1 Modeling of technological performance trends using design theory Subarna Basnet Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts Ave, Cambridge, Massachusetts 02139 Christopher L. Magee Massachusetts Institute of Technology, Institute for Data, Systems, and Society, 77 Massachusetts Ave, Cambridge, Massachusetts 02139 Abstract Functional technical performance usually follows an exponential dependence on time but the rate of change (the exponent) varies greatly among technological domains. This paper presents a simple model that provides an explanatory foundation for these phenomena based upon the inventive design process. The model assumes that invention ‐ novel and useful design‐ arises through probabilistic analogical transfers that combine existing knowledge by combining existing individual operational ideas to arrive at new individual operating ideas. The continuing production of individual operating ideas relies upon injection of new basic individual operating ideas that occurs through coupling of science and technology simulations. The individual operational ideas that result from this process are then modeled as being assimilated in components of artifacts characteristic of a technological domain. According to the model, two effects (differences in interactions among components for different domains and differences in scaling laws for different domains) account for the differences found in improvement rates among domains whereas the analogical transfer process is the source of the exponential behavior. The model is supported by a number of known empirical facts: further empirical research is suggested to independently assess further predictions made by the model. Keywords: Modeling, design, combinatorial Invention, technological performance
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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
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
Fig. 1a: Exponential growth of performance in sample domains – Electric motor and Magnetic resonance imaging (MRI). Adapted from Magee et al. 2014 with permission.
KJ(%)
Fig. 1b: Annual rate of performance improvement, KJ, for 28 domains. Adapted from Magee et al. 2014with permission.
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.
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.
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.
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a)
b)
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.
4.5.2 Generalization of scaling of design variables Thethreeexampleswehavepresentedillustratethenotionthatintensiveperformanceimprovedbydifferentdegreesdependinghowthedesignvariablesarescaled.Inthefirsttwocases,a10percentincreaseinadesignvariablewillimproveperformanceby10percentorless.However,inthecaseofcomputations,forthesame10percentchangeindesignvariable(featuresize),theperformancewouldimprovebyover33percent.Thisdependenceismodeledasapower‐law13:
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:
dln
∓1 1
(28)
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.
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