Althougharigoroustestofthisassumptionissuggestedforfurtherwork,wedonotesupportfortheassumptionintheexponentialgrowthofpatentsovertime(Younetal.2014,PackalenandBhattachayra,2015)14.ApproximatesupportisalsogivenbytheroughlyexponentialgrowthofR&D

14Bothofthesepapersshowmorerapidexponentialincreasesbefore1870andslowerbutstillexponentialincreasesovertimefrom1870tothepresentinthenumberofUSpatents.

spendingovertime(NSF,2014)andbytheroughlyexponentialgrowthofgraduateengineersglobally15overtime(NSF,2014)ThemodelassumesasimpleexchangebetweenUnderstanding(largelyscience)andOperations(largelytechnology)asdescribedbyEquations13and14.Thedetailsofthismechanismarenottestablebutinouropinionnotcriticalbecauseotherformalisms(basedupondifferencesratherthanratiosandbaseduponcountofunitsofunderstandingratherthanourchoiceofexplanatoryreach)leadtoresultscloselysimilartothosereportedhere.Therefore,thisassumptionremainsunverifiedbutisnotcriticaltoourconclusions.Similarly,theinitialvalueofIOI0choseninthesimulation(andtheexchangefrequencywithUnderstanding(α1/lnR))isessentialtoourfindingofhaltingslowgrowththatcantransitiontosustainedandmorerapidgrowth.Althoughthisfindingisconsistentwithdetailedobservationasnotedaboveandtheinitialnumberofusefulideasmustbesmall,thereisnoindependentmeansofassessingIOI0.Moreover,wehavemadeanumberofassumptionsinparametervaluestoconstructasimpleandoperationalsimulation.Thevaluesforparametersinthesimulation,suchasPIOI,numberoftimesteps,numberofscientificfields,R,fitnessvaluesarechosentokeepthecomputationalcostreasonable,withoutsacrificingtheessentialaspects.Simulationsshowthatresultsarerobusttodifferentcombinationsofparametervalueswithrespecttoexponentialtrendsandvariationinrates.Therefore,thesechoicesandsimplificationsdonotundercuttheexplanatoryorpredictivecapabilitiesofthemodelbutdolimitthepotentialfornon‐calibratedcalculationof,forexample,theimprovementrateforadomainsinceKisonlyapproximatelyknown.Tomakethemodeltractable,wehavemadenumberofsimplifyingabstractions,introducingseveralotherlimitationstothemodel.Sincethemodelisnotagent‐based,itdoesnotdistinguishbetweenorganizationsnorbetweeninventors.Sinceourgoalistoexplainthepatternsatthedomainlevel,weconsiderthedomainasoneentity.Forthisreason,variationsamongorganizationsoramonginventorswithinadomainarenottakenintoaccount,andhencethemodelisnotusefultounderstandorganizationalorindividualinventoreffectivenessinitscurrentformandanysystematicdifferencesamonginventorcapabilityacrossdomainsisignored.Second,onceIOIarecreatedbyanyinventor,themodelassumestheyareinstantlyavailableforcombinatorialanalogicaltransferacrossthepoolunderlyingalldomains.Thus,themodeldoesnottakeintoaccounttimedelaythatcanresultdueto,forexample,geography,secrecyandgovernmentalregulations,andhenceisnotusefulforstudyingsuchfactors’influenceintechnologicalchange.Third,themodelassumesthat2pre‐existingideasaresufficient(probabilistically)tocreateanotherideawhereasinventionsalsoresultfrombringingmorethan2pre‐existingideastogether.However,addingsuchcomplicationstothemodelandsimulationdoesnotchangethefundamentalfindingssincethecreationofnewideaswouldstillincreaseasthenumberofpre‐existingideasincreaseaslongaswestillassumeanincreasinginventioneffort.Fourth,althoughconceptuallythenotionoffitnessofscientificfieldsmakessense,howthe

15OthersupportingevidenceisalsopossibletoseeintheNSFmaterialathttp://www.nsf.gov/statistics/seind14/index.cfm/overview/c0s1.htm#s2

fitnesscanbemeasured,andwhomeasuresitforascientificfieldarecontested,especiallyforrapidlygrowingfields.ThisanalysisofthepredictionspointsoutthatsomekeyaspectsofEquation28havethepotentialtobeempiricallytestedandthusareclearfutureresearchactivitiessuggestedbythemodel.Amongthesefutureresearchactivities,oneimportantissuetodiscussistheextensionspossibletodesignresearchpotentiallyopenedupbythecurrentwork.Themodelinthispaperexplicitlyconsidersdesignchangesinsucceedingartifactsinaseriestobethecentralelementintechnologicalchangeovertime.Thus,itaddstothefewotherpapers(BaldwinandClark2006,Luoetal.2014)thathaveconnectedthesetwolargefieldsofresearch‐technologicalchangeanddesigntheory.Thispaperinparticularconnectsdesignconceptuallyandquantitativelytochangesinperformanceovertime.Sincethereissignificantdataofthistype(Moore,1965,Girifalco1991,Nordhaus1996,KohandMagee(2006,2008)andLeinhard2008),thispaperpointsthewayforfurtherquantitativecomparisonsofmodelsbasedupondesigntheorywithdata.Anotherlineofresearchthatthismodelsuggestsismoreexplicitconsiderationofinteractionsandscalingaspartofdesigntheories.Thecurrentmodelexploressimplemodelsforbothofthesethatarecapableofpredictingdifferencesintimedependenceofperformanceindifferingdomains.Designofartifactscouldconceptuallybechangedsothatthepotentialforimprovementwithongoingredesignisenhancedpossiblythroughreducedinteractionsormoreintensivescalingrelationships.Thus,thecurrentpapersuggeststhepotentialimportanceoffurtherresearchonspecificdifferencesindesignapproacheswithdifferentscalinglawsandwithdifferentlevelofinteractions.

6. Concluding remarks Themodelandsimulationsoftheimprovementsinperformanceduetoaseriesofinventions(newdesigns)overtimepresentedinthisworkarebaseduponasimpleversionofanalogicaltransferasacombinatorialprocessamongpre‐existingoperational/inventiveideas.Themodelissupportedbyanumberofempiricallyknownaspectsoftechnologicalchangeincluding:1. Thetransitionfromslow,hesitanttechnologicalchangetomoresustainedtechnological

progressastechnologicalideasaccumulate;2. Arolefortheemergenceofthescientificprocessinstimulatingthetransitioninpoint1;3. Theexponentialincreaseofperformancewithtime(generalizedMoore’sLaw)seen

quitewidelyempirically;4. Thatstochasticnoiseintheslopesofthelogperformancevs.timecurvesis

proportionaltotheslope;5. Thelevelofeffortindomainsisnotimportantintherateofprogress.

Themodelalsoindicatesthat:6. Therateofperformanceincreaseinatechnologicaldomainisatleastpartly(and

possiblylargely)duetofundamentaltechnicalreasons(componentinteractionsand

scalingofdesignvariables),ratherthancontextualreasons(suchasinvestmentinR&D,scientificandengineeringtalent,ororganizationalaspects).

Numerousmodelingassumptionsweremadeindevelopingthemodelbutonlysomeofthesearecriticaltotheconclusionsjustlisted.Furtherspecificresearchissuggestedtomovesomecriticalassumptionsintothetestablecategory,andtoconsiderinteractionsandscalingparametersinnewdesignapproaches.Thesearediscussedinthepaperparticularlyfortheassumptionsunderlyingpoint6above.Thetestsinvolvedetailedstudiesoftheinteractionandscalingparametersinavarietyofdomains.Allofthisfutureresearchcouldsupportorleadtomodificationofpoint6.

Acknowledgement

TheauthorsaregratefultotheInternationalDesignCenterofMITandtheSingaporeUniversityofTechnologyandDesign(SUTD)foritsgeneroussupportofthisresearch.WewouldalsoliketothankDr.JamesMcNerneyforhelpfuldiscussionaboutartifactinteractions.WewanttoalsoacknowledgevaluableinputonanearlierversionofthispaperbyDr.JamesMcNerneyandDr.Daniel.E.Whitney.

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