Optimization of Construction Duration and Schedule Robustness Based on Hybrid GreyWolf Optimizer with Sine Cosine Algorithm

2020·4.4 MB·

by  Mengqi Zhao

energies
Article
Optimization of Construction Duration and Schedule
Robustness Based on Hybrid Grey Wolf Optimizer
with Sine Cosine Algorithm
MengqiZhao1,XiaolingWang1,*,JiaYu1,LeiBi2,YaoXiao1andJunZhang1
1 StateKeyLaboratoryofHydraulicEngineeringSimulationandSafety,TianjinUniversity,
Tianjin300350,China;[email protected](M.Z.);[email protected](J.Y.);
[email protected](Y.X.);[email protected](J.Z.)
2 CCCCWaterTransportationConsultantsCo.,Ltd.,Beijing100007,China;[email protected]
* Correspondence: [email protected];Tel.:[phone]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received: 6December2019;Accepted: 30December2019;Published: 2January2020
Abstract: Constructiondurationandschedulerobustnessareofgreatimportancetoensureefficient
construction. However,thecurrentliteraturehasneglectedtheimportanceofschedulerobustness.
Relativelylittleattentionhasbeenpaidtoschedulerobustnessviadeviationofanactivity’sstarting
time, whichdoesnotconsiderschedulerobustnessviastructuraldeviationcausedbythelogical
relationshipsamongactivities. Thisleadstoapossibilityofdeviationbetweentheplannedschedule
andtheactualsituation.Thus,anoptimizationmodelofconstructiondurationandschedulerobustness
isproposedtosolvethisproblem. Firstly,durationandtworobustnesscriteriaincludingstartingtime
deviationandstructuraldeviationwereselectedastheoptimizationobjectives. Secondly,criticalchain
methodandstartingtimecriticality(STC)methodwereadoptedtoallocatebufferstotheschedulein
ordertogeneratealternativeschedulesforoptimization.Thirdly,hybridgreywolfoptimizerwith
sinecosinealgorithm(HGWOSCA)wasproposedtosolvetheoptimizationmodel.Themovement
directionsandspeedofgreywolfoptimizer(GWO)wasimprovedbysinecosinealgorithm(SCA)
sothatthealgorithm’sperformanceofconvergence,diversity,accuracy,anddistributionimproved.
Finally,anundergroundpowerstationinChinawasusedforacasestudy,bywhichtheapplicability
andadvantagesoftheproposedmodelwereproved.
Keywords: optimizationmodel;constructionduration;schedulerobustness;criticalchainmethod;
STCmethod;hybridgreywolfoptimizerwithsinecosinealgorithm
1. Introduction
Effective schedule management is becoming increasingly significant because of its social and
economicbenefits. Optimizationisanimportantissueinschedulemanagement. Bothconstruction
duration and schedule robustness, the latter of which is the schedule’s ability to resist duration
increases due to interference, are necessary to be taken as the optimization objectives. However,
schedulerobustnessisrarelyconsideredasanoptimizationobjectiveincurrentresearch,whichcauses
deviationsbetweentheactualscheduleandtheplannedone. Therefore,itisnecessarytoconduct
optimizationofconstructiondurationandschedulerobustness.
Thebasicsofoptimizationistodetermineobjectives.Besidestheobjectiveofconstructionduration,
it is also necessary to determine appropriate robustness criteria as optimization objectives. A few
generalrobustnesscriteriahavebeendefinedtomeasureschedulerobustness,includingthesummation
ofeachactivity’sfreefloat[1], ratioofeachactivity’sfreefloatandduration[2,3], linearweighted
summation of the total free float [4], and linear weighted summation of the time deviations [5].
For complex construction projects, the schedule is established by breaking down the construction
Energies2020,13,215;doi:10.3390/en13010215 www.mdpi.com/journal/energies
Energies2020,13,215 2of17
projectintobasicunits(whicharecalledactivities)anddeterminingthestartingtimeofactivitiesand
logicalrelationshipsamongthem. Startingtimeofactivitiesandlogicalrelationshipsamongthemare
twoindispensableparts,thustheschedulerobustnessshouldbemeasuredfromthesetwoaspects.
But,intheveryfewmulti-objectiveoptimizationstudiesconsideringbothconstructiondurationand
schedulerobustness,onlythestartingtimedeviationistakenasanobjective[6]. Theneglectofschedule
robustnessfromtheperspectiveoflogicalrelationshipsamongactivities—thatis, whenstructural
deviationisn’ttakenasanobjective—willcausetheconstructionsitetobechaotic. Therefore,itis
practicalandnecessarytotakeconstructionduration,startingtimedeviation,andstructuraldeviation
astheoptimizationobjectives.
The method to improve schedule robustness is a necessary condition to generate alternative
schedulesforoptimization. Bufferallocationmethodsareproveneffectivemethodstoimproveschedule
robustness. Thesemethodsaremainlybasedontwoideas: Allocatingbufferscentrallyattheendof
theschedulechainandallocatingbuffersscatteredlytotheschedule.Thecriticalchainmethod[7]
isthemainmethodtoallocatebufferscentrally[8–10]anditsadvantageshavebeenverified[11–13].
The methods of allocating buffers scatteredly mainly include virtual activity duration extension
(VADE)[14],resourceflowdependentfloatfactor(RFDFF)[15]andstartingtimecriticality(STC)[16],
whicharealsowidelystudiedandapplied. Theseaforementionedstudiesindicatethatallocating
buffers centrally improves the robustness of the whole project and allocating buffers scatteredly
improvestherobustnessofactivitieswithintheproject. Researcherstendtoallocatebufferscentrallyor
scatteredlyaccordingtotheresearchemphasis[17],ratherthanallocatingbothkindsofbuffersatthe
sametime. However,forcomplexconstructionprojects,theorderlyconstructionofeachactivityand
thetimelycompletionofthewholeprojectareequallyimportant. Therefore,itispreferabletoallocate
bothkindsofbufferstothescheduleinproperproportion,sothattheschedulesforoptimizationof
constructiondurationandschedulerobustnesscanbegenerated.
Metaheuristic algorithm is an effective tool for solving optimization problems. Several
metaheuristic algorithms have been developed, such as particle swarm optimization (PSO) [18],
evolutionaryalgorithm(EA)[19],multi-verseoptimization(MVO)[20],geneticalgorithm(GA)[21],ant
colonyoptimization(ACO)[22],andwhaleoptimizationalgorithm(WOA)[23]. GWOisaveryefficient
metaheuristicalgorithmduetoitssimplemathematicalmodel[24]. InGWO,thesearchdirectionis
determinedbythebestthreewolvesandfollowedbyothercandidatewolves. Ithasbeensuccessfully
appliedforsolvingoptimizationproblemsinthefieldsofflowshopscheduling[25–27],computer
science[28–30],mathematics[31–33],waterresources[34–36],energy[37–39],andsoon. Butthiskindof
exploitationmechanismmakesitpronetostagnationinalocaloptimuminmulti-objectiveoptimization.
Thesinecosinealgorithm(SCA)[40],whichhasbeenwidelyappliedinthefieldsofengineering[41–43],
computer science [44–46], control system [47,48], energy [49–51], and instrument [52,53], requires
the solutions to fluctuate towards or outwards of the true Pareto optimal front based on the sine
cosinefunction. SCAissuperiorinbalancingexplorationandexploitation,whichmakesitableto
approximatethetrueParetooptimalfront. Therefore,SCAisusefultoimprovethesearchingcapability
ofmulti-objectiveGWOsothattheglobaloptimalsolutionsoftheoptimizationmodelofconstruction
durationandschedulerobustnesscanbeobtained.
Insummary,thecurrentstudieshavepaidlittleattentiontoschedulerobustnessviastartingtime
deviation, whichneglectedtheschedulerobustnessviastructuraldeviation. Additionally, amore
efficientalgorithmisworthbeingdevelopedtosolvetheoptimizationmodel. Therefore,theaimofthis
researchistoproposeanoveloptimizationmodelofconstructiondurationandschedulerobustness
basedonahybridgreywolfoptimizerwiththesinecosinealgorithm(HGWOSCA),whichcanbe
dividedintothreedetailedpoints:
1. Constructionduration,startingtimedeviation,andstructuraldeviationaretakenastheobjectives
oftheoptimizationmodel.
2. Buffersarecentrallyallocatedtothescheduleusingthecriticalchainmethodandscatteredly
usingtheSTCmethodsoastogeneratealternativeschedulesforoptimization.
Energies2020,13,215 3of17
3. EnerTghiese 2h02y0b, 1r3id, xg FrOeRy PwEEoRlf RoEpVtIEimWi zerwithsinecosinealgorithm(HGWOSCA)isproposedtoso3 lovf e18t he
optimizationmodel.
3. The hybrid grey wolf optimizer with sine cosine algorithm (HGWOSCA) is proposed to solve
Thteher eomptaiminidzeartioonf tmhiosdpela. perisorganizedasfollows: Theframeworkofthispaperisshownin
Section2. Section3describesthedetailsoftheproposedmethodology. Section4presentsacasestudy
The remainder of this paper is organized as follows: The framework of this paper is shown in
anSdercetisounl t2s.. SSeecctitoionn 3 5depsrcorvibidese tshteh deedtaisilcsu osfs tihoen ptoroipmopseldy mtheethsuodpoelroiogryi.t Syeocftitohne 4p prroepseonstesd ao cpastiem stiuzdatyi on
moadnedl .reSseuclttiso. nSe6chtiiognh 5li gphrotsvitdheesc tohnec dluissciuosnssioonf ttoh iismppalpy etrh.e superiority of the proposed optimization
model. Section 6 highlights the conclusions of this paper.
2. ResearchFramework
2. Research Framework
In this research, an optimization model of construction duration and schedule robustness is
proposIend tshois trheasteaarcshc,h aend uolpetiwmiitzhatiboont hmsohdoerl tocfo cnosntsrturcutcitoionnd duurraattiioonn aanndd gscohoeddurloeb ruosbtunsetsnsescsa nis be
obptarionpeods.eTdh esor etsheaatr cah sfcrhaemdeuwleo wrkitihs sbhootwh nshionrFt icgounrsetr1u.cSttiaornt indgurtaimtioend eavnida tgioonodrs ,rostbruuscttnuersasl dcaenv iabtei on
rp,oabntadinceodn. sTtrhuec trieosneadruchra tfrioamneTwworekr eist hsehoowptni miniz aFtiigounreo b1j.e cSttiavretisn.gC rtiimticea ldcehviaaitniomn ertsh, osdtruacntudraSlT C
medtehvoidatiwone rrep, aadnodp ctoendsttroucatliloonc adtuerpatrioojne cTt wbuerffee trh(eP oBp)t,imfeiezdaitniognb oubffjeecrtiv(FeBs.) C,arintidcaslc cahtateinre mdebthuoffde rasn(dS B)
toSthTeCs mcheethdoudle wsoeraes atdoogpetende rtaot ealtlhoecastceh perdoujelcets bfuofrfeorp (tPimB)i,z faeteiodnin(gs ebeuAffeprp (eFnBd),i xanAd) .scHayttberreidd gbruefyfewrso lf
op(tSimB)i ztoe rthwe istchhesidnuelec osos ianse toa lggeonreitrhatme t(hHe GscWheOduSCleAs f)o,rw ohpetirmeitzhaetiopno s(isteieo nApuppednadtiixn gA)w. HayyborfidW gGreOy is
imwproolfv oepdtibmyizSeCr Aw,itwh assinpe rcoopsoinsee dalgtoorsitohlmve (tHhGeWopOtiSmCAiz)a, twiohnerme othdee lp.oTsihtieonn, uapnduatnindge rwgraoyu onf dWpGoOw er
is improved by SCA, was proposed to solve the optimization model. Then, an underground power
station in China was used as a case study and the applicability and advantages of the proposed
station in China was used as a case study and the applicability and advantages of the proposed
optimizationmodelarediscussedandverified.
optimization model are discussed and verified.
Optimization model Case study
ssentsubo Optrs,rp,T Trrsp==iii=n=nc00ritwwicaiil paStPhiiT--spiiiSubject to:mmaarxxssRii11++TTii--11SBSi,iFB1Si,i1 • ON12345po.timizPedB11122 /sd99900cahyedulesFB2/22222d2222ay SB1312/6d4023ay r77777s/56555d.....11020a23587y 00000.....66666r00010p2722272767 T22222/33433d99099a83169y
r eludehcs Acti7vGityeneraStBion of al…te…rnativeS Bschedul…es… for opAtcitminviityzationPB • Gantt chart of optimTi3z1eT6d-3in1 S6ti-acolhp=tei md4iu8ze ldde a=Ny 5os0. 1d ays IOnpittiiaml iszcehde dscuhleedule
dn T317-intial= 48 days
a noita Actii+vjity FB T318-intial= 66 days T317-optimized=49 days
rud n CriticcIardilte icnchatailf iycnh tmahiene thod DeatellromcainteedS t Th teoC l tomhceea tcthiroiotnidcsa ol fc bhuafinfers •MODPisScOu,ssMioOnGWOT,3M18-OopEtimAizaedn=d 6M8 OdMayVsOareselectedtocomparewith
o HGWOSCA.HGWOSCAapproachesclosertothePF,whichimpliesthat
itcu Hybrid grey wolf optimizer with sine cosine algorithm (HGWOSCA) tahlgeorHithGmWsO.SCA can obtain the bet8te4r solutions than other four
rtsnoc fo noitazimitpO SSCeAcoimnAdCpβ rfβoivtteesstht espoolusittPiioornenyu βpRdatingCwaDanyβdoifdαate solFuitDtitoαens tf soDorδ lnuetixAotα niCt UαeαAδTUrphapdtiACirdaodδδtane ::tf eiOIpmt ptoreiposgsrisiotnti ivsatoielo ondllon u lcoo atiucitnaiototssinnoii dndδee rrpp00000000000000..............345678934567898752r8s7(0d0)7754 HMMMM7MMMMH7G0rOOOOGOOOO6sWWPEMGPGME2(SAWOVSdAOOW3VOSOOC5)SOOA7C08A240800MMMMHT2GOOOO4(W8MPGEd5SA2OWV)0OSOO2CA58040 r(prds)0000000.......77777883456789024680222335500 HMMMMMMMMHGOOO2O2GOOOOWPGM4WEPMEG4SASWOAV0OWO0VOOSO0CSOO0ACTAT((dd))224455HMMMM00GMMMOOMOOHGWOOOOPGMEWPMEGSASAOWWOVVOOOSOSOOCCAA22550000
Figure1. Researchframework.
Figure 1. Research framework.
3. Methodology
3. Methodology
This section presents the description of the optimization model of construction duration and
This section presents the description of the optimization model of construction duration and
schedulerobustness. Inaddition,themethodologiesofrobustnesscriteria,generationofalternative
schedule robustness. In addition, the methodologies of robustness criteria, generation of alternative
schedulesforoptimization,andtheproposedHGWOSCAareintroduced.
schedules for optimization, and the proposed HGWOSCA are introduced.
Energies2020,13,215 4of17
3.1. ProblemFormulation
Theoptimizationmodelofconstructiondurationandschedulerobustnessisexpressedasfollows:

 rTOrpsp===t(rii(cid:80)=(cid:80)s=nn,00rwpw(cid:80),iiT((|(cid:12)(cid:12)(cid:12))SPii−−(spSi|iB)(cid:12)(cid:12)(cid:12))+T(i))+PB (1)
i∈criticalpath
subjectto:
 mm(cid:80)aarxx≤[[ssiiR−−11++TT((ii−−11))]+≤SSBi,+i≥FB1]≤Si,i≥1 (2)
where,threeoptimizationobjectivesareproposed. r istherobustnesscriterionproposedfromthe
s
aspect of deviation of an activity’s starting time, r is the robustness criterion proposed from the
p
aspectoflogicalrelationshipsamongactivities,andTrepresentsthetotalconstructiondurationofthe
schedule. ThedetaileddescriptionsofthesymbolsinthispaperareshowninAppendixA.
3.2. RobustnessCriteria
Robustnesscriteriaarethefoundationfortheoptimizationmodel. Sinceactivitiesandlogical
relationships among activities are two indispensable components of a construction schedule [6],
twocriteria(i.e.,startingtimedeviation(r )andstructuraldeviation(r )),areproposedtomeasure
s p
the schedule robustness from these two aspects. r is the robustness criterion proposed from the
s
aspectofactivities’startingtime,andr istherobustnesscriterionproposedfromtheaspectoflogical
p
relationshipsamongactivities. Thesetwocriteriaprovideamorecomprehensiveunderstandingof
measuringschedulerobustness.
Thestartingtimedeviationr istheabsolutevalueoftheweightedsummationofthedeviations
s
betweenthestartingtimesoftheactualactivitiesandtheplannedactivities.
(cid:88)n
r = w(|S −s|). (3)
s i i i
i=0
Thestructuraldeviationr istheabsolutevalueoftheweightedsummationofthedeviations
p
betweentheordersoftheactualactivityandtheplannedactivity.
(cid:88)n (cid:12) (cid:12)
rp = wi((cid:12)(cid:12)Pi−pi(cid:12)(cid:12)). (4)
i=0
Asseenfromtheaboveequations,thesmallerther andr values,thebettertherobustness.
s p
3.3. GenerationofAlternativeSchedulesforOptimization
Withtherobustnesscriteriaastheoptimizationobjectives,thenextstepwastoimproveschedule
robustnesstogeneratealternativeschedulesforoptimization. Thisincludedtwoparts(asshownin
Figure2): Identifyingthecriticalchainintheschedulebycriticalchainmethodandallocatingbuffers
toitbyusingtheSTCmethod.
Energies2020,13,215 5of17
Energies 2020, 13, x FOR PEER REVIEW 5 of 18
Legend
Activity Activity
3 4
Buffers
Acti1vity Acti2vity SB Acti5vity Acti6vity Acti7vity SB …… SB …… Actinvity PB Activity node
Activity ...... Activity FB Criticalchain
i i+j
Figure2. Generationofalternativescheduleswithbuffersallocatedbasedoncriticalchainmethodand
Figure 2. Generation of alternative schedules with buffers allocated based on critical chain method
thestartingtimecriticality(STC)method.
and the starting time criticality (STC) method.
Thecriticalchainisachaincomposedofactivitiesthatareinterdependentduetotimeconflicts
The critical chain is a chain composed of activities that are interdependent due to time conflicts
andresourceconflicts. Sincetheconstraintswhichexistineachschedulecorrespondtothebottleneck
and resource conflicts. Since the constraints which exist in each schedule correspond to the bottleneck
elementsthatpreventtheprojectfrombeingcompletedonschedule[7],itiseffectivetoidentifythe
elements that prevent the project from being completed on schedule [7], it is effective to identify the
concoflnicftliicntginagc aticvtiivtiietsiesso soa sasto toim impprorovveet hthees scchheedduullee rroobbuussttnneessss..
ThTehied eidnetnifiticfaictaiotinonp rporcoecsessso offa ac crirtiitcicaallc chhaaiinn wwaass aass ffoolllloowwss::
1. 1 CoCmompuptuetet htheei ninitiitaialls scchheedduullee vviiaa ssiimmuulalatitoionn. .
2 Identify the activities that can be postponed on the condition that the schedule fulfils the resource
2. Identifytheactivitiesthatcanbepostponedontheconditionthattheschedulefulfilstheresource
constraints and logical relationships. An activity which cannot be postponed was denoted as
constraints and logical relationships. An activity which cannot be postponed was denoted
NonAi; otherwise, it was denoted as MovAi. A MovAi which does not satisfy the resource
as NonAi; otherwise, it was denoted as MovAi. A MovAi which does not satisfy the resource
constraints was denoted as NonBi; otherwise, it was denoted as MovBi.
constraintswasdenotedasNonBi;otherwise,itwasdenotedasMovBi.
3 The critical chain was identified by combining NonAi and NonBi. If there was more than one critical
3. ThecriticalchainwasidentifiedbycombiningNonAiandNonBi. Iftherewasmorethanone
chain, the chain with the most activities was selected as the final critical chain, and the others were
crtirtiecaateldc haas inno,nth-cerictihcaali nchwaiinths. themostactivitieswasselectedasthefinalcriticalchain,andthe
othersweretreatedasnon-criticalchains.
After the critical chain was identified, three types of buffers, including project buffer (PB),
scaAttfetreerdt hbeucffreitri c(aSlBc)h, aanind wfeaesdiidnegn btiufifefedr, (tFhBre),e wtyepree salolfobcautffeder tso, iint.c PluBd winags parlloojceacttebdu affte trh(eP eBn),ds ocaf tttheer ed
bucffreirti(cSaBl c)h,aanind tofe eendsiunrgeb thueff perro(FjeBc)t, wwaesr ceoamllpolceatteedd otno tiitm. PeB. SwBsa wsaelrleo caalltoecdataetdt hine fernodnto offt hhiegchrliyt iccrailticchala in
activities to ensure that these activities started on schedule. FBs were allocated where non-critical
toensuretheprojectwascompletedontime. SBswereallocatedinfrontofhighlycriticalactivitiesto
chains feed into the critical chain in order to prevent a delay of the activity on the critical chain when
ensurethattheseactivitiesstartedonschedule. FBswereallocatedwherenon-criticalchainsfeedinto
the activities on the non-critical chain were delayed.
thecriticalchaininordertopreventadelayoftheactivityonthecriticalchainwhentheactivitieson
The sizes of FB and the sum of PB and SB were determined using the elasticity coefficient method
thenon-criticalchainweredelayed.
which was improved by the Dezert–Smarandache Theory (DSmT) [54,55].
ThesizesofFBandthesumofPBandSBweredeterminedusingtheelasticitycoefficientmethod
In the elasticity coefficient method, the (a, b, c) in the three-point estimation method is replaced
whichwasimprovedbytheDezert–SmarandacheTheory(DSmT)[54,55].
K ca/ ba
byI n(at, hbe) ealnadst icc iitsy ccoonesffiidceierendt mine tthhoed e,ltahsetic(ait,yb ,cco)eifnficthieentt:h ree-pointestimationme. tIhno dthiiss rwepalya, ctehde by
(a,ibm)paancdtsc oifs icnotenrsfiedreenreced fiancttohres eolna sstcihceitdyucleo ecaffinc bieen rte:flKec=ted( cin− bau)f/fe(rbs−. Tah).e Icnlotsheirs cw apayp,rothaechiems pa,a tchtse of
intleersfse rlieknecley fiat cisto tros doenlasyc htheed uacletivcaitny baendre tflheec stmedalilnerb tuhffee Krs v.aTlhuee.c Tlohsee crlcosaeprp cr oapacphroesacah,etsh eb,l tehses lmikoerley it
istloikdeleyl aiyt itsh teo adcetliavyit ythaen adcttihvietys manadll ethret hgereKatvear ltuhee. KT hvealculeo.s Terhecraepfoprreo, athche ebsubff,etrhse cmano bree lciaklecluylaitteids to
deltahyrotuhgeha Kct.i vityandthegreatertheKvalue. Therefore,thebufferscanbecalculatedthroughK.
FB(cid:88)b(cid:110)cacKc, (cid:111)
FB= (bc−ac)B×Kc , (5)( 5)
cB B
c∈B
where B denotes the set of activities related to the buffers.
whereBAddednitoiotensaltlhye, isne tthoef ealcatsivtiictiiteys croeelafftiecdientot mtheethboudff,e ar si.s the activity duration without considering
theA dimdpitaiocntsa lolyf ,iinnttehrfeeerelanscteic iftayctcooresffi acniden tism ceatlhcoudla,taedis bthye saicmtiuvliatytiodnu. raTthioen awctiitvhiotyu tdcuornastiidoenrsi nagret he
imrpeapcrtessoenftiendte rbfye rbe nacnedf acc taonrds acnodnsiisdcearl cvualraytiendg bdyesgirmeeusl aotfi oinm.pTahcetsa octfi vinittyerdfeurreantcioe nfsacatroersr eapnrdes eanret ed
bycbalacnudlacteadn ads:c onsidervaryingdegreesofimpactsofinterferencefactorsandarecalculatedas:
bb=aa×1(+1m+inmimnj(m1(,θm)j,m2(,θmj),m3(θ; ))); (6)( 6)
j 1 j 2 j 3
cc=aa×1(+1n3+1m(cid:88)j3mhj(,θ h)), (7)( 7)
n=1
Energies2020,13,215 6of17
where m (θ ) is computed using the DSmT method. Delay, downtime, and breakdown happen
j h
during construction as a result of interference factors. Hence, it is reasonable to determine the
sizeofbuffersaccordingtotheimpactofinterferenceonschedule.DSmTrecognizesthefusionof
occurrence probabilities of interference factors. It reasonably solves the problem of large conflicts
amonginterferencefactorsandachievestheunificationofoccurrenceprobabilitiesandimpactsof
interferencefactorsonschedule.
Interferencefactorj(j=1–6)representsthegeologicalfactor,technicalfactor,environmental
factor,managementfactor,materialfactor,andthemechanicalfactor,respectively. Theconstruction
parameterk(k=1–11)representssurveyingsettingout,chargeexplosion,safedisposal,explosionslag
cleaning,ventilation,disposalofunfavorablegeology,otherdowntime,handdrill,threearmtrolley,
dumptruckandloader,andtheconsequenceofinterferencefactors,respectively. Factorh(h=1–3),
representsdelay,downtime,andbreakdown,respectively. Factorm (θ )isdefinedas:
j h

 mmmjj(((θθθ12))) ===mmmjj(((γγγ17)))⊕⊕⊕....m..⊕⊕(mmγjj(()γγ69)) ; (8)
j 3 j 10 j 11
(cid:88)
∀A(cid:44)∅∈DU, [m +m ](A) = m (θ )m (θ ). (9)
1 2 1 1 2 2
θ1,θ2∈DU,θ1∩θ2=A
Then,theSTCmethodandgoldensectionrulewereusedtodeterminethelocationsofscattered
buffers. Theprocessisasfollows:
1. TheSTCmethodwasadoptedtomeasurethenecessityofallocatingthebufferbeforeanactivity.
1(cid:88)3
stc(i) = m (θ )×w, (10)
j h i
3
n=1
where 1 (cid:80)3 m (θ )denotesthepossibilitythattheactivityidoesnotstartonschedule.
3 j h
n=1
2. Basedonthegoldensectionrule,38.2%ofthebuffersthatneedtobeallocatedoncriticalchain
wereallocatedattheendofthecriticalchainastheprojectbuffer. Basedonthissamerule,61.8%
ofthebuffersthatneedtobeallocatedoncriticalchainwereallocatedinfrontofactivitiesasthe
scatteredbuffers.
(cid:88)(cid:110) (cid:111)
PB=0.382 (bc−ac)×Kc . (11)
B
c∈B
(cid:88)(cid:110) (cid:111)
SB=0.618 (bc−ac)×Kc . (12)
B
c∈B
3. Sorttheactivitiesoncriticalchainfromhightolowinorderofcriticality. Addtheirbuffersone
byone.
4. Judgewhetherthesumhasreached61.8%ofthebuffers. Itnot,gobacktostep3.
5. Ifthesumhasreached61.8%ofthebuffers,theaddedactivitiesaretheactivitiesthatscattered
buffersareallocatedbefore.
3.4. OptimizationofConstructionDurationandScheduleRobustnessBasedonHGWOSCA
Inthissection,ahybridmulti-objectivegreywolfoptimizerwithsinecosinealgorithmisapplied
tosolvetheoptimizationmodel. ThemovementdirectionsandspeedofGWOwasimprovedbySCA.
ThenthevalidityandsuperiorityoftheHGWOSCAwasdemonstratedbycomparisonwithfourother
algorithms. ThesolutionflowofHGWOSCAforT-r -r optimizationmodelispresented.
s p
Energies2020,13,215 7of17
3.4.1. TheoryofHGWOSCA
The HGWOSCA contains three phases: Encircling, hunting behavior improved by SCA,
andmodifiedsearchingandattackingbehavior. Fourtypesofgreywolvesnamedalpha(α),beta(β),
delta(δ),andomega(ω)wereemployedformathematicallymodelingtheirsocialhierarchy. Thefirst
threefittestsolutionsareα,β,andδ,respectively. Therestofthecandidatesolutionsareω. InGWO,
theoptimizationisguidedbyα,β,andδ,andtheωwolvesfollowthesethree.
Theencirclingbehaviorissimulatedbythefollowingequations:
→ (cid:12)(cid:12)→ → → (cid:12)(cid:12)
D=(cid:12)(cid:12)(cid:12)C·Xp(t)−X(t)(cid:12)(cid:12)(cid:12), (13)
→ → →→
X(t+1) = X (t)−A·D, (14)
p
→ →
wheretrepresentsthecurrentiteration, X isthepositionvectoroftheprey, and X representsthe
p
→ →
positionvectorofagreywolf,Aand C arecoefficientvectors,whicharecalculatedasfollows:
→ → → →
A =2a ·r − a, (15)
1
→ →
C =2·r , (16)
2
wherer ,r arerandomvectorsin[0,1].
1 2
Thehuntingbehaviorissimulatedbythefollowingequations:
 → (cid:12)(cid:12)→ → →(cid:12)(cid:12)
 DDD→→αβδ ===(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)CCC→→123···XXX→→αβδ−−−→→XXX(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)
 → → → →
 XXX→→123===XXX→→αβδ−−−AAA→→231···(((DDD→→αβδ))) , (18)
→ → →
→ X +X +X
X(t+1) = 1 2 3. (19)
3
GWOwasoriginallydevelopedforsolvingcontinuousoptimizationproblemsandcannotbe
usedtosolvethecombinatorialoptimizationproblemsinthispaper. Thus,thesearchoperatorwas
modifiedasfollows:
 → → →
→X(t+1) = ssshhhiiifffttt(((→→XXX(((ttt))),,,→→XXXαβδ−−−→→XXX(((ttt)))))),,, oiiftfhr13er<≤wi13rse< 23 . (20)
Iftheunitswerebeyondtheboundaryrange,theboundaryelementwasconnectedtoitinthe
otherdirection.
The position of α is determined by the position updating equation of SCA [40], by which the
position,speed,andconvergenceaccuracyofthegreywolfagentareimprovedandtheconvergence
performanceofGWOisextended.
Xt+1 =(cid:40) Xlt+r1×sin(r2)×(cid:12)(cid:12)(cid:12)(cid:12)r3Ptl−Xlt(cid:12)(cid:12)(cid:12)(cid:12), r4 <0.5 , (21)
l Xlt+r1×cos(r2)×(cid:12)(cid:12)r3Ptl−Xlt(cid:12)(cid:12), r4 ≥0.5
Energies 2020, 13, x FOR PEER REVIEW 8 of 18
The position of α is determined by the position updating equation of SCA [40], by which the
position, speed, and convergence accuracy of the grey wolf agent are improved and the convergence
performance of GWO is extended.
Xt+rsinr rPt Xt , r 0.5
Energies2020,13,215 Xt1  l 1 2 3 l l 4 , (281)o f17
l Xt+rcosr rPt Xt , r 0.5
 l 1 2 3 l l 4
where, r , r and r are random numbers, and r is a random number used for exploitation and
where, 1r1, 2r,2, and r33 are random numbers, and r4 4is a random number used for exploitation and
exploration.
exploration.
3.4.2. PerformanceoftheProposedHGWOSCA
3.4.2. Performance of the Proposed HGWOSCA
TheefficiencyoftheproposedHGWOSCAwastestedviathepatterson_pat1andpatterson_pat20
The efficiency of the proposed HGWOSCA was tested via the patterson_pat1 and
inpathtteerwsoenl_l-pkant2o0w inn thPeS wPLelIlB-kndoawtans PeStP[L5I6B] ,daatsa ssehto [w56n], ains sFhiogwunre in3 F.igMuurel t3i-. oMbujelctit-iovbejepctaivrtei cplaertsiwclea rm
opswtimarimza toiopntim(MizOatPioSnO )(,MmOuPltSiO-o)b, jemctuivltei-ogbrejeyctwivoel fgorpeyti mwiozelfr o(MptOimGizWerO ()M,mOuGltWi-oOb)j,e cmtiuvletie-ovbojleucttiiovne ary
alegvoorliuthtimona(rMy OaElgAor)i,thamnd (MmOuEltAi-o),b ajencdti vmeumltiu-oltbij-evcetirvsee moputlitmi-vizerasteio onpt(iMmOizMatiVonO )(MwOeMreVsOel)e cwteedre for
cosmelepcatreids ofonr wcoimthpathriesoHn GwWithO thSeC HAG. WThOeStCeAst.s Twhee treestism wpelreem imenptleedmeonntead coonm a pcoumtepruwteirt hwiItnht IenlteCl ore
i5C,2o.r3e9 i5G, H2.z3,91 G3H1,0z,7 123M1,0B7R2 AMMB ,RwAiMth, awWithin ad Wowinsd1o0wos p1e0r oaptienrgatsiynsgt esyms.tem.
1:6 8:4 1:3 4:7 10:4 14:5 17:8 19:4
4:1 9:3
5:7 11:2 15:1
0:0 2:4 5:6 11:3 12:5 13:0 0:0 2:2 16:1 21:0
7:2 8:1 12:2
6:2 10:2 Legend
Number:Duration
3:3 7:1 3:9 6:3 9:2 13:3 18:6 20:2
(a) patterson_pat1 (b) patterson_pat20
FFiigguurree 33.. NNeettwwoorrkk ooff ppaatttetersrosonn__ppata1t 1anadn dpaptatettresorsno_np_apt2a0t2 [056[]5. 6].
FoFuourrm meetrtircicssi nincclluuddiinngg ggeenneerraattiioonnaall ddiissttaannccee (G(GDD),) i,nivnevresres geegneenreartiaotnioanl adlisdtaisntcaen (cIeG(DIG), Dra)t,iroa otifo of
nonno-nd-odmomininaateteddi ninddiviviidduuaall ((RRNNII)),, aanndd SSpprreeaadd wweerree eemmpploloyyeedd. .
(1) GD represents the distance between the Pareto front and the optimal Pareto front and is
(1) GD represents the distance between the Pareto front and the optimal Pareto front and is
formulated as:
formulatedas:
(cid:115)
(cid:80)PF |PF|
i1dip2=1 d2p (22)
GGDD= , , (22)
PF |PF|
where PF is the number of solutions in the Pareto front. The lower the GD value, the better the
where PF is the number of solutions in the Pareto front. The lower the GD value, the better the
convergenceperformanceofanalgorithm.
convergence performance of an algorithm.
(2()2I) GIGDDr erepprreesseennttsst thhee ddiissttaanncceess bbeettwweeeenn eeaacchh sosolulutitoionn ini nopotpimtimal aPlaPreatroe tforofnrot nantda nisd foisrmfourmlatuelda ted
asa:s: (cid:115)
(cid:80)
d2
 x∈dO2PF p
IGIGDD= xO|OPFPF|p , (23)( 23)
,
OPF
whereOPFisthenumberofsolutionsinoptimalParetofront. ThelowertheIGDvalue,thebetterthe
cownhveerreg eOnPcFe pise trhfoe rnmuamnbceero offa snolaulgtioornist hinm o.ptimal Pareto front. The lower the IGD value, the better
the (c3o)nRveNrgIernecper peseerfnotrsmtahnecer aotfi oano aflgthoeritnhumm. ber of non-dominated individuals to the size of the
popula(3ti)o RnNanI dreipsrfeosremntus lathteed raasti:o of the number of non-dominated individuals to the size of the
N
population and is formulated as: RNI= non ×100%, (24)
N
where N is the number of non-dominated individuals and N is the number of all individuals.
non
ThehighertheRNIvalue,thebetterthesolutionquality.

(4)Spreadrepresentstheextentofspreadachievedamongthefrontandisformulatedas:
(cid:80) n0 de +(cid:80) |PF| (cid:12)(cid:12)(cid:12)d −d(cid:12)(cid:12)(cid:12)
q=1 q p=1 (cid:12) p (cid:12)
Spread= , (25)
(cid:80) n0 de +|PF|·d
q=1 q
Energies 2020, 13, x FOR PEER REVIEW 9 of 18
Energies 2020, 13, x FOR PEER REVIEW 9 of 18
N
RRNNII NNnnoonn110000%%,, ((2244))
N
wwhheerree NNnnoonn iiss tthhee nnuummbbeerr ooff nnoonn--ddoommiinnaatteedd iinnddiivviidduuaallss aanndd NN iiss tthhee nnuummbbeerr ooff aallll iinnddiivviidduuaallss.. TThhee
higher the RNI value, the better the solution quality.
higher the RNI value, the better the solution quality.
(4) Spread represents the extent of spread achieved among the front and is formulated as:
(4) Spread represents the extent of spread achieved among the front and is formulated as:
n PF
SSpprreeaaddqnq0011nddqeqeppPF11ddppdd ,, ((2255))
Energies2020,13,215 qnq0011ddqeqePPFFdd 9of17
wwhheerree nn00 iiss tthhee nnuummbbeerr ooff oobbjjeeccttiivveess.. TThhee lloowweerr tthhee sspprreeaadd vvaalluuee,, tthhee bbeetttteerr tthhee ddiissttrriibbuuttiioonn ooff
whssoeolrluuettniioonnisss.. t henumberofobjectives. Thelowerthespreadvalue,thebetterthedistributionofsolutions.
0
FiFFgiiuggruuerrseess4 44 aaannnddd 555 sshhsohowwow tthheeth ffoeouufrro mmureettrmriiccesst oroinnc stthhoee n aallggthooerriittahhlmmgoss rmmitehennmttiisoonnmeedde naabtbiooovvneee odovveaerrb 5o500v reruunonsvs..e TTrhh5ee 0 GGDrDu ns.
value, IGD value, and spread value of HGWOSCA were the lowest among the five algorithms, and
ThvealGueD, IGvaDl uvea,luIeG, Dandv aslpuree,ada nvadlusep oref aHdGvWaOluSeCAof wHerGeW thOe SloCwAeswt aemreontgh ethleo fwivees talgaomriotnhmgst,h aendfi ve
the RNI value of HGWOSCA was the highest among the five algorithms. This result indicates the
algthoer itRhNmI sv,aalnude tohfe HRGNWIOvaSlCuAe owfaHs GthWe OhiSgChAestw aamsothneg hthigeh feivste aamlgoonrigthtmhes.fi Tvheisa lrgeosuriltth imndsi.cTatheiss trhees ult
good performance of the proposed HGWOSCA. Thus, the HGWOSCA is a good fit for multi-
indgoicoadte psethrfeorgmooandcpe eorff otrhme apnrcoepoofsethd eHpGroWpoOsSeCdAH. GTWhuOs, StChAe .HTGhWusO,tShCeAH Gis Wa OgSoCodA fiist afogro modulfitit-for
objective optimization.
muobltjie-cotbivjeec otipvteimopiztaimtioinza. tion.
000000......344344505505 1222222.......70250255050050
0.30 1.75
DGDG0000000000000.............00112200112230505050505050 mmooppssoo mmooggwwooAAllggmmoororoiieetthahammmmoommvvoo hhggwwoossccaa DGIDGI00001110000111..............0257025025702505050500505050 mmooppssoo mmooggwwooAAllggomomrroioittehehaamm mmoommvvoo hhggwwoossccaa
(a) GD-patterson_pat1 (b) IGD-patterson_pat1
(a) GD-patterson_pat1 (b) IGD-patterson_pat1
1.10 2.00
1.10 2.00
11..0000 11..8800
00000.....7898900000 d1111....46460000
INRINR000000000.........345634567000000000 aerpSdaerpS00110011........6802680200000000
00..2200 00..4400
0.10 0.20
0.10 0.20
00..0000 mmooppssoo mmooggwwoo mmooeeaa mmoommvvoohhggwwoossccaa 00..0000 mmooppssoo mmooggwwoo mmooeeaa mmoommvvoo hhggwwoossccaa
Algorithm Algorithm
Algorithm Algorithm

(c) RNI-patterson_pat1 (d) Spread-patterson_pat1
(c) RNI-patterson_pat1 (d) Spread-patterson_pat1
Figure 4. Metrics value of patterson_pat1.
FFiigguurree 44.. MMeettrriiccss vvaalluuee ooff ppaatttetersrosonn_p_pata1t.1 .
11..2200 44..0000
11..0055 33..5500
0.90 3.00
0.90 3.00
0.75 2.50
0.75 2.50
DGDG00..6600 DGIDG22..0000
00..4455 I11..5500
00..3300 11..0000
00..1155 00..5500
00..0000 mmooppssoo mmooggwwooAAllmmggoooorreeiiatathhmm mmoommvvoo hhggwwoossccaa 00..0000 mmooppssoo mmooggwwoo AAlmlgmgoooorreeiiatathhmm mmoommvvoo hhggwwoossccaa
Energies 2020, 13, x FOR PEER REVIEW 10 of 18
(a) GD-patterson_pat20 (b) IGD-patterson_pat20
(a) GD-patterson_pat20 (b) IGD-patterson_pat20
1.00
2.25
0.90
2.00
0.80
1.75
0.70
1.50
0.60 d
INR00..4500 aerpS11..0205
0.75
0.30
0.50
0.20
0.25
0.10
0.00 mopso mogwo Almgooreiathm momvo hgwosca 0.00 mopso mogwo Almgooreiathm momvo hgwosca

(c) RNI-patterson_pat20 (d) Spread-patterson_pat20
FFiigguurree 55.. MMeettrriiccss vvaaluluee oof fppaatttetresrosno_np_apta2t02. 0.
3.4.3. Solution Flow of HGWOSCA for T-rs-rp Optimization Model
The proposed HGWOSCA was applied to solve the optimization model of construction duration
T, starting time deviation rs, and structural deviation rp. A basic flowchart of this is shown in Figure
6.
(1) Encode the schedule.
(2) Initialization: To ensure a good diversity of solutions, one schedule was generated using the
method proposed in Section 3.3 and other schedules were generated randomly.
(3) Search for the prey. The search operator of GWO was originally developed to solve continuous
optimization problems [57], which was not suitable to solve the optimization problems of
discrete variables. Thus, in this paper, the search operator was modified as shown in Section
3.4.1. Secondly, the position updating way of α was improved by SCA to improve the searching
capability of GWO as shown in Section 3.4.1.
(4) Compute the three criteria for each schedule and sort the schedules using Pareto relationship.
According to the Pareto dominance nature of multi-objective optimization problems, there is no
unique optimal solution but a set of non-dominated solutions. Thus, the population can be
divided into several ranks according to the Pareto dominance relationship. The solutions at the
first level rank are denoted as α solutions, the solutions at the second level rank are denoted as
β solutions, the solutions at the third level rank are denoted as δ solutions, and the other
solutions are donated as ω solutions.
(5) Evaluate: The newly generated individuals were evaluated based on the fitness values. The best
individuals were saved and combined with the previous population to form the new population.
Then the populations were sorted according to the social hierarchy and the good population was
selected as the new population.
(6) Judgment: Determine whether the algorithm satisfies the termination condition. If yes, output
the best solutions; otherwise, let t = t+1 and go back to step 5.
(7) Output: Output the best solutions which are the schedules with both short construction duration
and good schedule robustness.
Energies2020,13,215 10of17
3.4.3. SolutionFlowofHGWOSCAforT-r -r OptimizationModel
s p
TheproposedHGWOSCAwasappliedtosolvetheoptimizationmodelofconstructionduration
T,stEanrertginiesg 2t0i2m0, e13d, xe vFOiaRt iPoEnERr sR,EaVnIdEWst ructuraldeviationrp. AbasicflowchartofthisisshowninF1i1g ouf r1e8 6.
Start
Encoding and decoding for HGWOSCA
Initialization:
a. One initial schedule is generated usingcritical chain method;
b. Other schedulesare generated randomly.
The search operator is modified to solve
optimizationofdiscrete variables
Searchforthe prey
ThepositionupdatingwayofαisimprovedbySCA
Compute the three criteria for each schedule and sort the schedules using Pareto.
Evaluate newly generated individuals.
Combinepreviouspopulationandnewlygenerated
individualstoformthenewpopulation.
Sort the population according to the social hierarchy
and select the good population as the next population.
Yes
t = t + 1 t<maximumiteration?
No
Output the best solutions

Figure6. Aflowchartofthehybridgreywolfoptimizerwithsinecosinealgorithm(HGWOSCA).
Figure 6. A flowchart of the hybrid grey wolf optimizer with sine cosine algorithm (HGWOSCA).
(1) Encodetheschedule.
4. Case Study
(2) Initialization: Toensureagooddiversityofsolutions, oneschedulewasgeneratedusingthe
meAth oredalp-lriofep ousneddeirngrSoeucntido npo3w.3earn sdtaotitohne risnc hSeoduuthlwesewst eCrehginean ewraatse dserleacntdedo masly .a case study to
demonstrate the rationality of the research in this paper. The layout of the underground power station
(3) Searchfortheprey. ThesearchoperatorofGWOwasoriginallydevelopedtosolvecontinuous
is shown in Figure 7. It is composed of a main workshop, main transformer, last chamber, and several
optimizationproblems[57],whichwasnotsuitabletosolvetheoptimizationproblemsofdiscrete
branch tunnels.
variables. Thus, in this paper, the search operator was modified as shown in Section 3.4.1.
Secondly, the position updating way of α was improved by SCA to improve the searching
capabilityofGWOasshowninSection3.4.1.