Environment-Aware Production Scheduling for Paint Shops in Automobile Manufacturing

32 Pages·2017·1.06 MB·English

by  Rui Zhang

International Journal of
Environmental Research
and Public Health
Article
Environment-Aware Production Scheduling
for Paint Shops in Automobile Manufacturing:
A Multi-Objective Optimization Approach
RuiZhang
SchoolofEconomicsandManagement,XiamenUniversityofTechnology,Xiamen361024,China;
[email protected];Tel.:[phone]
Received:22November2017;Accepted:20December2017;Published:25December2017
Abstract:Thetraditionalwayofschedulingproductionprocessesoftenfocusesonprofit-drivengoals
(suchascycletimeormaterialcost)whiletendingtooverlookthenegativeimpactsofmanufacturing
activitiesontheenvironmentintheformofcarbonemissionsandotherundesirableby-products.
Tobridgethegap,thispaperinvestigatesanenvironment-awareproductionschedulingproblem
that arises from a typical paint shop in the automobile manufacturing industry. In the studied
problem,anobjectivefunctionisdefinedtominimizetheemissionofchemicalpollutantscausedby
thecleaningofpaintingdeviceswhichmustbeperformedeachtimebeforeacolorchangeoccurs.
Meanwhile,minimizationofduedateviolationsinthedownstreamassemblyshopisalsoconsidered
becausethetwoshopsareinterrelatedandconnectedbyalimited-capacitybuffer. First,wehave
developed a mixed-integer programming formulation to describe this bi-objective optimization
problem. Then,tosolveproblemsofpracticalsize,wehaveproposedanovelmulti-objectiveparticle
swarmoptimization(MOPSO)algorithmcharacterizedbyproblem-specificimprovementstrategies.
Abranch-and-boundalgorithmisdesignedforaccuratelyassessingthemostpromisingsolutions.
Finally, extensive computational experiments have shown that the proposed MOPSO is able to
matchthesolutionqualityofanexactsolveronsmallinstancesandoutperformtwostate-of-the-art
multi-objectiveoptimizersinliteratureonlargeinstanceswithupto200cars.
Keywords: greenmanufacturing;automobileindustry;pollutionreduction;sustainableproduction
scheduling;particleswarmoptimization
1. Introduction
Inrecentyears,theChinesegovernmenthasenforcedstrictregulationstodealwithpollutions
inthemanufacturingindustry[1]. Theregulatorypressureurgesrelevantcompaniestopaymore
attention to sustainability aspects of their operational systems with an aim of reducing pollutant
emissions. Thelatestresearchhasrevealedthatproductionschedulingcouldserveasacost-effective
tool for realizing the goal of sustainable manufacturing [2]. For example, Zhang et al. [3] develop
a time-indexed integer programming formulation to identify production schedules that minimize
energyconsumptionunderTOU(time-of-use)tariffs. LiuandHuang[4]investigateabatch-processing
machineschedulingproblemandahybridflowshopschedulingproblemwithcarbonemissioncriteria.
Zhouetal.[5]applyageneticalgorithm(GA)fortheoptimizationofproductionschedulesintextile
dyeingindustrieswithaclearaimofreducingtheconsumptionoffreshwater.
Productionschedulingisasystem-leveldecisionwhichdeterminestheprocessingsequenceof
jobs(orders)ineachproductionunit. Conventionalschedulingresearchhasmostlybeenfocusedon
profit-drivenperformanceindicatorssuchasmakespan(formeasuringproductionefficiency)andtotal
flowtime(formeasuringwork-in-processinventory).Toincorporateenvironmentalconsiderations,itis
possibletointroducesustainability-inspiredobjectivesintotheschedulingmodelssothattheresulting
Int.J.Environ.Res.PublicHealth 2018,15,32;doi:10.3390/ijerph15010032 www.mdpi.com/journal/ijerph
Int.J.Environ.Res.PublicHealth 2018,15,32 2of32
jobprocessingsequencecanachieveasatisfactorytrade-offbetweenthetraditionalperformancegoal
andthepollutionreductiongoal.
This paper reports a new study based on the motivation and methodology stated above.
In particular, we consider the scheduling of a paint shop in automotive manufacturing systems,
where pollutant emissions are mainly caused by the frequent cleaning operations performed on
paintingdevicessuchassprayguns.Thecleaningprocessinevitablyleadstodischargeofunconsumed
paintsanddetergentswhichcontainhazardouschemicals. Therefore,theschedulingfunctionshould
attempttominimizethefrequencyofcleaningbyarrangingcarsthatrequireidenticalorsimilarcolors
tobeprocessedinaconsecutivemanner(becauseadeepcleaningisneededwheneverthepainting
equipmentispreparingtoswitchtoadrasticallydifferentcolorforpainting). However,considering
therequirementonpollutionreductionaloneisnotfeasibleinpractice,duetothefactthatthepaint
shopiscoupledwiththesubsequentassemblyshopviaabuffersystemwithlimitedresequencing
capacity, which means the sequencing decision for the paint shop will have a strong impact on
possibleprocessingsequencesfortheassemblyshop. Inthiscase,thepreferencesoftheassembly
shopmustbeconsideredsimultaneouslyandthusshouldbeintegratedintotheschedulingproblem
forthepaintshop. Thisclearlydefinesabi-objectiveoptimizationproblem,inwhichoneobjective
functionisconcernedwithminimizationofpollutantemissionswhiletheotherobjectivefunction
reflectsthemajorcriterionadoptedbytheassemblyshop(wewillconsiderduedateperformance
inthispaperbecausetheassemblyshopmuststrivetodeliverfinishedproductsontimetothefinal
testingdepartment). Tosolvesuchacomplicatedproductionschedulingproblemwithreasonable
computationaltime,wewillpresentahighlymodifiedmulti-objectiveparticleswarmoptimization
(MOPSO)algorithmwithenhancedsearchabilities.
Theremainderofthispaperisorganizedasfollows. Section2providesabriefliteraturereview
onthepublicationsrelatedwiththeschedulingofautomotivemanufacturingprocesses. Section3
introducestheproductionenvironmentconsideredinourresearch(withafocusonthebuffersystem)
andthenformulatesthestudiedbi-objectiveproductionschedulingproblemusingamixed-integer
programmingmodel. Section4dealswiththesubproblemofschedulingtheassemblyshopunder
agivenscheduleforthepaintshopandtheintermediatebuffer. Section5presentsthemainalgorithm,
i.e., the proposed MOPSO for solving the bi-objective integrated production scheduling problem.
Section 6 gives the computational results together with statistical tests to compare the proposed
algorithmwithanexactsolverandtwohigh-performinggenericmulti-objectiveoptimizerspublished
in recent literature. Finally, Section 7 concludes the paper and discusses potential directions for
futureresearch.
2. LiteratureReview
2.1. TheColor-BatchingProblem
A line of research that is closely related to our study deals with the color-batching problem,
whichconcernstheuseofabuffersystemtoadjustthecarsequenceinheritedfromtheupstreambody
shopsothattheresultingsequencebestsuitstheneedofthepaintshop. Inparticular,theobjectiveis
tominimizethenumberofcolorchanges(orequivalently,maximizingtheaveragesizeofcolorblocks)
intheoutputsequence.
Spieckermannetal.[6]presentaformulationofthecolor-batchingprocessasasequentialordering
problemandproposeabranch-and-bound(B&B)algorithmtofindtheoptimaloutputsequencewith
the minimum number of color changes. Moon et al. [7] conduct a simulation study for designing
andimplementingacolorreschedulingstorage(CRS)inanautomotivefactoryandsuggestsome
simplefillandreleasepoliciesforoperatingtheselectivitybank. HartmannandRunkler[8]present
twoantcolonyoptimization(ACO)algorithmstoenhancesimplerule-basedcolor-batchingmethods.
ThetwoACOalgorithmsareusedforhandlingthetwostagesofonlineresequencing,i.e.,fillingand
releasing,respectively. Sunetal.[9]proposetwoheuristicprocedures(namely,arrayingandshuffling
Int.J.Environ.Res.PublicHealth 2018,15,32 3of32
heuristics) for achieving quick and effective color-batching. The arraying heuristic is applied in
the filling stage, while the shuffling heuristic is used in the releasing stage. Experiments show
thatthetwoproposedheuristicscanworkjointlytoobtaincompetitivecolor-batchingresultswith
very short computational time. Ko et al. [10] investigate the color-batching problem on M-to-1
conveyorsystems,withmotivationsfromtheresequencingproblematamajorKoreanautomotive
manufacturer. The authors first develop a mixed-integer linear programming (MILP) model and
a dynamic programming (DP) algorithm for a special case of the problem (i.e., 2-to-1 conveyor
system),andthenproposetwoefficientgeneticalgorithms(GAs)tofindnear-optimalsolutionsforthe
generalcase.
Inadditiontotheabove-mentionedmethodofusingabuffersystemtoresequencecarsphysically,
anotherwayofimplementingcolor-batchingistoperformavirtualresequencingofcars. Inthiscase,
carpositionsinthesequenceremainunchanged,butthepaintingcolorsarereassignedamongthose
cars which share identical product attributes (and therefore are indistinguishable) at the moment.
Thecolor-batchingproblembasedonthevirtualresequencingstrategyisalsoreferredtoasthepaint
shopproblemforwords(PPW),whichhasbeenshowntobeNP-complete[11]. XuandZhou[12]
presentfourheuristicrulesforsolvingthisproblemandalsoproposeabeamsearch(BS)algorithm
based on the best heuristic rule. Some researchers, for example Amini et al. [13] and Andres and
Hochstättler[14],presentheuristicmethodsforsolvingaspecialcaseofthePPW,i.e.,PPW(2,1)orthe
binaryPPW(involvingonlytwopaintingcolors),whereeachtypeofcarbodyappearstwiceinthe
sequenceandhavetobepaintedwithdifferentcolors. Recently,SunandHan[15]showthatintegrated
physicalandvirtualresequencingcangenerallyobtainnoticeablybettercolor-batchingresultsthan
theconventionalphysicalresequencing.
2.2. TheCarSequencingProblem
Theresearchonschedulingofautomobilemanufacturingprocesseshasmainlyfocusedonthe
car sequencing problem (first described in [16]). The problem concerns the sequencing of cars in
theassemblyshop,wheredifferentoptions(e.g.,sunroof,air-conditioning)aretobeinstalledonthe
carsbythecorrespondingworkstationsdistributedalongtheassemblyline. Topreventoverloadfor
aworkstation,thecarswhichrequirethesameoptionhavetobespacedoutintheprocessingsequence.
Such restrictions are modeled as ratio constraints. Regarding the r-th option, the ratio constraint
n : p indicatesthatinanysubsequenceof p cars,thereshouldbenomorethann carsrequiring
r r r r
thisoption. Theschedulingobjectiveisthereforetofindasequencethatminimizesthenumberof
constraintviolations(NP-hardinthestrongsense[17,18]).
Golleetal.[19]presentagraphrepresentationofthecarsequencingproblemanddevelopsan
exactsolutionapproachbasedoniterativebeamsearch. Usingimprovedlowerbounds,theproposed
approach is shown to be superior to the best known exact solution procedure, and can even be
appliedtoproblemsofreal-worldsize. Inadditiontoexactsolutionmethods,therearealsosome
approachesbuiltonthehybridizationofmeta-heuristicsandmathematicalprogrammingtechniques.
Zinflouetal.[20]proposethreehybridapproachesbasedonageneticalgorithmwhichincorporates
crossover operators using an integer linear programming model for the construction of offspring
solutions. Itisshownthatthehybridapproachesoutperformageneticalgorithmwithlocalsearchand
otheralgorithmsfoundintheliterature. Thiruvadyetal.[21]designahybridalgorithm(alsocalled
amatheuristic)byintegratingLagrangianrelaxation(LR)andantcolonyoptimization(ACO)forthe
car sequencing problem. According to the experiments on various LR heuristics, ACO algorithm,
anddifferenthybridsofthesemethods,itisfoundthatthetwo-phaseLR+ACOmethodwhereACO
uses the LR solutions to produce its pheromone matrix is the best-performing method for up to
300cars.
A very well-known variation of the car sequencing problem is the one proposed by the
French car manufacturer Renault and used as subject of the ROADEF’2005 challenge [22]. The
Renaultproblemdiffersfromthestandardprobleminthat,besidesratioconstraintsimposedbythe
Int.J.Environ.Res.PublicHealth 2018,15,32 4of32
assembly shop, it also introduces paint batching constraints and priority classifications. The aim
istofindacommonprocessingsequenceforboththepaintshopandtheassemblyshopsuchthat
a lexicographically defined objective function is minimized. Due to the large number of cars and
tightcomputationaltimelimit, thealgorithmsthatrankthefirst10placesinthefinalcompetition
allbelongtotheheuristiccategory. Estellonetal.[23]describeafirst-improvementdescentheuristic
usingavarietyofneighborhoodoperatorsrandomly,whichisfurtherenhancedbyastrategytospeed
upneighborhoodevaluationsthroughtheuseofincrementalcalculation. Ribeiroetal.[24]design
asetofheuristicsmostlybasedonvariableneighborhoodsearch(VNS)anditeratedlocalsearch(ILS).
Quickneighborhoodevaluationsandad-hocdatastructuresarealsoakeyfeatureintheirmethod.
Briantetal.[25]presentasimulatedannealing(SA)algorithminwhichtheprobabilitiesofacceptance
are computed dynamically so that the search process tends to favor the moves that have the best
successratesofaramongallthepossiblemoves.Gavranovic´[26]presentsaheuristicbasedonvariable
neighborhoodsearch(VNS)andtabusearch(TS).Theauthoralsoproposesadatastructuretospeed
uppenaltyevaluationforratioconstraintsandexploitstheconceptofanalphabettoimprovethe
numberofbatchcolors.Asamatteroffact,theRenaultversionofcarsequencingproblemhasattracted
long-lastingresearchinterest. Mostrecently,JahrenandAchá[27]revisitthisproblemtoinvestigate
howtoclosethegapbetweenexactandheuristicmethods. Theauthorsreportnewlowerboundsfor
7outofthe19instancesusedinthefinalroundofthecompetitionbyapplyinganimprovedinteger
programmingformulation. Inaddition,anovelcolumngeneration-basedexactalgorithmisproposed
forsolvingtheproblem,whichoutperformsanexistingbranch-and-boundapproach.
BesidesthestandardversionandtheRenaultversionofcarsequencingproblems,researchersare
alsostudyingextendedcarsequencingproblemswithadditionalconstraints,objectivefunctionsor
decisiontypes. Zhangetal.[28]proposeanartificialecologicalnicheoptimization(AENO)algorithm
foracarsequencingproblemwithanadditionalobjectiveofminimizingenergyconsumptioninthe
sortingprocess,whichisanimportantissuebutoftenneglectedinpreviousresearch. Computational
experimentsshowthattheproposedAENOalgorithmachievescompetitiveresultscomparedwiththe
mosteffectiveheuristicsfortheconventionalobjectives,andinthemeantime,itrealizesconsiderable
reductionofenergyconsumption.Yavuz[29]studiesthecombinedcarsequencingandlevelscheduling
problemwhichaimsatfindingtheoptimalproductionschedulethatevenlydistributesdifferentmodels
overtheplanninghorizonandmeanwhilesatisfiesallratioconstraintsfortheoptions. Theauthor
proposesaparametriciteratedbeamsearchalgorithmforthecombinedschedulingproblem,whichcan
beusedeitherasaheuristicorasanexactsolutionmethod. ChutimaandOlarnviwatchai[30]apply
aspecialversionoftheestimationofdistributionalgorithm(EDA)calledextendedcoincidentalgorithm
(COIN-E)toamulti-objectivecarsequencingproblembasedonamorerealisticproductionsetting,
namely,two-sidedassemblylines. ThreeobjectivesareminimizedsimultaneouslyintheParetosense,
includingthenumberofpaintcolorchanges,thenumberofratioconstraintviolationsandtheutility
work(i.e.,uncompletedoperationswhichmustbefinishedbyadditionalutilityworkers).
Basedontheliteraturereview, thelimitationsofpreviousresearchcanbesummarizedinthe
following two aspects. Firstly, the existing scheduling models require that the same processing
sequence is adopted by both the paint shop and the assembly shop, without considering the
resequencing opportunity provided by a buffer connecting the two shops. Secondly, the ratio
constraints for installing options in the assembly shop have been overemphasized while the
environmentalimpactofthecleaningoperationsinthepaintshophavenotbeenpreciselymeasured.
Infact,mostofthemodernautomobilemanufacturersadoptabuffersystemtoconnecttheshops,
and the increased level of automation suggests that the ratio constraints are not so binding as
before. On the other side, environmental protection has evidently become a major concern in the
manufacturingsector. Undersuchabackground,thispaperaimsatdealingwiththenewchallengeon
sustainability-consciousproductionschedulinginthecontemporarycarmanufacturingindustry.
Int.J.Environ.Res.PublicHealth 2018,15,32 5of32
3. ProblemFormulation
3.1. TheProductionSetting
Inatypicalautomotivemanufacturingsystem,paintingandassemblyoperationsareperformed
intwosequentialworkshopsconnectedwitharesequencingbuffer. Figure1providesanillustration
oftheproductionsettingconsideredinthisstudy.
Paint Shop
Buffer
(Selectivity Bank)
Assembly Shop
Figure1.Illustrationofaproductionsystemforautomotivepaintingandassembly.
Suppose that a set of n cars {1,2,...,n} are to be processed successively in the paint shop.
Whenthenextcarintheprocessingsequencerequiresadifferentcolorthanthepreviouscar,asetup
operation is needed to clean the painting equipment (e.g., spray guns) thoroughly. The cleaning
procedure is accompanied by the use of a chemical detergent, and the resulting discharge of
unconsumed paint will directly lead to sewage emissions. Therefore, it is desirable to have the
carswithidenticalorsimilarcolorsprocessedconsecutively(asblocks)soastoreducethefrequency
ofcolorchangesintheprocessingsequenceofpaintshop. Formally,foranytwocolors(e ande )from
1 2
thesetofallpossiblecolors{1,2,...,E},letδ denotetheamountofpollutantemissionscaused
e1e2
bythesetupoperationthatisneededbetweenthepaintingofe andthepaintingofe . Forexample,
1 2
whenalightcolorimmediatelyfollowsadarkcolor,thepaintingdevicesneedadeepcleaningand
consequentlytheemissionlevelishigher. Theaimofpaintshopschedulingistominimizethetotal
amountofpollutantemissionsproducedbythecleaningoperations.
Oncethecarshavebeenpainted,theywillbereleasedfromthepaintshoponeafteranotherin
theiroriginalorder. Then,therewillbeanopportunitytoresequencethecarsbyutilizingthebuffer
systeminordertomeetthepreferencesofthesubsequentassemblyshop. Inthisstudy,weconsider
theduedatepreferences. Formally,foreachcari,aduedated isgivenaccordingtotherequirement
i
ofcustomers,whichisexpressedintermsoflatestpositionintheproductionsequence. Forexample,
if car i is preferred to be sequenced in the first 10 positions in the assembly shop, we set d = 10,
i
whichmeansatardinesscostwillbeincurredincasethecarisplacedafterthe10thpositioninthe
processingsequence. Position-basedduedatespecificationmakessensebecausetheproductionin
assembly shop is organized according to a fixed cycle time (also known as paced assembly line).
Oncetheprocessingsequenceisfixed,thetimeofcompletingandreleasingeachcarfromtheassembly
Int.J.Environ.Res.PublicHealth 2018,15,32 6of32
shop is also known. In addition, a priority weight w is assigned to each car i, which reflects the
i
relativevalueandimportanceofitsrelatedcustomer. Theaimofassemblyshopschedulingistherefore
to minimize the total weighted tardiness defined as (cid:80)n w(πˆ −d )+, where (x)+ = max{x,0}
i=1 i i i
and πˆ representstheactualpositionofcar i intheprocessingsequenceofassemblyshop. Inthis
i
study,wehaveignoredtheratioconstraints(cf.Section2.2)basedonthefollowingtwoobservations
(particularlyforChinesecarmanufacturers). First,theadvancedautomationtechnologyappliedin
contemporary car assembly lines significantly reduces the occurrence of workstation overloading.
Second,manufacturingofmedium-gradecarshasgraduallytransformedtoamassproductionmode,
whichmeansthenumberofindependentoptionalfeatureshasdecreased. Despitethesefacts,however,
itmustbenotedthatratioconstraintsarestillimportantconcernsforEuropean(andmorespecifically
German)manufacturersofhigh-endcars.
According to the above descriptions, it is very clear that paint shop and assembly shop have
theirindividualgoalswhichareinfactmutuallyindependent. Themajordifficultyintheintegrated
schedulingofbothshopsarisesfromthefactthattheresequencingbufferconnectingthemhasalimited
capacity,whichmeansthencarscannotbecompletelyresequencedafterleavingthepaintshopand
beforeenteringtheassemblyshop. Therefore,itisnecessarytomakeacompromisebetweenthegoal
of paint shop (total pollutant emissions) and the goal of assembly shop (total weighted tardiness)
by building a bi-objective optimization model that is able to produce a set of Pareto solutions for
thedecision-makers.
3.2. TheResequencingBufferSystem
Thebuffersystemthatconnectspaintshopandassemblyshopoffersanopportunitytopartially
resequencethecars. Amongthevarioustypesofbuffersystemsmentionedin[31],selectivitybank
(alsoreferredtoasmixbank)isthemostcommonlyusedsystemforphysicalresequencinginthe
automobilemanufacturingindustrybecauseofitslowcostandhighflexibility.
AselectivitybankconsistsofLparallellanes,wherethel-thlanehasc spacesforstoringcars.
l
Attheentranceofthebuffer,acarmaychoosetoenteranyofthelaneswithunoccupiedspacesand
jointhequeueinthatlane. Attheexitofthebuffer,thefirstcarinanynonemptylanemaybereleased
intotheprocessingsequenceforassemblyshop. Clearly,theresequencingabilityofaselectivitybank
dependsonthenumberoflanes. IfL = n,thenitcanrealizeacompleteresequencingofncarsand
outputanysequenceasneeded. Inreality,however,Liscertainlymuchlessthanthenumberofcarsto
beresequenced,andthereforetheselectivitybankcanonlyimplementapartialresequencingofcars.
Thegeneralruleis: itisnotpossibletochangetherelativeorderoftwocarsiftheyhaveenteredthe
samelane(becauseeachlaneisequivalenttoafirst-infirst-outqueuestructure).
Figure2givesanexampletoillustratethefunctionofaselectivitybankwithtwolanes(L =2)
andtwospacesineachlane(c = c = 2). Initially, thefourcarsaresequencedaccordingtotheir
1 2
indexes,i.e.,intheorderof1,2,3,4(Step(a)). Tomovecar3totheveryfirstposition,wemustletcar3
enteradifferentlanethantheonechosenbycar1andcar2becausecar3needsto“jump”overthetwo
cars(Step(b)). Finally,car3isfirstreleasedfromtheselectivitybank,andconsequentlythesequencing
ofthefourcarsisalteredto3,1,2,4(Step(c)). Notethatitisnotpossibletorealizeasequencelike3,2,
1,4becauseofthelimitednumberoflanes.
Int.J.Environ.Res.PublicHealth 2018,15,32 7of32
(a) 1 2 3 4
1 2
(b)
3 4
(c) 3 1 2 4
Figure2.Anexampleofselectivitybankwithtwolanes.
3.3. TheMILPModel
Wewillformulatethepaintshopschedulingproblemasamixed-integerlinearprogramming
(MILP)model. First,agroupof0−1decisionvariablesareintroducedasfollows.
(cid:40)
1 ifcariisprocessedinthek-thpositioninpaintshop,
x = (1)
ik
0 otherwise.
(cid:40)
1 ifcariisprocessedinthek-thpositioninassemblyshop,
xˆ = (2)
ik
0 otherwise.
(cid:40)
1 ifthecarprocessedinthek-thpositioninpaintshoprequirescolore,
y = (3)
ek
0 otherwise.
(cid:40)
1 ifcarientersthel-thlaneinthebufferarea,
z = (4)
il
0 otherwise.
Withthesedecisionvariables,thecompleteMILPmodelcanbeestablished. Notethattheother
decision variables in the following model (e.g., Y , Φ , T) are all defined on the basis of these
e1e2k ij i
fundamentalvariables. Weuse Mtodenoteaverylargepositivenumber.
n E E
(cid:88)(cid:88) (cid:88)
Minimize TPE = (δ ·Y ) (5)
e1e2 e1e2k
k=2e1=1e2=1
n
(cid:88)
TWT = (w ·T) (6)
i i
i=1
n n
(cid:88) (cid:88)
subjectto: x = xˆ =1, k =1,...,n (7)
ik ik
i=1 i=1
n n
(cid:88) (cid:88)
x = xˆ =1, i =1,...,n (8)
ik ik
k=1 k=1
L
(cid:88)
z =1, i =1,...,n (9)
il
l=1
n
(cid:88)
z ≤ c , l =1,...,L (10)
il l
i=1
Int.J.Environ.Res.PublicHealth 2018,15,32 8of32
n
(cid:88)
y = (β ·x ), k =1,...,n, e =1,...,E (11)
ek ei ik
i=1
Y ≥ y +y −1, k =1,...,n, e ,e =1,...,E (12)
e1e2k e1(k−1) e2k 1 2
Y ≥0, k =1,...,n, e ,e =1,...,E (13)
e1e2k 1 2
n n
(cid:88) (cid:88)
(k·x )− (k·x ) ≤ M·Φ , i,j =1,...,n (14)
ik jk ij
k=1 k=1
n n
(cid:88) (cid:88)
(k·x )− (k·x ) ≥ M·(Φ −1), i,j =1,...,n (15)
ik jk ij
k=1 k=1
n n
(cid:88) (cid:88)
(k·xˆ )− (k·xˆ ) ≤ M·(2−z −z )+M·Φ , i,j =1,...,n, l =1,...,L (16)
ik jk il jl ij
k=1 k=1
n
(cid:88)
T ≥ (k·xˆ )−d, i =1,...,n (17)
i ik i
k=1
T ≥0, i =1,...,n (18)
i
x ,xˆ ,y ,z ,Φ ∈ {0,1}, i,k =1,...,n, e =1,...,E, l =1,...,L (19)
ik ik ek il ij
Equation(5)definesthefirstobjective,i.e.,minimizingthetotalpollutantemissions(TPE)caused
by the setup operations in paint shop (δ represents the amount of emissions during a setup
e1e2
operation to switch from color e to color e ). The binary variable Y is defined in such a way
1 2 e1e2k
thatY =1ifandonlyifthecarinthe(k−1)-thpositionispaintedwithcolore (i.e.,y =1)
e1e2k 1 e1(k−1)
andmeanwhilethecarinthek-thpositionispaintedwithcolore (i.e.,y =1);Y =0otherwise
2 e2k e1e2k
(i.e., eithery =0or y = 0). Equation(6)definesthesecondobjective, i.e., minimizingthe
e1(k−1) e2k
totalweightedtardiness(TWT)incurredbylatefinishingofcarsinthesubsequentassemblyshop
(w representsthepriorityweightofcari). Equations(7)and(8)reflecttheassignmentconstraints,
i
i.e.,eachcarshouldbeassignedtoexactlyonepositionintheprocessingsequenceandeachposition
inthesequencemustbeoccupiedbyexactlyonecar. Likewise,Equation(9)specifiesthateachcar
canonlychoosetoenteronelaneoftheselectivitybank. Equation(10)requiresthatthenumberof
carsenteringlanel shouldnotexceeditscapacitydenotedbyc (wecanassumec = ∞ifthecars
l l
movethroughthebufferinadynamicmanner). Equation(11)definesy ,whichequals1ifandonly
ek
ifthecarinthek-thpositionshouldbepaintedwithcolore. Notethat β isaparameterknownin
ei
advance (β = 1 if car i requires color e and β = 0 otherwise). Equations (12) and (13) provide
ei ei
thedefinitionforY basedony andy (notethatthetwoinequalitiesareenoughtomake
e1e2k e1(k−1) e2k
Y binary variables). Equations (14) and (15) are used to define Φ , which depicts the relative
e1e2k ij
orderoftwocarsiand jinthepaintshop: Φ = 1ifcariisprocessedaftercar j,andΦ = 0ifcar
ij ij
i isprocessedbeforecar j. Weneed Φ asintermediatevariablestoreflecttheimpactofthepaint
ij
shopsequenceonthepossiblesequencesforassemblyshop(notethat Φ isusedinthefollowing
ij
Equation(16)). Equation(16)describestheconstraintimposedbytheselectivitybank: therelative
orderoftwocarscannotbealterediftheytravelthroughthesamelane. Inparticular,ifcarihasbeen
scheduledbeforecarjinthepaintshop(i.e.,Φ =0)andbothcarshaveenteredthel-thlaneofthe
ij
buffer(i.e., z +z = 2), thencar i isdefinitelypositionedbeforecar j whentheyleavethebuffer
il jl
forthenextproductionstep. Equations(17)and(18)evaluatethetardinessofcariintheassembly
shop,whichisdefinedasT =max{πˆ −d,0},withπˆ denotingthepositionofcariintheprocessing
i i i i
sequenceofassemblyshopandd representingthepreferredlatestpositionofcari.
i
3.4. FurtherDiscussion
Thestudiedproductionsystemconsistsoftwosequentialstageswithanadditionalintermediate
buffer. Minimizing the TPE in the first production stage is equivalent to minimizing the sum of
sequence-dependentsetuptimes(moreaccurately,itisthesameastheproblem1|s |C [32]whichis
ij max
Int.J.Environ.Res.PublicHealth 2018,15,32 9of32
further equivalent to the traveling salesman problem and thus strongly NP-hard). The problem
of the second stage is minimizing TWT under precedence constraints (more precisely, 1|chains;
(cid:80)
p = 1| w T),whichisalsoNP-hardinthestrongsense(moredetailsaregiveninSection4).
j j j
In fact, our problem is much more complicated than the two subproblems mentioned above
becauseoftheresequencingbufferlocatedbetweenthetwoproductionstages. Thebufferisused
forpartiallyresequencingthecarsaftertheyleavethefirststageandbeforetheyenterthesecond
stage. Inotherwords,thetwostagescanprocessdifferentcarsequences. However,theproduction
systemdoesnotfallunderthecategoryofflowshopsbecauseitisimpossibletogenerateanarbitrary
processingsequenceforthesecondstagegiventheprocessingsequenceinthefirststage(thebuffer
hasalimitednumberoflanesandconsequentlyitcanonlyrealizepartialresequencingofthecars).
Whatcomplicatestheproblemisthefactthatthebuffersystemalsorequiresanoptimizeddecision
regardingtheallocationofcarstoeachlane.
Nowitisfairlyclearthatschedulingdecisionsforthetwoproductionstagesaretightlycoupled.
Solvingthefirst-stageproblemtooptimalitymayleadtopoorperformanceintermsofthesecond-stage
criterion,andviceversa,whichmeansthestrategyofsolvingeachsubproblemindividuallyforeach
productionstageisapparentlyinfeasibleforaddressingthewholeproblem. Theonlywayofresolving
theproblemistobuildanintegratedschedulingmodelbyincorporatingtheconstraintsimposedbythe
resequencingbuffer. Inthisway,itispossibletotakethepreferencesofbothstagesintoconsideration
andobtainwell-balancedschedulesfortheentireproductionsystem. Thisisexactlythemotivation
behindtheintegratedproblemformulation.
4. TheAssemblyShopSequencingSubproblem
ThemainoptimizationalgorithmwhichwillbedetailedinSection5dealswiththedecisionson
carsequencinginthepaintshopaswellastheallocationofcarstothedifferentlanesoftheselectivity
bank. A critical issue that arises in the meantime is how to sequence the cars in the downstream
assemblyshopunderafixedplacementofcarsintheselectivitybank. Thissubproblemneedstobe
solvedproperlyfortheevaluationofsolutionsinthemainalgorithm.
(cid:80)
Theassemblyshopsequencingsubproblemcanbedescribedas1|chains;p =1| w T according
j j j
tothethree-fieldnotationsystem,basedonthefollowingobservations:
• Thepacedproductionmodeinassemblyshopmeansthateachjobhasidenticalprocessingtime
(p = p). Inaddition,theposition-basedduedateassignmentschemesuggeststhatthesituation
j
canbefurthersimplifiedas p =1.
j
• Eachlaneoftheselectivitybankisactuallyimposingasetofprecedenceconstraints(inthechain
form) on the relevant cars traveling through the lane. These precedence constraints must be
respectedwhenschedulingtheassemblyshop.
(cid:80)
Lemma1. Theproblem1|p =1| w T ispolynomiallysolvable.
j j j
Proof. ItcanbeshownthatthisproblemisequivalenttotheAssignmentProblem(concerningthe
assignmentofnjobstonconsecutivepositionsonasinglemachine):
n n
(cid:88)(cid:88)
min C(i,j)x (20)
ij
i=1 j=1
n
(cid:88)
s.t. x =1, i =1,...,n (21)
ij
j=1
n
(cid:88)
x =1, j =1,...,n (22)
ij
i=1
x ∈ {0,1}, i =1,...,n, j =1,...,n (23)
ij
Int.J.Environ.Res.PublicHealth 2018,15,32 10of32
The equivalence is established by setting the cost of assigning job i to the j-th position as
C(i,j) = w ·max{j−d,0}. Therefore, the Hungarian algorithm can solve this problem within
i i
O(n3)time.
(cid:80)
Lemma2. Theproblem1|chains;p =1| w T isNP-hardinthestrongsense.
j j j
Forproofofthelemma,readersaresuggestedtorefertotheworkofLeungandYoung[33].
4.1. ABranch-and-BoundAlgorithm
In view of the complexity results presented above, we propose a branch-and-bound (B&B)
(cid:80)
algorithm to solve the problem 1|chains;p = 1| w T, using solutions of the relaxed problem
j j j
(cid:80)
1|p = 1| w T asthebasisforbounding. Schedulesareconstructedfromtheendtothefront,i.e.,
j j j
backwards in time, considering the fact that the larger values of weighted tardiness are likely to
correspondtothejobsthatarescheduledmoretowardstheendoftheprocessingsequence. Therefore,
itappearstobebeneficialtoschedulethesejobsfirstinthebranch-and-boundprocedure. Attheq-th
levelofthesearchtree,jobsareselectedforthe(n−q+1)-thposition. UndertheLsetsofchain-based
precedenceconstraints, thereareatmost L branchesgoingfromeachnodeatlevel q tolevel q+1
(becauseonlythelastunscheduledjobineachchainmaybeconsidered). Itfollowsthatthenumberof
nodesatlevelqisboundedbyLq. Thesolutionoftherelaxedproblemwithoutprecedenceconstraints
(cid:80)
providesalowerboundfortheoriginalproblem1|chains;p = 1| w T. Thisboundingstrategy
j j j
isappliedtothesetofunscheduledjobsateachnodeofthesearchtree. Ifthelowerbound(LB)is
largerthanorequaltotheobjectivevalueofanyfeasibleschedule,thenthenodewillbediscarded.
ThecompletealgorithmisformallydescribedasAlgorithm1.
In the following, we make some comments for better explaining the algorithm. In Line 1,
thevariableforrecordingthebestobjectivevalueobtainedsofar(TWT )isinitializedtobeavery
min
largepositivenumber,andthesetofnodes(N)isinitializedwiththerootnodewhichcorresponds
to a null sequence N0 (the job to be put in each position is pending). The lower bound for N0 is
unimportantandthusLB(N0)isassignedwith0. Thetree-typesearchisthenstartedandthesearch
processwillbecontinueduntilthenodesetbecomesempty. Ineachiteration,threemajorstepsare
performed,i.e.,nodeselection,branching,andhandlingofoffspringnodes.
1. The algorithm always selects the node with the smallest lower bound from N for further
exploration(Line3). Themotivationistofocusonthepromisingsubregionofthesearchspace
so that it is likely to discover a feasible solution with lower objective value, leading to more
opportunitiesofpruningthesearchtree.
2. IftheselectednodeNchasalowerboundbelowthecurrentbestobjectivevalue(upperbound),
thealgorithmhastofurtherexplorethenodebycreatingbranchesonit. Thisisimplementedin
Line6. Inparticular,thealgorithmisattemptingtoinsertjobsintothelastvacantpositionofthe
currentpartialsolutioncorrespondingtoNc. Constrainedbytheprecedencerelationsgivenin
theformofP ,...,P ,onlythelastunscheduledjobineachprecedencechainP (l =1,...,L)is
1 L l
applicableforthispurpose. Hence,atmostLdescendantnodeswillbecreated.
3. For each descendant node Nc, the lower bound LB(Nc) is first obtained by employing
l l
a Hungarian algorithm to solve the relaxed scheduling problem (neglecting precedence
constraints) which consists of the unscheduled jobs with respect to the partial solution of Nc
l
(Line8). Then,threecasesareidentifiedandhandledseparately.
• If the schedule obtained after solving the relaxed problem (πc) turns out to be a feasible
l
solution for the original problem (which means respecting all the precedence relations),
thenthealgorithmfurtherinvestigateswhetherthissolutiondefinesanewupperboundand
updatestherelevantvariableswhennecessary(Lines11–14).
• If the obtained schedule is not feasible for the original problem but its objective value
(i.e.,thelowerboundLB(Nc))turnsouttobelargerthanorequaltothecurrentupperbound
l