Constrained, mixed-integer and multi-objective optimisation of building designs by NSGA-II with fitness approximation

ContentslistsavailableatScienceDirect

Applied

Soft

Computing

jo u r n al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e/ a s o c

Constrained,

mixed-integer

and

multi-objective

optimisation

of

building

designs

by

NSGA-II

with

fitness

approximation

Alexander

E.I.

Brownlee

_,

_Jonathan

_A.

_Wright

a_{DivisionofComputingScienceandMathematics,UniversityofStirling,UK}

b_{SchoolofCivilandBuildingEngineering,LoughboroughUniversity,UK}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received5July2013 Receivedinrevisedform 15December2014 Accepted2April2015 Availableonline17April2015

Keywords: Simulation-basedoptimisation Multi-objective Constraints Surrogate NSGA-II

a

b

s

t

r

a

c

t

Reducingbuildingenergydemandisacrucialpartoftheglobalresponsetoclimatechange,and evolution-aryalgorithms(EAs)coupledtobuildingperformancesimulation(BPS)areanincreasinglypopulartool forthistask.FurtheruptakeofEAsinthisindustryishinderedbyBPSbeingcomputationallyintensive: optimisationrunstakingdaysorlongerareimpracticalinatime-competitiveenvironment.Surrogate fitnessmodelsareapossiblesolutiontothisproblem,butfewapproacheshavebeendemonstrated formulti-objective,constrainedordiscreteproblems,typicaloftheoptimisationproblemsinbuilding design.Thispaperpresentsamodifiedversionofasurrogatebasedonradialbasisfunctionnetworks, combinedwithadeterministicschemetodealwithapproximationerrorintheconstraintsbyallowing someinfeasiblesolutionsinthepopulation.Differentcombinationsoftheseareintegratedwith Non-DominatedSortingGeneticAlgorithmII(NSGA-II)andappliedtothreeinstancesofatypicalbuilding optimisationproblem.Thecomparisonsshowthatthesurrogateandconstrainthandlingcombinedoffer improvedrun-timeandfinalsolutionquality.Thepaperconcludeswithdetailedinvestigationsofthe constrainthandlingandfitnesslandscapetoexplaindifferencesinperformance.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The building and construction industry is one which offers largepotentialforreduced environmentalimpactandimproved sustainability.Buildingsarea largecontributor toworldcarbon emissions,particularlyduetotheirheating,coolingandlighting energydemands(forexample,over50%ofUKcarbonemissionsare relatedtobuildingenergyconsumption[1]).Usualpractice,driven byexpectationandbuildingfunction,isforeachindividualbuilding tobedesignedafresh,losingthebenefitofmass-productionwhere asingleoptimisationprocessappliestothousandsormillionsof units.Thismeansthatimprovementsinthebuildingdesign pro-cesstoaiddesignersinreducingbuildingcarbonemissionsarean importantareaofresearch.Computer-basedoptimisationmethods areofrapidly-increasingimportanceinthisarea.

Severalcharacteristicsofbuildingenergyoptimisationpresent a challenge. Each specific problem has a large decision space, withamixtureofcontinuousanddiscretevariablesdetermining variedaspectsofabuilding’sdesign.Typicallytherearemultiple

∗Correspondingauthor.Tel.:+4401786467462.

E-mailaddresses:[email protected](A.E.I.Brownlee),[email protected]

(J.A.Wright).

conflictingobjectivesand constraints: reducing energydemand conflictswithreducingthebuilding’scapitalcost,andbothconflict withincreasing the comfort of thebuilding’s occupants. These factorsmotivate theuseof evolutionary algorithms (EAs), suc-cessfullyappliedtoanumberofapplicationsinthisdomain[2], including:heating,ventilating,andairconditioning(HVAC)system sizing[3];fenestration[4];buildingenvelope[5,6];andbuilding form[7,8].Furthertotheissuesabove,thebuildingperformance simulations(BPS)usedtoestimateenergyconsumption,occupant comfort and factors affecting capital cost are time-consuming, takingminutestohoursdependingonbuildingcomplexity and themodellingaccuracyrequired.ThislimitsthepracticalityofEAs forreal-worlduseinthisdomain;ahighlycompetitiveindustry wheresubmissionofdesignsolutionsareusuallytime-criticaland optimisationrunsmustbecompletedoveranightorweekend.

Therearemanyapproachestoimprovingtherun-timeofEAs: parallelisation [9], reducing simulation complexity, and fitness approximationmethodssuchassurrogates[11].Parallelisationof fitnessevaluations iseasily adoptedand can beapplied along-sidethelattermethods,foramultiplicativespeedgain.Reducing thesimulationcomplexityrequiresproblem-specificknowledge, and hasthenegative impactofless accuratesimulation results possiblyleadingtheEAtofalseoptima.Surrogatefitness approxi-mation,wheresomesolutions(includingthefinalreturnedoptima) http://dx.doi.org/10.1016/j.asoc.2015.04.010

areevaluatedby thefullsimulation whileothers areevaluated byamuchcheapermodelofferspotentialtoimproveruntimes withoutlosing accuracy. Thereare,however, a limited number ofworksfeaturingsurrogatesforproblemswithdiscretedecision spaces,andstillfewerlookingatproblemswithconstraintsor mul-tipleobjectives.Thishasledtominimaladoption ofsurrogates inbuildingoptimisation,wheretheproblemsoftenexhibitthese characteristics.Theneedfor greaterefficiency ofEAsfor build-ingoptimisationmotivatesfurtherresearchintosurrogatesthat cansimultaneouslyhandleconstraints,multipleobjectivesand dis-creteormixedvariabletypes.Ifsuccessful,suchsurrogateswill improvethepracticabilityofEAsforthisapplicationarea.

Giventheabovepoints,this paperexplorestheuseofradial basisfunctionnetworks(RBFN)toreducethenumberofcallsto thefullbuildingperformancesimulation,byapproximating objec-tiveandconstraintvalues.Severalcontributionsaremade.Thefirst isthatanexistingapproach[12]isadaptedformulti-objectiveand constrainedproblems,withachangetothewaythatcategorical variables arehandled. A separateRBFNforeach constraint and objectiveallowsforretrainingoneachindependently.Thepaper’s secondcontributionisDeterministicInfeasibilitySorting(DIS),a schemewhereinfeasiblesolutionsareretainedinthepopulation todealwithtwoissues.Oneissueisthatestimationofconstraints is difficult because an approximation error can mean that the algorithmdiscardsfeasiblesolutionsthatareintruthfeasible: par-ticularlyproblematicwithmanyconstraints.Thesecondissueis thatapproximationerrorscanalsoaffectthedominance relation-shipbetweensolutionsinmulti-objectiveproblems.Thepaper’s thirdcontributionisthecomparisonofseveralpossibleapproaches tointegratingthesurrogateandDISwithinNon-DominatedSorting Genetic Algorithm II (NSGA-II) [13]. Several variants of NSGA-IIareappliedtothreeversionsofa mixed-integer,constrained, multi-objectivebuildingoptimisationproblem. Comparisonsare madein both thequalityof thefinal solutionsetsand conver-gencespeed.ItisfoundthatthebestperformingNSGA-IIvariant offersaperformanceimprovementfortwoofthethreeproblems (withnosignificantdifferenceonthethird).Thefinalcontributions areadetailedanalysisoftheconstrainthandling,toallowgreater understandingoftheimpactofthetechniques,andastudyofthe problem’sfitnesslandscapetoaidanalysisoftheresults.

Therestofthepaperisstructuredasfollows.Section2describes relatedworkinthisarea.Sections3and4describethebuilding optimisationproblemandtheproposedmodificationstoNSGA-II: theRBFNsurrogate;theconstrainthandlingapproach;anumberof minorchangestoimproveperformance;andtheresultingvariants ofNSGA-II.Section5describesanddiscussesexperimentstuning andcomparingthealgorithmvariants.Sections6and7provide deeperanalysisoftheconstrainthandlingandfitnesslandscape, andSection8concludesthepaper.

2. Relatedwork

Thereisarapidlygrowingbodyofworkonapproximating fit-ness[14,15,11],whetherto:reducecallstotheevaluationfunction anddecreaserunningtimeorothercost;counteractnoise;or over-comediscontinuitiesinthefitnessfunction.Techniquesinclude: fitness inheritance [10,16]; co-evolved fitness predictors [17]; fuzzymatchingagainstanarchive[18];artificialneuralnetworks [19–21];linearandpolynomialregression[22,23];Gaussian pro-cessesorKriging[24,25]andprobabilisticdistributions[26].

Ofparticularinterestforthepresentworkisthecombination of fitness approximation for problems with discrete variables, multipleobjectivesandconstraints.Severaloftheabovepapers reported successwith surrogates for multi-objective problems. A recent review [27] on constraint handling approaches in

evolutionaryalgorithms(EAs)suggeststhattheuseofsurrogate approachesforconstraintsremainsquiterare,andallforproblems withcontinuousdecisionvariables.

Therehas, however, beensomeworkonfitness approxima-tionfordiscreteproblemswithoutconstraints,including[28–32]. Tresidderetal.[33]andZemellaetal.[34],makeuseofsurrogates inEAsfordiscretebuildingdesignproblems.BajerandHole ˇna[35] alsousedaRBFNsurrogate,foramixedcontinuousanddiscrete problem.Incontrastwiththepresentpaper,theRBFsweredefined overonlythecontinuousvariables,withcentreslocatedin clus-tersdeterminedbyHammingorJaccarddistanceonthediscrete variables.

In[12],anRBFNisusedasasurrogateforamixed-integersearch space.Themodelisusedasafilter;toomanyoffspringare cre-atedeachgenerationandthemodelisusedtochoosepromising oneswhicharethenevaluatedwiththetruefitnessfunctionand retained.Thetechniquewasappliedtointravascularultrasound imageanalysis;thisapproachisthebasisofthemethodproposed inthispaper.

Thispaperalsodescribesanoperatorwhichallowsinfeasible solutionstoremaininthepopulationasameansofimproving per-formanceforconstrainedproblems.Therearemanyexamplesof workinthisarea,including[36–40].

3. Application

VariantsofNSGA-II wereappliedtoareal-world simulation-basedbuildingoptimisationproblem;thiswillnowbedescribedin detail.Theaimistofindthetrade-offbetweenoperationalenergy use(forheating,coolingandlighting)and constructioncostfor amid-floorofasmallcommercialofficebuilding(Fig.1),subject toconstraintsrelatingtothecomfortofthebuilding’soccupants. Thefloorisdividedintofiverooms,treatedasseparatezonesfor energyand thermalcomfort simulation.Thetwo endzonesare 24m×8m,andthethreemiddlezones30m×8m.Floortoceiling heightthroughoutis2.7m.Buildingenergyconsumptionstrongly dependsonlocalclimateandtheexperimentsinthisworkplaced thebuildinginthreelocations:Birmingham(UK),Chicago(IL,USA), andSanFrancisco(CA,USA).AbuildingsituatedinChicago(hot summer,coldwinter)wouldbeexpectedtoneedbetterinsulation andmoreartificialheatingandcoolingthanBirmingham,andeven moresoSanFrancisco(moretemperate,flatterclimates). Conse-quentlytheserepresentthreequitedifferentinstancesofthesame optimisationproblem,albeitsharingthesameobjectives,variables andconstraints.

Theoptimisationtakesplaceatthebuilding’sdetaileddesign stage,withoverallformandlayoutfixed.Constructionvariablesto beoptimisedarebuildingorientation,glazingtypeand window-to-wall ratio, and wall/floor material types. The optimisation alsoincludesvariablesfor theoperationofthebuilding’sHVAC systems.Thisallowsforthefinaldesigntobetunedtoaparticular operationalpattern,forexampleexchanginganincreasedcapital

Table1

Summaryofconstraints;theabbreviationsareusedinsubsequentreferencesto individualconstraintsinthepaper.

Abbreviation Description Limit

NSOH Summeroverheating:numberofhoursperyear wherezonetemperatureexceeds28◦C,forNorth, West,East,SouthandInternalzonesrespectively

≤30 WSOH

ESOH SSOH ISOH

NPMV Predictedmeanvote:numberofhoursperyear wherethepredictedmeanvoteforzoneoccupants fallsbelow−0.5,forNorth,West,EastandSouth zonesrespectively ≤30 WPMV EPMV SPMV IPMV

NCO2 CO2:maximumconcentrationofcarbondioxide (partspermillion)intheairofNorth,West,East andSouthzonesrespectively

≤1500 WCO2

ECO2 SCO2

NACH Maximumairchangerate:apracticallimittothe airchangeratearisingfromnaturalventilation; theconstraintsareonthenumberofhoursper yearwhereazonerequiresanairchangerateof morethan8airchangesperhour

≤0 WACH

EACH SACH

costforabuildingwhichcanbeeconomicallyrunatalower tem-peraturerange.Thesearethetemperatureandairflowset-points (thresholdsbeyondwhich HVAC systems areswitched on) and start-stoptimesforheatingandcoolingsystems.Overallthereare 8integer,30 continuousand12categoricalvariables (thelatter takingoneof 2or 3values), summarisedinTable 14(A).Gray codingisusedforcrossoverandmutationoperations,allowingthe sameoperatorstoapplyacrossallvariables.Thismeansthatthe continuousvariablesarediscretisedformutationandcrossover; thestepsizeisalsonotedinthetable.

Energyuseisdeterminedbythefreely-availablewhole-building energy simulation package EnergyPlus (E+) [41]. This models heating,coolingandlightingloadsrequiredtomeetpredefined set-pointsformeasuressuchastemperatureandlightlevel.Energyuse coversawholeyearofoperationgiventheweatherpatternsateach location.Theconstructioncost objectiveuses asimple function basedontheSpon’s[42]costsforthematerialsusedandequipment size(determinedbyE+).ArunoftheE+simulationtakes1–2minon anInteli72.7GHzCPU,dependingonthespecificbuilding config-uration.Allowinguptosixsimulationsinparallel,abudgetof5000 evaluationscompletesinapproximately24h(apracticalduration forreal-worlduse).

Theoptimisation is alsosubject to18 inequalityconstraints basedonoccupantcomfort,determinedbyE+,andsummarisedin Table1.Theycoverhoursofoverheatingduringsummermonths, carbondioxide(CO2)concentration(“freshness” ofinternalair), predictedmeanvote,andmaximumairchangeratedueto nat-uralventilation.Predictedmeanvoteisanestimationofthemean responseofagroupofpeopleaccordingtotheAmericanSocietyof Heating,RefrigeratingandAir-Conditioning Engineers(ASHRAE) thermalsensationscale,inwhich+3ishot,zeroisneutraland−3 iscold.

Severalassumptions are madeabout thefinal working con-ditionsfor the buildingthat impacton operationalenergy use andcomfort.Workinghoursare9:00to17:00,outsidewhichthe buildingisunoccupied.Eachzonehastypicaldesignconditionsof 1occupantper10m2_{floorareaandequipmentloadsof11.5W/m}2 floor area. Lighting is controlled to provide an illuminance of 500luxattworeferencepointslocatedineachperimeterzone; visibleaspointsonthefloorofFig.1.Infiltrationis0.1airchanges perhour,andventilation ratesare8l/sperperson.Heatingand cooling are modelled using an idealised system that provides sufficient energy to offset the zone loads and meet the zone

temperatureset-pointduringhoursofoperation;free-coolingis availablethroughnaturalandmechanicalventilation.

A simplified version of the same problemformed the basis forexperimentsdescribinganapproachtomulti-criteriadecision makingandvisualisationin[43].

4. Algorithms

This section describes the proposed surrogate, constraint handlingtechnique,anumberofminorchangestoimprove perfor-mance,andmethodsbywhichallofthesecanbeintegratedwithin NSGA-II.

4.1. RBFNsurrogate

Radialbasisfunctionnetworks(RBFN)arefullyconnected feed-forwardartificialneuralnetworkswhichcanbeusedforfunction interpolation[44],providingamappingfromasetofreal-valued inputstoasetofreal-valuedoutputs.Theyhavebeenappliedas surrogatesforevolutionaryalgorithmsinseveralpreviousworks including[45,12,46,47].Lietal.[12]notedthatanRBFNcanbeused forapproximationinmanydifferentdecisionspaces,aslongasan appropriatedistancemeasurecanbedefined.ThismakesRBFNs particularlysuitableforaproblemwithmixedvariabletypessuch astheexamplebuildingproblems.

Forthedistancemeasure,thepresentresearchadoptsthe Het-erogeneousEuclidean-OverlapMetric(HEOM)approachsuggested byWilsonandMartinezin[48]tocombinedistancesfordifferent parametertypes.Lietal.[12]alsousedthemeasurewithinaRBFN meta-modelforamixed-integersearchspace.Solutionsaredivided intothreecomponents;real/continuous,integerandnon-numeric orcategoricalparameters.Continuousandintegerparametersare normalisedtotherange(0,1)priortocalculatingthedistance met-ric.ThetotalrawdistanceisthecombinationoftheEuclideanr,

ManhattanzandHammingddistancesbetweenthese

param-etersrespectively:

=

Innormalisingthevariablesto(0,1),categoricalvariablesare over-weighted compared to integer and continuous variables. Whilethemeandistancebetweentwonormallydistributed ran-dom variables witha range of (0,1) is 1/3, ifthe variables are restrictedto0or1,themeandistanceis1/2.Tocorrectforthis,a weightthatwillbetermedtheHammingweightingωisintroduced, changingthedistancemeasureto(2).

=

ωis2/3sothemeancontributiontothetotaldistancethata cat-egoricalvariablemakesis1/3,equaltothatfortheothervariable types.Thiscouldalsoapplytotheintegervariablesbutthe prob-lemismuchreducedbecausethelowestcardinalityfortheinteger variablesintheexamplebuildingoptimisationproblemis7,for whichthemeandifferencebetweenapairofvariablesis0.38.An experimentexploringtheimpactofHammingweightingfurtheris describedinSection5.4.

4.1.1. Training

TrainingoftheRBFNtakesthreesteps:determiningthecentres for the hidden layernodes, the widths for the Gaussian func-tions,andtheoutputlayerweights.Thisworkadoptsthecommon practicethatthefirsttwostepsareunsupervisedandthelaststepis supervised.Fordeterminingtheidealclustersizeanddetermining whentoretraintheRBFN,thefitnesspredictioncorrelation(FPC) [49]isusedtomeasurethemodel’saccuracy.TheFPCisSpearman’s

Fig.2. CoreofNSGA-II[13]. rankcorrelation[50],betweenthetrueconstraint/objectivevalues

andthesurrogatepredictedvalues,forasolutionset.

Typically, a clusteringalgorithm suchas k-meansis usedto locatethecentresforRBFNhiddenlayernodes,orthecentresare simplythesetoftrainingpointsasin[12].However,trainingtime fortheoutputlayerweightsincreasesrapidlywiththenumberof trainingpoints.Karakasisand Giannakoglou[51]alsosuggested thatusingthetrainingpatternsascentresissuboptimalfor multi-objectiveproblems.Thepresentworkusesaclusteringalgorithm todeterminethecentres.k-medoidratherthank-meansclustering isused,sothecentresrepresentrealpointsinthediscretedecision space.Thisisnecessaryasitisimpossibletointerpolatebetween pointsseparatedbycategoricalvalues.

Tofindasuitablenumberofclusters,1000solutionswere gener-atedatrandom,andthenetworkwasrepeatedlytrainedonrandom subsets ofsolutions (subsetsizesmatched thepopulationsizes inSection5).TheFPCwascalculatedforobjectiveandconstraint predictionsoftheremainingsolutions,andwashighestwhenthe numberofclusterswasonethirdofthepopulationsize.

ThewidthforeachhiddennodeRBFistherootmeansquare distancefromthatnode’scentretothecentresofthetwonearest neighbours.Leastsquaresfittingisusedtodeterminetheoutput layerweights.

4.1.2. IntegratingthesurrogatewithNSGA-II

Forconvenience,thecoreofNSGA-II[13]isreproducedinFig.2. Theimportantmodificationisatstep11:toomanyoffspringare generated,andthesurrogatefilterspromisingsolutionsfor inclu-sioninthenewpopulation(Fig.3).

Thesurrogateisinitiallytrainedusingthestartingpopulation

P0.Then,inplaceoftheoriginaloffspringcreationstep,standard crossoverandmutationoperatorsareusedtogenerateasurrogate populationS,ofsize|S|,amultipleofthepopulationsize|P|.The surrogateapproximatesobjectiveandconstraintvaluesinS,and NSGA-II’sfast-non-dominated-sortranksSbyfeasibilityand non-dominance,withthehighestrankingsolutionsinsertedintothe offspringpopulationR.Risevaluatedusingthefullfitness func-tion(thebuildingperformancesimulation),andmodelaccuracyis checkedtodetermineifretrainingisneeded.NSGA-IIcontinuesto thenextgenerationasusual.

4.1.3. Modelupdates

A separate RBFN is used for each objective and constraint, allowingeach toberetrainedasneeded.FPCdeterminesmodel accuracyforeachobjectiveandconstrainteverygeneration,using thefilteredoffspringsolutions(Qt+1),afterevaluationbythefull

simulation.(Thisapproachistermedevolutioncontrol[14].)The thresholdFPCforretrainingwas0.7,representingastrong statisti-calcorrelationbetweenpredictedandtruevalues.Thesolutionsof

themostrecentgeneration(bothparentsandoffspring)wereused astrainingdata.1

4.2. Constraints

InapplyinganEAtoaconstrainedproblem,animportant con-siderationishowconstraintsarehandledalongsidetheobjective valuesofsolutions.Thisisfurthercomplicatedbythepossibilityof approximationerrorcausedbythesurrogateresultinginsolutions beingincorrectlyclassifiedas(in)feasible.

SurveysofEAconstrainthandlingapproaches[52,53,27]have identified four categories: penalty functions, decoders, special operators and separation ofobjectives and constraints,thelast beingusedinNSGA-II (classedas afeasibilityrulein[27]).The concept(originallyproposedbyDeb[13,54])issimple;when com-paringsolutions(eitherintournamentselection,orelitism): 1.Oftwofeasiblesolutions,choosetheonewiththebestobjective

function(s).

2.Ofaninfeasibleandafeasiblesolution,choosethefeasibleone. 3.Oftwoinfeasiblesolutions,choosetheonewiththelowestsum

ofnormalisedconstraintviolations.

Mezura-MontesandCoello[27]describesseveralnewcategories ofapproach,andofthese,thepresentworkbuildsonstochastic rankinganduseofmulti-objectiveconcepts.

Anumber ofstudieshave shownthat inclusionofinfeasible solutionsinthepopulationisbeneficial(seeSection2).The argu-mentcanbemadethatofteninfeasiblesolutionswhicharealmost feasiblearevaluablesteppingstonesbetweenfeasibleregionsof thesearchspace.Further,whenapproximatingconstraintvalues, thereisariskthatsomesolutionswillbeincorrectlyclassifiedas infeasible,anditisbettertoincreasethechancethattheywillbe retained.

The method proposed here takes a deterministic approach, referredtoasDeterministicInfeasibilitySorting(DIS).Theelitist componentofNSGA-IIwasmodifiedtoensurethatafixed propor-tion˛ofthepopulationistakenfromthesetofinfeasiblesolutions, unlessthesetissmallerthanthe˛-proportion,inwhichcaseallare copiedintothenewpopulation.Like[38,39],afterthepopulationis evaluated,itisdividedintofeasibleandinfeasiblesolutions,andthe feasiblesolutionsarenon-dominatingsorted(withcrowding dis-tanceapplied).TheinfeasiblesolutionsaresortedusingNSGA-II’s fastnon-dominatedsort,ignoringthemagnitudeoftheirconstraint

1_{AsmallnumberofrunsofNSGA-II-S(seeSection4.3)withretrainingbasedon} thepreviousfourgenerations’worthofsolutionsfoundnosignificantdifference betweenthatandusingonlythepreviousgeneration.

Fig.3. Amendedstep11inNSGA-II.

violations(so,usingonlyobjectivevalues).Thetop˛|P|solutions fromtherankingarecombinedwiththehighestrankingfeasible solutionstoformthenextpopulationofsize|P|.

Thisissimilartotheapproachin[38,39],althoughsimpler.DIS doesnotsubstitute constraintviolationwithanextraobjective, insteadignoringconstraintviolationcompletelyintheinfeasible population.Additionally, selection forreproduction stillfavours feasiblesolutions.DIScanbeappliedintwoplaces;intheelitist stepofthecorealgorithm,andinthesurrogatefilteringstep.Both approachesareexplored.

4.3. Algorithmvariants

TheexperimentscoveredseveralvariantsofNSGA-II,including differentapproachestosurrogateandconstrainthandling.NSGA-II

referstotheoriginalalgorithmofDebetal.[13].Useofthesurrogate isdenotedwithsuffix-S.DIS,ifpresentintheelitiststepofNSGA-II, isdenotedbysubscriptc,andifpresentinthesurrogatefiltering step,isdenotedbysubscriptd.Overall,wehave:

1.NSGA-II-S:NSGA-II,withRBFNsurrogate,andoriginalNSGA-II constrainthandling;

2.NSGA-IIc:NSGA-II,withDISatelitiststep;

3.NSGA-II-Sc:NSGA-II,withRBFNsurrogateandDISatelitiststep;

4.NSGA-II-Sd:NSGA-II,withRBFNsurrogateandDISatsurrogate

filteringstep;

5.NSGA-II-Scd:NSGA-II,withRBFNsurrogateandDISatbothelitist

andsurrogatefilteringsteps.

MinorchangesweremadetothebaseNSGA-IIalgorithmandall variants.Tomaximiseefficientuseofthe5000evaluationquota, populationsizewassmall.Anexternalarchivewasaddedtothe algorithmtostoretheParetofront,allowingabiggerfrontthanthe populationsizetobereturned.Thealgorithmattemptstoinsert everysolutionevaluatedintothearchive,andthesamedominance rulesusedbyNSGA-IIareusedtodeterminewhichsolutionsstay inthearchive.

Aninternalcacheofpreviously-evaluatedsolutionswas main-tained for each run, to avoid re-evaluating identical solutions.

Solutionsdrawnfromthecachedidnotcounttowardsthelimit of5000.Sixthreadswereusedtoconductevaluationsinparallel.

Optimisation variables were handled as Gray-encoded bit-stringsduringcrossoverandmutationtosimplifyoperatorchoice (bit-flipmutationanduniformcrossover).Thesurrogateoperated directlyontheunencodedcontinuous,integerandcategorical vari-ablevalues.

5. Experimentsandresults

Threesetsofexperimentscomparedtheperformanceofthe dif-ferentapproachestoconstrainthandlingandthesurrogate.Firstly, thealgorithmvariantsweretunedontheBirminghamproblem. Secondly,allvariantswerecomparedontheBirminghamproblem. Thirdly,asubsetwerecomparedontheChicagoandSanFrancisco problems.Allrunswererestrictedto5000trueevaluations,and comparisonsweremadeofthefinal Paretofrontsreturned.The goalwastofindthealgorithmsetupreturningthehighest hypervol-umeandlowestspreadmetricforthesamenumberofevaluations (i.e.,betterqualitysolutionsforthesameeffort).Alsogivenare comparisonsoftheeffortrequired toreachthesamequalityof solutions.

5.1. Parametertuning

Preliminaryrunsaimedtofindgoodparametersforthe algo-rithmandvariants:theparameterstunedandthevalueforeach yieldingthehighestmeanhypervolumeisgiveninTable2.Each runwaslimitedtofiverepeats(limitedbythetimeneededfora singlerun:aroundoneday).

5.2. ComparisonsforBirmingham

EachNSGA-IIvariantwasrun30times:Table3givesthemean hypervolumeandspreadfortheParetofrontsfound.Toavoidissues withmultiplet-tests,a Dunnetttestforsignificanceusingplain NSGA-IIasthebase-casewasperformed:theresultingp-values arealsogiven.AnalysisofvariancealsofoundthatP<0.05forthe nullhypothesisthattheresultsfromallrunsweredrawnfromthe

Table2

Parametersettingexperiments.Therangeofparameters,andthevalueproducing thebesthypervolumeforallalgorithms.

Parameter Valuestried Bestvalue

Populationsize 15,20,30,60 20

Mutationratea _{0.5/n,1.0/n,2.0/n,4.0/n 4.0/n(NSGA-II,}

NSGA-IIc),1.0/n (NSGA-II-Svariants) ˛forconstraintmethodat

elitiststep

0.1,0.2,0.3,0.4,0.5 0.2 ˛forconstraintmethodat

surrogatefiltering

0.1,0.2,0.3,0.4,0.5 0.3 Numberofoffspringfor

filteringbysurrogate (multipleofpopsize)

3,5,10,20,100 3

a_{Mutationratesare1/n,nbeingthenumberofbitsintheGrayencoding.}

Table3

OptimisationresultsfortheBirminghamproblem.Themeanhypervolumeand spread(over30runs)oftheParetofrontfoundbytheNSGA-IIvariants.Standard deviationsareinsubscript.Alsogivenarethep-valuesfromaDunnetttest com-paringtheresultsofthevariantswithplainNSGA-II.Boldfiguresindicatestatistical significance(p<0.05).

Algorithmvariant Hypervolume p Spread p

NSGA-II 0.8490.028 n/a 0.7960.088 n/a NSGA-IIc 0.8450.022 0.969 0.7920.070 1.000 NSGA-II-S 0.8560.028 0.783 0.8020.092 0.998 NSGA-II-Sc 0.8600.024 0.338 0.7890.076 0.996 NSGA-II-Sd 0.8810.031 <0.001 0.8270.052 0.342 NSGA-II-Scd 0.8670.027 0.034 0.7790.059 0.841

samedistribution.Fig.4givesthecorrespondingmediansummary attainmentcurves[55]:thecurveofpointsintheobjectivespace reachedbyatleasthalfoftherepeatruns.Acomparisonwasalso madewithrandomsearch,seekingthesetofnon-dominated feasi-blesolutionsfrom5000randomlygeneratedsolutions.Noneof30 runsofrandomsearchfoundafeasiblesolution,whereasallrunsof theNSGA-IIvariantsfoundsolutionsmeetingallconstraintswithin 1984evaluations.

Althoughanimprovementinfinalsolutionqualityisuseful,at leastasimportantisreducedoverallrun-timewhileachievingthe samequalityofsolutions.Thisisparticularlytrueforthebuilding industry,wheretimelyproductionofplansfortenderingiscritical, andaspeedupof10–20%couldallowanoptimisationrunto com-pleteovernightratherthanpostponingamorning’swork.Forthis reason,theoriginalresultswerereprocessed,takingthemedian hypervolumereachedbyNSGA-II(0.8486)asatarget,and measur-ingthenumberofevaluationstakenbyeachalgorithmtofinda

360000 365000 370000 375000 380000 385000 390000 395000 400000 40000 45000 50000 55000 60000 65000 70000 75000 80000 Cost (£) Energy (kWh) NSGA-II

NSGA-II_c NSGA-II-SNSGA-II-S_c

NSGA-II-S_cd NSGA-II-S_d Fig.4.SummaryattainmentcurvefortheBirminghamproblem.

Table4

OptimisationresultsfortheBirminghamproblem.SRisthepercentageofruns foreachalgorithmvariantthatweresuccessful(reachingthetargethypervolume withinthe5000evaluationbudget),andEvalsisthemeannumberofevaluations takenbysuccessfulrunstoreachafrontwiththetargethypervolume.

Algorithmvariant SR(%) Evals

NSGA-II 50 4017 NSGA-IIc 40 4026 NSGA-II-S 83 3817 NSGA-II-Sc 63 4002 NSGA-II-Sd 83 3184 NSGA-II-Scd 73 3340

Paretosetwiththathypervolume.TheresultsaregiveninTable4; wherea rundidnotreachthetargethypervolumewithin5000 evaluations,itwasclassedasfailed.

There are several key points to note. None of the methods improveorworsenspreadsignificantly.Thisisimportantbecauseit wouldbelessusefultohaveasetofsolutionswithimproved hyper-volumeiftheywereunevenlydistributed, reducingthechoices availabletothedecisionmaker.

DIS (in NSGA-IIc) reducestheperformance of NSGA-II, with

lowermeanhypervolumeandhighermeanspread,although nei-therdifferenceisstatisticallysignificant.Thisisreflectedinalower successrate(fewerrunsreachedthetargethypervolumewithin 5000evaluations).Further,themedianattainmentcurvefor NSGA-IIc,whiledominatingpartofthatforNSGA-II,isdominatedatthe

highcapitalcostendbythelatter.Thispointofthefrontiswhere thecomfortconstraintsaremorelikelytobebroken,further ana-lysedinSection7.ThisindicatesthatDISimprovessearchefficiency inpartsofthesearchspacewithabalancedmixtureoffeasibleand infeasiblesolutions,butDISdecreasesefficiencywherethelocal searchspaceismostlyinfeasible.Giventhatallalgorithmvariants foundfeasiblesolutionswithinafewhundredevaluations,itwould appearthatthefeasibleregionofthesearchspaceislargeand con-tiguousenough(furtherobservedinSection7)toavoidneedingto retaininfeasiblesolutionstohelpjumpgapsinthefeasibleregion. Instead,DISaddsdragtothesearch:removingfeasiblesolutions fromthepopulationeffectivelyreducesitssizewithnocountering benefit,loweringthesearchspeed.

ThemeanhypervolumeofthesetsfoundbyNSGA-II-Sishigher thanforNSGA-II,thoughnotsignificantly.Thisisreflectedbyan increasedsuccessrate,andthemedianattainmentcurveof NSGA-II-SdominatingthatofNSGA-II.

Moreimportantly,NSGA-II-Swasfurtherimprovedbyadding DIS–particularlyatthefilteringstage(NSGA-II-Sd),wherethe

dif-ferenceoverNSGA-IIissignificant.Thiswouldappeartobebecause DIScompensatesforconstraintapproximationerrorsmadebythe surrogate: furtherexplored inSection 6.Adding DIS atthe eli-tiststep (NSGA-II-Sc)also improvesfinal hypervolume,but not

significantly. Adding DIS at both the filtering and elitist steps (NSGA-II-Scd)is sub-optimal,exceptat thelow-costend where

thecurveforNSGA-II-ScddominatesthatforNSGA-II-Sd(similarto

theslightbenefitshownforNSGA-IIcvsNSGA-II).Asimilareffect

occurswhenDISisappliedattheelitiststep,butwhenitisalready appliedatthefilteringstagethisbenefitislostduetothedrageffect described earlier.Themedian attainmentcurves showno simi-larbenefitforNSGA-II-Sc.Itwouldappearthattheissuesrelated

toconstraintapproximationhavemoreimpactonefficiencythan retaininginfeasiblesolutionsinthepopulation.

Inpracticalterms,thereal-worldbenefitoftheimprovedfronts foundbyNSGA-II-Sd issubstantial.Takingthemediansummary

attainmentcurvesforNSGA-IIandNSGA-II-Sd,andchoosinga

cap-italcostabouthalfwayalongthecurve(

£

Table5

OptimisationresultsforChicago,formattingmatchesFig.3.

Algorithmvariant Hypervolume p Spread p

NSGA-II 0.6250.107 n/a 0.7790.099 n/a NSGA-II-S 0.7160.090 0.001 0.8050.057 0.345 NSGA-II-Sd 0.7840.095 <0.001 0.7030.055 <0.001 NSGA-II-Scd 0.6950.093 0.016 0.7790.060 0.999

Table6

OptimisationresultsforSanFrancisco,formattingmatchesFig.3.

Algorithmvariant Hypervolume p Spread p

NSGA-II 0.7270.067 n/a 0.7590.070 n/a NSGA-II-S 0.7300.063 0.992 0.8100.100 0.069 NSGA-II-Sd 0.7330.064 0.869 0.7550.069 0.999 NSGA-II-Scd 0.7270.041 0.990 0.7950.075 0.422

savingof6531kWh(13%)perannumwillreducecarbonemissions andrunningcostsoverthelifetimeofthebuilding.

5.3. ComparisonsforChicagoandSanFrancisco

Furtherexperiments appliedNSGA-II, NSGA-II-S, NSGA-II-Sd

andNSGA-II-Scdtothesamebuilding,usingtheChicagoandSan

Franciscoweather,for30independentruns.AsnotedinSection3, whilethebasicproblemissimilar,thesearchspaceandoptima for theseproblems are expected tobe quite differentto those forBirmingham.ComparedtotheBirminghamclimate,Chicagois moredemanding,requiringhighlevelsofheatinginwinterand active/mechanicalcoolingin summer(thebuildinghastowork acrosstherange).SanFranciscohasalocalmicro-climateresulting inalmostnoneedforheatingandmechanicalcooling,the build-ingremainingcomfortablewithpassiveoperation.Tables5and6 givethemeanhypervolumeandspreadoftheParetosetsfoundby eachalgorithmwithinthe5000evaluationbudgetforthetwo prob-lems.Table7givethesuccessrateforeachalgorithm(proportion ofrunsachievingthemedianhypervolumereachedbyNSGA-II: 0.6095forChicagoand0.7334forSanFrancisco).Summary attain-mentcurvesforChicagoaregiveninFig.5;thoseforSanFrancisco areexcludedastheywerealloverlaidovereachotherandaddlittle tothediscussion.

ForChicago,allNSGA-II-Svariantsfoundfrontswith statisti-callysignificantlyhighermeanhypervolumesthanNSGA-II, and NSGA-II-Sd foundfrontswithsignificantlylower(better)spread

values.Thisisparticularlyclearinthemedianattainmentcurves; thecurvesforallvariantsdominatethatforNSGA-II.Aswiththe Birminghamproblem,NSGA-II-Sd performsbest,withNSGA-II-S

andNSGA-II-Scd performingsimilarly,both betterthan NSGA-II.

The issueseen with Birmingham where the attainment curves coincidedforlowcostsolutionsdoesnotappearforChicago.This isbecausethemoreextremeweatherconditionsrequireallthe

Table7

OptimisationresultsforChicagoandSanFrancisco.SRisthepercentageofrunsfor eachalgorithmvariantthatweresuccessful(i.e.,reachingthetargethypervolume withinthe5000evaluationbudget),andEvalsisthemeannumberofevaluations takenbysuccessfulrunstoreachafrontwiththetargethypervolume.

Algorithmvariant Chicago SanFrancisco

SR(%) Evals SR(%) Evals NSGA-II 50 3892 50 3620 NSGA-II-S 83 3557 50 3684 NSGA-II-Sd 97 3085 57 3747 NSGA-II-Scd 80 3603 47 3898 400000 405000 410000 415000 420000 425000 110000 115000 120000 125000 130000 135000 140000 Cost (£) Energy (kWh) NSGA-II NSGA-II-S NSGA-II-S_cd NSGA-II-S_d

Fig.5. SummaryattainmentcurvefortheChicagoproblem.

buildingstohaveheavierconstructiontypes(e.g.allsolutionshave heavyweightexternalwalls,ratherthanavariety)inordertomeet thecomfortconstraints.Thismoves thebuildingsfromrunning closetoconstraintviolationwiththelighterconstructiontypesto runningsafelywithintheconstrainedregionwithheaviertypes, makingtheimpactofDISlessthanforBirmingham.Thisalsocomes withahighercapitalcostforallsolutionsinthetrade-offthanfor Birmingham.

ForSanFrancisco,allalgorithmvariantsperformedsimilarly. Thereisnostatisticallysignificantdifferenceinthehypervolumes ofthefrontsfound, andsuccessratesarealsosimilar(although NSGA-II-Sdfoundthetargethypervolumewithinthe5000

evalua-tionbudgetslightlymoreoftenthantheothervariants).Indeed,all thealgorithmsconvergeatapproximatelythesamerate, demon-stratedforNSGA-IIandNSGA-IIdinFig.6.

Inpart,thedifferenceinalgorithmbehaviourforthethree prob-lemsisduetotheconstraints.Table8showsthemeannumberof evaluationstakenbyrunsofNSGA-IItofindsolutionsmeetingthe individualconstraints(thisisnotthetimetakentofindfeasible solutions,whichismuchlongerasallconstraintsmustbe simulta-neouslymet).MostconstraintsforChicagoarehardertosolvethan forBirmingham,andinturnareharderthanthoseforSanFrancisco. Thisimpliesthatwheretheconstraintsareeasiest,thereisleastto gainfromuseofthesurrogate.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Mean Hypervolume Evaluation Count NSGA-II NSGA-II-S_d

Fig.6. HypervolumereachedoverfunctionevaluationsfortheSanFrancisco prob-lem.

Table8

ThenumberoffunctionevaluationstakenbyNSGA-IItofindsolutionsmeetingthe individualconstraintsforthethreeproblems(B)irmingham,(C)hicagoand(S)an Francisco.(constraints,lefttoright,areintheordergiveninTable1).

Constraints1–9 B 14 25 23 37 24 89 142 129 82 C 31 38 33 44 27 382 475 528 350 S 11 11 9 18 8 23 39 28 9 Constraints10–18 B 13 8 15 16 8 112 38 28 43 C 43 8 8 10 8 120 82 64 58 S 9 8 13 16 8 61 62 142 61 Table9

Themeanhypervolumeandspread(over30runs)oftheParetofrontfoundby plainNSGA-II,andNSGA-II-ShavingHammingweightingsof2/3and1.Standard deviationsareinsubscript.ThefirsttworowsduplicatepartofTable3,repeated hereforconvenience.

Algorithmvariant Hypervolume Spread

NSGA-II 0.8490.028 0.7960.088 NSGA-II-Swithω=2/3 0.8560.028 0.8020.092 NSGA-II-Swithω=1 0.8510.024 0.8210.078 NSGA-II-Sdwithω=2/3 0.8810.031 0.8270.052 NSGA-II-Sdwithω=1 0.8760.037 0.8100.075 Table10

p-valuesforunpairedt-testsbetweenhypervolumesforNSGA-IIandNSGA-II-Swith thetwoHammingweightings.Differentp-value(toTable3)forNSGA-IIvsNSGA-II-S arisesfromuseofunpairedt-testratherthanDunnetttest.

Algorithmvariants p NSGA-IIvsNSGA-II-S(ω=2/3) 0.347 NSGA-IIvsNSGA-II-S(ω=1) 0.710 NSGA-II-S(ω=2/3)vsNSGA-II-S(ω=1) 0.524 NSGA-IIvsNSGA-II-S(ω=2/3) <0.001 NSGA-IIvsNSGA-II-Sd(ω=1) 0.003 NSGA-II-Sd(ω=2/3)vsNSGA-II-Sd(ω=1) 0.545 5.4. Hammingweighting

TheweightingforallNSGA-II-Svariantswas2/3,justifiedin Section 4.1. The optimisation experiment for Birmingham was repeatedwithNSGA-II-SandNSGA-II-Sd,usingasurrogatewith

aHammingweightingof1(thesameas[12]).Themean hypervol-umeandspreadoftheParetofrontsfoundin30independentruns aregiveninTable9,alongsideresultsforNSGA-IIandNSGA-II-S withtheHammingweightingof2/3,repeatedforconvenience.t -testsbetweenthehypervolumesaregiveninTable10(noneofthe differencesforspreadweresignificant).

Whilehypervolumeishigherfortheapproacheswithω=2/3 comparedtothosewithω=1,thedifferenceisnotsignificant;the changealsodoesnotaffectwhetherthehypervolumereachedby NSGA-II-SorNSGA-II-SdissignificantlyhigherthanforNSGA-II.

Thereisnosignificantdifferenceinthespreadforthefrontsfound byanyofthealgorithms;forNSGA-II-S,settingω=2/3improves spread,butforNSGA-II-Sdsettingω=2/3resultsinpoorerspread.

Thealgorithmswithω=2/3didconvergemorereliablytoafront withthetargethypervolumeof0.8486withinthe5000evaluations budget.NSGA-II-SandNSGA-II-Sdreachedthetargetin83%ofthe

30runs.Withω=1,thisdroppedto57%and67%respectively.This providessomejustificationbeyondthetheoryinSection4.1for usingtheHammingweightingof2/3.

6. Analysisoftheconstrainthandling

Thissectionexploresinmoredetailthesuccessofthe surro-gateinhandlingconstraints.Section4.1.2describedhow,within

NSGA-II-S, the surrogate assigns values to each objective and constraintforeachsolutioninthesurrogatepopulation.A fast-non-dominated-sortisconducted,andthetoprankingNsolutionsare evaluatedusingthefullsimulation,thencombinedwiththeparent populationbytheelitiststepofNSGA-II.Toexplorethisfurther, foronerunofNSGA-II-SontheBirminghamproblem,thefull sim-ulationwasrunforallsolutionsevaluatedbythesurrogate.This amountedtoanassessmentofthesurrogate’sperformanceover 14,940solutions.Thesesolutionsweredividedintofourcategories foreachconstraint:

•Correctfail–predictedtohaveviolatedtheconstraint,whichthe simulationconfirmed;

•Incorrectfail–predictedtohaveviolatedtheconstraint,buthad not;

•Correctpass–predictedtohavemettheconstraint,whichthe simulationconfirmed;

•Incorrectpass–predictedtohavemettheconstraint,buthadnot. Thesecanbecomparedwith:

•Truefail–violatesconstraint,accordingtothesimulation; •Truepass–meetsconstraint,accordingtothesimulation; •Predictedfail–violatesconstraint,accordingtothesurrogate; •Predictedpass–violatesconstraint,accordingtothesurrogate.

Thecountsofsolutionsineachcategorywereusedtocalculate, foreachconstraint,twomeasuresborrowedfromtheinformation retrievalcommunity,precisionPandrecallR[56]:

Precision= Correctpassesorfails

Totalpredictedpassesorfails (3) Recall= _TotalCorrect_truepasses_passesor_orfails_fails (4) Precisionistheproportionofsurrogatepredictionswhichwere correct,andrecallistheproportionoftruepassesorfailsthatwere predictedassuch.Ideally,bothmeasuresshouldbe1meaningthat alltruepassesandfailswerecorrectlypredictedassuch.PpandRp

refertoprecisionandrecallforpasses,andPfandRfforfails.The

resultsaregiveninTable11,togetherwiththeFPCandnumberof timesthesurrogateforeachconstraintwasrebuiltovertherun. TheabbreviatedconstraintnamesaregiveninSection3.

Ppexceeds0.85forallconstraints,exceptthePMVconstraints,

whichhavealowmeanFPC(<0.3)andahighnumberofrebuilds (>95:aroundonerebuildeverythirdgeneration).Incontrast,Pfis

lowformostconstraints;exceeding0.5foronlythreeoftheACH constraints(notethatforthese,FPCishighandrebuildsarelow). Thisisinpartbecausetherearefarmoretruepassesthanfails,so withasimilarerrorrate,morepasseswillbeincorrectlyclassedas failsthantheotherwayround.Theconverseofthisisshownfor theWPMVandEPMVconstraints,wherethenumberoftruepasses ismuchclosertothenumberoftruefails,andPpdrops(withan

increaseinP_f).

Thesurrogateclearlyperformsbetterwithsomecategoriesof constraintthanothers:

•ForPMVs,passprecisionisworsethanfortheotherconstraints butnotlow,andfailprecisionisnothighbutbetterthanforSOH andCO2.

•ForSOHandCO2,passprecisionishigh,andfailprecisionislow. •ForACH,passprecisionishigh,andfailprecisionisnothighbut

betterthanforSOHandCO2.

Table11

SummaryofthesurrogatepredictionsoftheconstraintsoveronerunofNSGA-II-S.TfandTparethenumberoftruefailsandpassesfortheconstraint.Tp/Tisthefractionof allsolutionsevaluatedthatpassedtheconstraint(intendedtogiveanimpressionoftheconstraint’sdifficulty).Pf,pandRf,paretheprecisionandrecallforsolutionspassing andfailingtomeettheconstraintrespectively.FPCisthemeanfitnesspredictioncorrelationfortherawconstraintvalue,andCisthemodelrebuildcountforthesurrogate onthatconstraint. Con Tf Tp Tp/T Pf Rf Pp Rp FPC C NSOH 892 14,048 0.94 0.07 0.239 0.943 0.798 0.534 59 WSOH 1381 13,559 0.91 0.107 0.482 0.918 0.591 0.518 61 ESOH 1340 13,600 0.91 0.104 0.293 0.915 0.752 0.723 22 SSOH 2680 12,260 0.82 0.18 0.671 0.822 0.332 0.24 97 ISOH 1029 13,911 0.93 0.122 0.277 0.941 0.852 0.92 1 NPMV 3179 11,761 0.79 0.252 0.608 0.828 0.511 0.165 104 WPMV 6475 8465 0.57 0.447 0.78 0.609 0.262 0.149 99 EPMV 6235 8705 0.58 0.46 0.788 0.689 0.336 0.179 106 SPMV 2335 12,605 0.84 0.183 0.561 0.868 0.536 0.287 97 IPMV 542 14,398 0.96 0.097 0.314 0.972 0.89 0.545 53 NCO2 800 14,140 0.95 0.06 0.216 0.948 0.809 0.071 104 WCO2 134 14,806 0.99 0.018 0.06 0.991 0.97 0.045 109 ECO2 275 14,665 0.98 0.039 0.076 0.982 0.964 0.135 107 SCO2 409 14,531 0.97 0.03 0.11 0.973 0.9 0.139 108 NACH 1066 13,874 0.93 0.44 0.093 0.934 0.991 0.889 5 WACH 1308 13,632 0.91 0.825 0.255 0.933 0.995 0.841 9 EACH 1087 13,853 0.93 0.389 0.064 0.931 0.992 0.867 6 SACH 1121 13,819 0.92 0.213 0.06 0.928 0.982 0.856 8

FortheACHconstraints,whichhavehigherPfandaroundthe

samePpcomparedwiththeotherconstraints,themeanFPCis

high-estandthenumberofmodelrebuildsislow.Theotherconstraints, whicheitherhaveverylowPf(SOHandCO2)orlowerPp(PMV),

havelowermeanFPCvalues,andcorrespondinglyhigherrebuild counts.Thismeansthatwherethemodelisaccuratelypredicting constraintpassesandfailsforsolutionsitisbeingkept,andwhere itis lesssuccessfulit isbeingrebuiltin anattempttoimprove accuracy.TheseresultsprovidefurtherevidencethattheFPCis areasonablemeasureofthesurrogate’spredictioncapability,and suitableforchoosingwhentorebuildit.

6.1. ImpactofDIS

Asnotedabove,precisionandrecallaremuchhigherforpasses thanfails.Insimpleterms,forthisproblemthesurrogateis conser-vativeinitspredictionofconstraintvalues,tendingtofalselyfail feasiblesolutionsratherthanallowinginfeasiblesolutionsintothe population.Whilethismaybepreferable,itmeansalargenumber ofsolutionswithusefulgeneticmaterialareincorrectlylabelledas

infeasibleanddiscarded.ThismotivatedtheintroductionofDIS, whichallowssomeofthesolutionspredictedinfeasible,butalso predictedhighintheobjectivevalues,throughthefilteringonthe goodchancethattheyare(inreality)feasible.

ThetwoDISvariantswere:allowingsomeinfeasiblesolutions topassfromonegenerationtothenextattheelitiststep (NSGA-IIc);andallowingsomesolutionspredictedinfeasiblethroughthe

surrogatefilteringstep(NSGA-II-S_d).Toexploretheimpactofthe latter,theexperimentdescribedatthebeginningofthissection wasrepeated,butlimitedtosolutionsonlyincludedinthe pop-ulationbecauseofDIS(“filtered-in”solutions),andthesolutions whichtheyreplaced(“filtered-out”solutions).Astherewerefar fewersolutionsaffectedbyDISthanwerepresentedtothe sur-rogateoverall,theseresultswereaggregatedfromall30runsof NSGA-II-Sd.Table12showsthesuccessofthemodelin

approxi-matingconstraintsforthesetwogroupsofsolutionsasprecision andrecallfigures.Figuresinboldshowwherethevalueislower thanthatforthewholeruninTable11.

Among filtered-out solutions, relative to the overall figures for precisionand recall:14/18 constraints had lower Pf; 12/18

Table12

SummaryofthesurrogatepredictionsoftheconstraintsforsolutionsthatwerelostbecausetheywerereplacedwithinfeasiblesolutionsbyDIS.Pf,pandRf,paretheprecision andrecallrelatedforsolutionspassingandfailingtomeettheconstraintrespectively.ValuesinboldarelowerthanthecorrespondingvaluesinTable11.

Constraint Pf Rf Pp Rp Pf Rf Pp Rp NSOH 0.143 0.125 0.978 0.981 0.143 0.125 0.978 0.981 WSOH 0.096 0.302 0.941 0.796 0.095 0.295 0.939 0.793 ESOH 0.047 0.150 0.971 0.902 0.047 0.150 0.971 0.902 SSOH 0.091 0.329 0.952 0.802 0.091 0.329 0.952 0.802 ISOH 0.000 0.000 0.974 0.984 0.000 0.000 0.974 0.984 NPMV 0.226 0.686 0.818 0.375 0.226 0.671 0.805 0.371 WPMV 0.429 0.732 0.516 0.227 0.429 0.732 0.516 0.227 EPMV 0.423 0.725 0.552 0.254 0.424 0.738 0.565 0.253 SPMV 0.267 0.695 0.811 0.407 0.267 0.686 0.803 0.404 IPMV 0.124 0.416 0.956 0.813 0.126 0.413 0.954 0.810 NCO2 0.048 0.122 0.963 0.905 0.048 0.122 0.963 0.905 WCO2 0.000 0.000 0.991 0.975 0.000 0.000 0.990 0.975 ECO2 0.042 0.077 0.991 0.982 0.042 0.071 0.990 0.982 SCO2 0.000 0.000 0.991 0.954 0.000 0.000 0.991 0.954 NACH 0.143 0.077 0.932 0.965 0.143 0.076 0.931 0.965 WACH 0.500 0.317 0.921 0.962 0.500 0.319 0.922 0.962 EACH 0.143 0.090 0.934 0.960 0.143 0.091 0.935 0.960 SACH 0.029 0.012 0.934 0.973 0.029 0.012 0.935 0.973

Table13

Meantrue(ratherthansurrogatepredicted)valuesforoverallconstraintviolation andtheobjectives,forfiltered-inandfiltered-outsolutions.Standarddeviationsare omittedbecausetheymeanlittle–thesolutionsarefromvariouspartsofthePareto frontandhaveawiderangeofobjectivevalues.

Filtered-insolutions Filtered-outsolutions

Trueenergy 97,378 98,468

Truecost 399,910 40,224

Trueoverallconstraintviolation 631,602 135,273

hadlowerRf;8/18hadlowerPp and9/18hadlowerRp.Among

filtered-insolutions,relativetotheoverallfiguresforprecisionand recall:14/18constraintshadlowerPf;13/18hadlowerRf;8/18had

lowerPpand9/18hadlowerRp.Thismeansthatforthesolutions

nearesttothefeasible/infeasibleboundary,mostconstraintswere lessaccuratelypredictedbythesurrogatethanwasthecaseforall solutions.Thisishardlyasurprise:boundarysolutionswerelikely tobethepointatwhichanyerrorinapproximationwouldmanifest itself.It does,however,providesomejustificationfor usingDIS toswapsomesolutionsatthefeasible/infeasibleboundary:ifthe modelislesslikelytoaccuratelyclassifythesesolutionsthenthey mayaswellberandomlylabelledfeasibleorinfeasible,avoiding anyunhelpfulbiasintroducedbymodellingerror.

Asnotedabove,themostcommonerrorispredictingfeasible solutionsasinfeasible.Aneffectofthisisthatnotenough solu-tionsarepredictedfeasibleforDIStofilterthemout(enoughofthe filteredpopulationisalreadyinfeasible).WhereDISdoesfilter solu-tionsinandout,comparingtruevaluesforfiltered-insolutionswith thosefiltered-outshowedthattheformerwerehigherinoverall constraintviolation(asintended),higherinthecostobjectiveand lowerintheenergyobjective.71%oftheDISfilteringoperations ledtoanincreaseinthemeantrueoverallconstraintviolationfor thepopulation;52%ledtoadecreaseinthemeantrueenergyuse ofbuildingsinthepopulation;53%ledtoanincreaseinthemean truecapitalcost.Meanfiguresforallfiltered-inandfiltered-out solutionsaregiveninTable13.Thelowerenergyobjectivevalues amongfiltered-insolutionsappearstocontributetoimproved per-formancewhenusingDIS(NSGA-II-Sd),comparedwithusingthe

surrogatealone(NSGA-II-S).Thesummaryattainmentcurveforthe Birminghamproblem(Fig.4)showedthatNSGA-II-Scwasableto

findfrontswithlowerenergyvaluesbutnotaslowcapitalcostsas

NSGA-II-S.Whilethedifferenceisslight,theimpactofslightly bet-tersolutionswouldbeamplifiedasthealgorithmrunsovermany generations.

7. Analysisofthefitnesslandscape

Itishelpfultounderstandthefitnesslandscapefortheproblem todeterminehowdifficultitis tomodel accurately.For single-objectiveproblems,commonapproachesincludefitnessdistance plots[57]andautocorrelation[57].Bothuseasinglesolution, typ-icallytheglobaloptimum,asastartingpoint.Formulti-objective problems,therearemultiplesolutionsintheParetofrontrather thanonesingleglobaloptimum,sothepictureismorecomplex. Inthepresent study,a modifiedfitnessdistanceplotis usedto illustratetheareaimmediatelyaroundtheParetofront.

AsthetrueParetofrontisunknown,aglobalParetofrontfor theproblemwasconstructedbytakingthesetofnon-dominated solutionsfromtheunionofallParetofrontsgeneratedduringthe originalexperimentalruns.Theseexperimentscovered approxi-matelytwomillionfunctionevaluations(480experimentalrunsof 5000solutions,withsomeduplicatesolutions).Theresultingfront consistedof120solutions.Foreachmemberofthefront,allpossible valuesweretriedforeachofitsvariables(giventhediscretisation causedbyGrayencoding),onevariableatatime.Thisamountsto anexhaustivewalkaroundthelocalityoftheParetofront, cover-ing55,101solutions.TheresultsaregiveninFig.7.Thepointsare separatedusingshapeandcolourintofeasible,infeasibleandthe originalfront.

Adeeperanalysisoftheconstraintswasperformedbysorting thesolutionsgeneratedbythestepwisetraversalbyascending cap-italcost.Thissetwasdividedinto10bandsrepresentingincreasing cost.Thefractionofsolutionsineachbandthatwerefeasibleare plottedinFig.8.

Anumberofobservationsaboutthelandscapecanbemade. 1.Nolargeimprovementsineitherobjectivecanbefoundnearthe

globalParetofrontwithoutbreakingaconstraint.Thismeans thatthe“global”Paretofrontisatleastneartolocallyoptimal andisusefulforcomparisons.However,withoutan(impractical) exhaustivesearchitisimpossibletosaythatthefrontistruly, globally,optimal. 340000 360000 380000 400000 420000 440000 460000 480000 20000 40000 60000 80000 100000 120000 140000 160000 Cost (£) Energy (kWh)

Infeasible Feasible GlobalPF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 6 7 8 9 10 Fraction feasible

Capital cost decile

Fig.8. Fractionofsolutionsthatwerefeasible,whensolutionsweregroupedinto binsofascendingcapitalcost.

2.27,896(50.6%)ofthe55,101solutionsintheimmediate neigh-bourhoodof thePareto front areinfeasible. Thismeans that whileconsiderableeffortcouldbewastedexploringinfeasible solutions,alargeproportionofsolutionsarefeasible.The prob-lemwould bemuchharderweretheproportionofinfeasible solutionshigher.

3.ThefeasibleregionaroundtheParetofrontismostly contigu-ous.Thisprovidessomeevidencefortheconclusion(Section5) thatNSGA-IIcdidnotofferimprovedperformancebecausethere

werenotenoughgapsinthefeasiblepartofthesearchspace forincludinginfeasiblesolutionsinthepopulationtogivemuch benefit.

4.Onevariable(ceiling/floorconstructiontype)hasalargeimpact oncost,resultingintwoduplicatesoftheParetofrontwhichare muchhigherincostthanmostoftheothersolutionsintheplot. 5.Thereare large changes in theobjective values (particularly energy)bysimplychangingonevariable.Thismeansitisharder tomodelthana“flatter”landscapewouldbe;somoredifficult forasurrogatetoreproduce.

6.WithinthevicinityoftheParetofront,ascapitalcostincreases, solutionsaremorelikelytobeinfeasible.Thisisofrelevanceto thediscussionofDISinSection5.2.

ThisexperimentwasrepeatedfortheSanFranciscoproblem,to attempttofurtherexplainthedifferentoptimisationresults.The plotisnotshownforspacereasons,butrevealedtwopoints.Firstly, therangeintheenergyobjectiveacrossthesolutionsismuchlower fortheSanFranciscoproblemmeaningthatonceclosetothePareto front,crossoverandmutationarelesslikelytogeneratesolutions farfromit (whichwould bewastedeffort).Secondly, a slightly higherproportion(51.7%)ofthesolutionsaroundtheParetofront arefeasible.Ifthesefactorssimplifytheproblem,theremaybelittle roomforimprovementinefficiencyoverplainNSGA-II.

8. Conclusions

Thispaperhasdescribedaconstrained,multi-objective prob-lem,withmixedvariabletypes,fromthebuildingdesigndomain. Thisapplicationareaformsa crucialpartoftheglobalresponse toclimatechange,andgiventhetime-constrainednatureofthe industryitisimportantthatoptimisationapproachesareableto findahigh-qualitysolutionorsetofsolutionsasquicklyaspossible. Tomeetthisdemand,thepaperhasproposedseveralvariantsof NSGA-II,incorporatingRBFNsurrogatesandanapproachto allow-inginfeasiblesolutionstoremaininthepopulation.Thesevariants wereappliedtothreeinstancesofthebuildingproblem(using dif-ferentlocationsforthebuildingweatherdata).Optimisationruns werelimitedtoabudgetof5000functionevaluations,and com-paredbothonthequalityofsolutionsfound(howclosethesetwas tothePareto-optimalset)andontherateofconvergencerelative toplainNSGA-II.

Thepaperhasmadeanumberofcontributions.Ithasproposed usingaseparateRBFNsurrogateforeachconstraintandobjective toallowforretraining.TocontroltheretrainingtheFitness Predic-tionCorrelation(FPC)isused.AnexistingapproachtousingRBFNs assurrogateshasbeenmodifiedusingtheHammingweightingand

k-medoidclustering,tobetteraccommodatecategoricalvariables. DeterministicInfeasibilitySorting (DIS), a methodforincluding infeasiblesolutionsinthepopulation,hasbeenproposed,andalso usedtomitigatesurrogateapproximationerrors.Itwasfoundthat theoptimalconfigurationforthisproblemwasNSGA-II-Sd:

NSGA-II,withthesurrogate,andDIS appliedatthefilteringstep.This reliablyfoundbettersetsofsolutionsthanNSGA-II,morequickly, fortwoofthebuildingprobleminstances.Noneofthealgorithms showedmuchdifferenceinperformanceforthethird(easier) prob-leminstance.Thepaperhasalsoexploredinmoredetailtheeffect onoptimisationoftheconstrainthandlingapproachandthe Ham-mingweighting,andexploredthelandscapeofthefitnessfunctions toaddcontexttotheoptimisationresults.

Thereremainsomeopenareasforfutureworktoexpandon thatdescribed here.Aside fromtheobviousroutesofexploring moreapplicationsoralternativealgorithmstoNSGA-II,onearea whichwillbemoredifficulttoconsiderisadaptingthesurrogate basedalgorithmtocopewithequalityconstraints.Therearevery fewexamplesofthisintheliterature,oneexceptionbeing[58].It wouldbeinterestingtoexplorefurtherthefrequencywithwhich modelbuildingforeachoftheobjectivesandconstraintsrecurs andeitherusethistoswitchtodifferentsurrogatemodeltypesor toweighttheconstraintsatthesurrogatefilteringstagetohelp mitigateforapproximationerror.

Acknowledgement

ResearchfundedundertheUKEPSRCgrantTS/H002782/1.

AppendixA. Decisionvariables

Table14

Summaryofvariables;IDnumbersareusedinsubsequentreferencestoindividualvariablesinthepaper.

ID Name Notes Range&stepsize

Continuousvariables

1 HVACheatingset-point Appliestoallzones 18–21,step0.5

2 HVACcoolingset-point 22–29,step0.5

3 MechVentMinSPFrac Controlsthebandoftemperatureswheremechanicalratherthannaturalventilationisused; appliestoallzones

0–1.0,step0.1

4 MechVentMaxSPFrac 0–1.0,step0.1

5 NatVentDelta Minimumdifferencebetweenindoorandoutdoortemperatureforventilationtobeswitchedon 0–10,step0.5 6 Orientation Buildingissymmetrical,sorotationof90degreescoversallorientations 0–90,step5 7 Northupperheightratio Eachperimeterzonehastwohorizontalwindows,tosimulateasinglewindowwhichcan

openattopandbottomtoproducea“stackeffect”todrivenaturalventilation.Thesecontrol theheightproportionaltothefloor-to-ceilingheight,andtheglazedareaasaproportionof thewall,foreachwindow.

0.4–0.7,step0.05

8 Northlowerheightratio 0.5–1.0,step0.05

9 Northupperarearatio 0.1–0.9,step0.05

10 Northlowerarearatio 0.1–0.9,step0.05

11 Southupperheightratio 0.4–0.7,step0.05

12 Southlowerheightratio 0.5–1.0,step0.05

13 Southupperarearatio 0.1–0.9,step0.05

14 Southlowerarearatio 0.1–0.9,step0.05

15 Eastupperheightratio 0.4–0.7,step0.05

16 Eastlowerheightratio 0.5–1.0,step0.05

17 Eastupperarearatio 0.1–0.9,step0.05

18 Eastlowerarearatio 0.1–0.9,step0.05

19 Westupperheightratio 0.4–0.7,step0.05

20 Westlowerheightratio 0.5–1.0,step0.05

21 Westupperarearatio 0.1–0.9,step0.05

22 Westlowerarearatio 0.1–0.9,step0.05

23 Northopenarea Theproportionofeachwindowwhichcanopenforventilation 0.1–2.3,step0.05

24 Southopenarea 0.1–2.3,step0.05

25 Eastopenarea 0.1–2.3,step0.05

26 Westopenarea 0.1–2.3,step0.05

27 Northmechventrate Mechanicalventilationrateinl/s/person 0.06-0.24,step0.02

28 Southmechventrate 0.06–0.24,step0.02

29 Eastmechventrate 0.06–0.2,step0.02

30 Westmechventrate 0.06–0.2,step0.02

Integervariables

31 Wintermorningstart OperationtimesforHVACsystemsinperimeterzones(24hclock) 1–8,step1

32 Wintereveningstop 17–23,step1

33 Summermorningstart 1–8,step1

34 Summereveningstop 17–23,step1

35 Wintermorningstart OperationtimesforHVACsystemsininternalzone(24hclock) 1–8,step1

36 Wintereveningstop 17–23,step1

37 Summermorningstart 1–8,step1

38 Summereveningstop 17–23,step1

Categoricalandbinaryvariables

39 Externalwalltype From[59] Heavy,medium,lightweighta

40 Internalwalltype From[59] Heavy,lightweight

41 Floorandceilingtype From[59] Heavy,medium,lightweight

42 Northupperoverhang Presenceoromissionofshadingoverhangsabovethetworowsofglazingineachzone Present,notpresent

43 Northloweroverhang Present,notpresent

44 Southupperoverhang Present,notpresent

45 Southloweroverhang Present,notpresent

46 Eastupperoverhang Present,notpresent

47 Eastloweroverhang Present,notpresent

48 Westupperoverhang Present,notpresent

49 Westloweroverhang Present,notpresent

50 Glazingtype Typesfrom[59];appliestoallzones Standard,low-emissivity

a_{Theweightreferstotheeffectthatthematerialshaveonthethermalresponseofthebuilding,heavybeingslowandlightbeingfast.}

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