ContentslistsavailableatSciVerseScienceDirect
Applied
Soft
Computing
jo u r n al h om epa g e :w w w . e l s e v i e r . c o m / lo c a t e / a s o c
Genetic
algorithms
for
a
two-agent
single-machine
problem
with
release
time
Wen-Chiung
Lee
a,∗, Yu-Hsiang
Chung
b,
Mei-Chia
Hu
a aDepartmentofStatistics,FengChiaUniversity,Taichung,TaiwanbDepartmentofIndustrial&EngineeringManagement,NationalChiaoTungUniversity,Hsinchu,Taiwan
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:Received6February2012
Receivedinrevisedform11May2012 Accepted11June2012
Availableonline6July2012 Keywords: Scheduling Totaltardiness Two-agent Single-machine Releasetime Maximumtardiness
a
b
s
t
r
a
c
t
Schedulingwithtwocompetingagentshasdrawnalotofattentionlately.However,itisassumedthatall thejobsareavailableinthebeginninginmostoftheresearch.Inthispaper,westudyasingle-machine probleminwhichjobshavedifferentreleasetimes.Theobjectiveistominimizethetotaltardinessofjobs fromthefirstagentgiventhatthemaximumtardinessofjobsfromthesecondagentdoesnotexceedan upperbound.Threegeneticalgorithmsareproposedtoobtainthenear-optimalsolutions.Computational resultsshowthatthebranch-and-boundalgorithmcouldsolvemostoftheproblemswith16jobswithina reasonableamountoftime.Inaddition,itshowsthattheperformanceofthecombinedgeneticalgorithm isverygoodwithmeanerrorpercentagesoflessthan0.2%forallthecases.
©2012ElsevierB.V.Allrightsreserved.
1. Introduction
Recently,thereis agrowinginterestin multi-agent
schedul-ing where jobs might come from severalcustomers who have
theirownobjectivefunctions.Forexample,BakerandSmith[1]
gave an example of a prototype shop where the
manufactur-ingdepartmentmightbeconcernedaboutfinishing jobsbefore
theirdue dates, andtheresearchanddevelopmentdepartment
mightbemoreconcernedaboutquickresponsetime.Kubzinand
Strusevich[2]presented anotherexampleinwhich the
mainte-nanceactivitiescompetewithreal jobsfor machineoccupancy
inmaintenanceplanning.MeinersandTorng[3]gavea
telecom-municationexamplewherevarioustypesofpacketsandservice
competefortheradioresourceusage.SoomerandFranx[4]gave
atransportationexamplewheretheagentsowntheir
transporta-tionresources,andcompetefortheusageoftheinfrastructures.
Leung et al. [5] pointed out that several important classes of
schedulingproblems,suchasreschedulingproblemsor
schedul-ingwithavailabilityconstraints,canbeformulatedastwo-agent
schedulingproblems.BakerandSmith[1]and Agnetisetal.[6]
pioneeredtheschedulingproblemswithtwo competingagents.
Sincethen,two-agentschedulinghasdrawnresearchers’attention
[7–16].
Recently,Leungetal.[5]generalizedthesinglemachine prob-lemsofAgnetisetal.[6]tothecaseofmultipleidenticalparallel
∗ Correspondingauthor.
E-mailaddress:wclee@fcu.edu.tw(W.-C.Lee).
machineswherejobpreemptionisallowed.Theyalsoconsidered
certainsingle-machineproblemswherethejobsmayhave
differ-entreleasedates,andjobpreemptionsmayormaynotbeallowed.
Leeetal.[17] consideredatwo-agentscheduling problemona
two-machinepermutation flowshop.Theirobjective isto
mini-mizethe total tardiness ofjobs fromthefirst agentgiven that
thenumberof tardyjobsofthesecond agentiszero.Liu etal.
[18] broughtthe aging and learning effects intothe two-agent
scheduling.Their objective isto minimizethetotal completion
timeofjobsfromthefirstagentgiventhatthemaximumcostof
jobsfromthesecondagentcannotexceedagivenupperbound.
Wanetal.[19]consideredseveraltwo-agentschedulingproblems
withcontrollablejobprocessingtimesinwhichtwoagentshaveto shareeitherasinglemachineortwoidenticalmachinesin paral-lelwhileprocessingtheirjobs.MorandMosheiov[20]considered
atwo-agentschedulingproblemonasingle-machineproblemto
minimizethemaximumearlinesscostortotal (weighted)
earli-nesscostofjobsfromoneagent,subjecttoanupperboundon
themaximumearlinesscostofjobsfromtheotheragent.They
introduceda polynomial-timesolutionfor the maximum
earli-nessproblemandprovedNP-hardnessfortheweightedearliness
case.Lee etal.[21] considereda two-agentproblemwherethe
objectiveistominimizethetotal completiontime ofjobsfrom
thefirstagentgiventhatnotardyjobisallowedforthesecond
agent.Liuetal.[22]developedtheoptimalsolutionsforcertain two-agentproblemswithincreasinglineardeteriorationona sin-glemachine.Theirgoalistominimizetheobjectivefunctionofthe
firstagentgiventhattheobjectivefunctionofthesecondagent
cannotexceedagivenbound.Nongetal.[23]consideredatwo
1568-4946/$–seefrontmatter©2012ElsevierB.V.Allrightsreserved. http://dx.doi.org/10.1016/j.asoc.2012.06.015
agentproblemonasinglemachinewheretheobjectiveisto
min-imizetheweightedsumofthemaximumcompletiontimeofjobs
fromoneagentandthetotalweightedcompletiontimeof jobs
fromtheotheragent.Theyprovideda2-approximationalgorithm
andshowedthecaseisNP-hardwhenthenumberofjobsofthe
firstagent isfixed.Yinet al.[24] studiedthree single-machine
problemswithdeterioratingjobs.Theobjectivesarethemaximum
earlinesscost, total earlinesscost, and total weightedearliness
cost,whilekeepingthemaximumearlinesscostofjobsfromthe
otheragentbelowafixedlevel.MorandMosheiov[25]
consid-eredasingle-machineproblemwithbatchschedulingtominimize
thetotalcompletiontimeofjobsfromoneagent,giventhatthe
maximumcompletiontimeofjobsfromtheotheragentdoesnot
exceedan upper bound.Wu et al. [26] studiedsingle-machine
schedulingwithlearning effects. Theirobjective is tominimize
thetotaltardinessofjobsfromthefirstagent,giventhatnotardy jobisallowedforthesecondagent.Chengetal.[27]considered
single-machineschedulingwithtruncatedlearningeffects.Their
objectiveis tominimizethetotal weightedcompletiontime of
jobsfromthefirstagent,giventhat notardy jobisallowedfor
thesecondagent.LiandHsu[28]investigatedasingle-machine
problemwithlearningeffectwheretheobjectiveistominimize
thetotalweightedcompletiontimeofbothagentswiththe
restric-tionthatthemakespanofeitheragentcannotexceedanupper
bound.
Mostoftheresearchinschedulingwithtwocompetingagents
assumesthat jobs are ready to beprocessed in the beginning.
However, customer orders might not arrive simultaneously in
manyrealisticsituations.Thus,itismorepracticaltoconsiderjobs releasetimes.Leungetal.[5]weretheonlyauthorswho
consid-eredtwo-agentschedulingwithjobreleasetimes.Inthispaper,
westudy atwo-agent schedulingproblemona singlemachine
withrelease time where theobjective is tominimizethe total
tardiness of jobsfromthe firstagent given that themaximum
tardinessofjobsfromthesecondagentcannotexceedanupper
bound. To the best of our knowledge, this problem has never
beenstudied.The restof this paper is organizedasfollows. In
thenextsection,theformulationofourproblemisdescribed.In
Section3,a branch-and-bound algorithm withseveral
elimina-tion rules and a lower boundis developed. In Section 4, three
geneticalgorithmsareproposedtosolvethisproblem.InSection5,
computationalexperimentsareconductedtoevaluatethe
perfor-manceofthegeneticalgorithms.Aconclusionisgiveninthefinal section.
2. Problemdescription
Theproblemformulationisdescribedasfollows.Therearen
jobs,eachbelongstoeitheragentAG1orAG2.Foreachjobj,there isaprocessingtimepj,aduedatedj,areleasetimerj,andanagent codeIj,whereIj=1ifj∈AG1orIj=2ifj∈AG2.Underaschedule S,letCj(S)bethecompletiontimeofjobjandletTj(S)=max{0, Cj(S)−dj}bethetardinessofjob j.Inthis paper,weconsidera
singlemachineproblemtominimizethetotaltardiness of jobs
fromagentAG1 giventhatthemaximumtardinessofjobsfrom
agentAG2doesnotexceedanupperboundM.Usingthethree-field notationextendedbyAgnetisetal.[6],thisproblemisdenotedby 1|r1
j;r2j|
Tj;Tmax.
3. Abranch-and-boundalgorithm
WhenallthejobsarefromagentAG1 and thereleasetimes
arezero,theproblemreducestotheclassicalsingle-machinetotal tardinesstimeproblemwhichisNP-hard[29].Therefore,a branch-and-boundalgorithmisproposedtoderivetheoptimalsolution.
3.1. Dominanceproperties
First,weprovidearesulttospeedupthesearchprocess.We
thendevelopseveraladjacentdominancepropertiestoreducethe
searchingscope.
Theorem1. Ifthereisajobisuchthatri+pi≤rjforalltheremaining jobsj,thenjobiisscheduledfirstintheoptimalsequence.
Proof. Theproofisomittedsinceitisstraightforward.
SupposethatSandS aretwoschedulesofjobswiththeonly
differencebetweenthemapairwiseinterchangeoftwoadjacent
jobsiandj.Thatis,S=(,i,j,)andS=(,j,i,),whereand eachdenoteapartialsequence.Inaddition,lettbethecompletion timeofthelastjobin.ThecompletiontimesofjobsiandjinS are
Ci(S)=max{t,ri}+pi (1)
and
Cj(S)=max{Ci(S),rj}+pj (2)
Similarly,thecompletiontimesofjobsjandiinSare
Cj(S)=max{t,rj}+pj (3)
and
Ci(S)=max{Cj(S),ri}+pi (4)
Dependingonwhetherjobsare fromagentsAG1 orAG2,we
dividethesituationintothefollowingthreecases.
Case1. BothjobsiandjarefromagentAG1.
To show that S dominates S, it suffices to show that
Cj(S)−Ci(S)≤0,andTi(S)+Tj(S)<Tj(S)+Ti(S)inthiscase.
Property1.1. Ift≥max{ri,rj}anddi≤t+pi<dj,thenSdominates S.
Proof. Sincet≥max{ri,rj},wehave Ci(S)=t+pi
Cj(S)=t+pi+pj Cj(S)=t+pj and
Ci(S)=t+pj+pi
Therefore,wehaveCj(S)≤Ci(S).Sincet+pi≥di,wehave
Ti(S)=t+pi−di (5)
and
Ti(S)=t+pj+pi−di (6)
SupposethatTi(S)isnotzero.Notethatthisisthemore restric-tivecasesinceitcomprisesthecasethatTi(S)iszero.FromEqs.(5)
and(6),wehave
Tj(S)+Ti(S)−Ti(S)−Tj(S)=dj−t−pi>0 sincet+pi<dj.Thus,SdominatesS.
Property1.2. Ift≥max{ri,rj}anddi<t+pi+pj≤dj,thenS domi-natesS.
Property1.3. Ift≥max{ri,rj},t+pi≤di≤t+pj+pi,anddj>di,then SdominatesS.
Property1.4. If ri≤t≤rj≤t+pi,dj≥t+pi+pj,anddi<rj+pj+pi, thenSdominatesS.
Property1.5. Ifri≤t≤rj≤t+pi,t+pi≤di≤rj+pj+pi,anddj>di, thenSdominatesS.
Property1.6. Ifri≤t≤rj≤t+pianddi≤t+pi≤dj,thenSdominates S.
Property1.7. Ift≤ri≤rj≤ri+pi,ri+pi+pj≤dj,andrj+pj+pi>di, thenSdominatesS.
Property1.8. Ift≤ri≤rj≤ri+pi≤di≤rj+pj+piandri+di<rj+dj, thenSdominatesS.
Property1.9. Ift≤ri≤rj≤ri+pi<djandri+pi≥di,thenS domi-natesS.
Property1.10. Ifmax{t,ri}+pi≤rjandrj+pj+pi>di,thenS dom-inatesS.
Case2. JobiisfromagentAG1,butjobjisfromagentAG2. ToshowthatSdominatesS,itsufficestoshowthatTj(S)≤M, Ti(S)<Ti(S)andCj(S)−Ci(S)≤0.
Property2.1. Ift≥max{ri,rj}andt+pi+pj−dj≤M,thenS domi-natesS.
Property2.2. Ifri≤t≤rj≤t+piandt+pi+pj−dj≤M,thenS dom-inatesS.
Property2.3. Ift≤ri≤rj≤ri+piandri+pi+pj−dj≤M,thenS dom-inatesS.
Property2.4. Ift≥ri,t+pi≤rj,andrj+pj−dj≤M,thenSdominates S.
Property2.5. Ift≤ri,ri+pi≤rj,andrj+pj−dj≤M,thenSdominates S.
Case3. BothjobsiandjarefromagentAG2.
ToshowthatSdominatesS,itsufficestoshowthatTi(S)≤M, Tj(S)≤MandCj(S)−Ci(S)<0.
Property3.1. Ift≥max{ri,rj},t+pi−di≤M,t+pi+pj−dj≤M,and di<dj,thenSdominatesS. Property 3.2. If ri≤t<rj≤t+pi, t+pi−di≤M, and t+pi+pj−dj≤M,thenSdominatesS. Property 3.3. If t≤ri<rj≤ri+pi, ri+pi−di≤M, and ri+pi+pj−dj≤M,thenSdominatesS. Property3.4. Ift≥ri,t+pi≤rj,t+pi−di≤M,andrj+pj−dj≤M, thenSdominatesS. Property3.5. Ift≤ri,ri+pi≤rj,ri+pi−di≤M,andrj+pj−dj≤M, thenSdominatesS.
Tofurtherfacilitatethesearchprocess,weprovideaproposition todeterminethefeasibilityofapartialschedule.Assumethat(, c)isasequenceofjobswhereisthescheduledpartandcis theunscheduledpart.
Proposition1. Ifthereisajobj∈c∩AG
2suchthatt+pj>dj+M, then(,c)isnotafeasiblesequence.
3.2. Alowerbound
Inthissubsectionwedevelop alowerboundforthe
branch-and-boundalgorithm.LetPSbeapartialsequenceinwhichsjobs
arescheduled.Supposethat,amongtheunscheduledsetUSwith
n− sjobs,therearen1jobsfromagentAG1andn2jobsfromagent
AG2, where n1+n2=n−s. For these unscheduledjobs, we have
p(s+1)≤p(s+2)≤···≤p(n)whentheyarearrangedinnon-decreasing orderoftheirprocessingtimes andr(s+1)≤r(s+2)≤···≤r(n) when
theyare arranged in thenon-decreasing order of their release
times.Notethatp(i)andr(i)maynotbefromthesamejob. Fur-thermore,theduedatesofthen1(n2)unscheduledjobsfromagent AG1(AG2)aredenotedasd1(1)≤d1(2)≤···≤d1(n1)(d2(1)≤d2(2)≤···≤ d2
(n2))whentheyareinnon-decreasingorderoftheirduedates.The
ideaoftheproposedlowerboundisthatwefirstderivealower
boundonthecompletiontimesoftheunscheduledjobsbasedon
theSPTrule,andthenweassignthemtoagentsAG1andAG2
with-outviolatingtheconstraintthatthemaximumtardiness ofjobs
fromagentAG2 doesnotexceedtheupperboundM.Inthefirst
step,thecompletiontimeofthe(s+1)thjobis C[s+1]=max{C[s],r[s+1]}+p[s+1]≥C[s]+p(s+1)
Byinduction,thecompletiontimeofthe(s+i)thjobis C[s+i]≥C[s]+
i
l=1
p(s+l) (7)
On the other hand, this lower bound might not be tight if
therelease times are large. Thus, C[s+1]=max{C[s], r[s+1]}+p[s+1] ≥r(s+1)+p(s+1). Byinduction,wehave C[s+i]=max 1≤k≤i
r[s+k]+ i−k+1l=1 p[s+k+l] ≥max 1≤k≤i r(s+k)+ i−k+1
l=1 p(s+l) (8)
FromEqs.(7)and(8),alowerboundonthecompletiontimeof the(s+i)thjobis
C[s+i]≥max{t+ i
l=1 p(s+l),max 1≤k≤i{r(s+k)+ i−k+1l=1 p(s+l)}}
fori=1,2,...,n− s.Inthesecondstep,theremainingtaskisto assigntheestimatedcompletiontimestothejobsfromagentAG1 orAG2.Theprincipleistoassignthecompletiontimestothejobs fromagentAG2aslateaspossiblewithoutviolatingtheassumption thatthemaximumtardinessofthejobsofagentAG2cannotexceed theupperbound.Inaddition,letC1
(1)≤C 1 (2)≤···≤C 1 (n1)andC 2 (1)≤ C2 (2)≤···≤C 2
(n2)denotetheestimatedcompletiontimesofthejobs
fromagentsAG1andAG2,respectively,whentheyarearrangedin
non-decreasingorder.Theassignmentprocedureisinabackward
mannerstartingfromthejobwiththeremaininglargestduedate
untilallthejobsareassigned.Thedetailsaregivenasfollows:
Algorithmofthelowerbound:
Step 1: Set ic=n−s, i1=n1, i2=n2, and C(s+i)=max{t+ i
l=1 p(s+l),max 1≤k≤i{r(s+k)+ i−k+1 l=1 p(s+l)}}fori=1,2,...,n−s. Step2:IfC(s+ic)≤d2 (i2)+M,thensetC 2 (i2)=C(s+ic)and i2=i2−1. Otherwise,setC1 (i1)=C(s+ic)andi1=i1−1.Step3:Setic=ic−1.Ific≥1,thengotoStep2.
Therefore,a lowerboundonthetotaltardiness ofjobsfrom
agentAG1forPSis
LB=
j∈AG1 Tj(PS)+ n1 j=1 max{0,C1 (j)−d1(j)}3.3. Descriptionofthebranch-and-boundalgorithms
Adepth-firstsearchisusedinthebranchingprocedurestarting fromthefirstposition.Wechooseabranchandsystematicallywork downthetreeuntilweeithereliminateitorreachitsfinalnode, inwhichcasethissequenceeitherreplacetheinitialsolutionoris
eliminated.Theoutlineofthebranch-and-boundalgorithmisas
follows.
Step 1. {Initialization}Implement the geneticalgorithms
(dis-cussed inthe nextsection)toobtaina sequence asthe initial
incumbentsolution.
Step2.{Branching}ApplyTheorem1,Properties1.1to3.5,and
Proposition1toeliminatethedominatedpartialsequence.
Step3.{Bounding}Forthenon-dominated nodes,computethe
lower bound of the total tardiness of jobsfrom agent AG1 of
the unfathomed partial sequences or that of the completed
sequences.Ifthelowerboundontheobjectivefunctionforthe
partialsequenceisgreaterthantheinitialsolution,eliminatethat
nodeand all the nodesbeyond it in the branch.If the
objec-tivefunctionof thecompletedsequenceis lessthantheinitial
solution, replace it as the new solution. Otherwise, eliminate
it.
4. Geneticalgorithms
Evolutionaryalgorithmshavebecomepopularinobtaininggood
approximatesolutionsformanyNP-hardproblems[30–35].Inthis paper,weutilizethegeneticalgorithm(GA).Itisanintelligent
ran-domsearchstrategywhichhasbeenusedsuccessfullytofindnear
optimalsolutionstomanycomplexproblems[36–38].TheGA
usu-allystartswithapopulationoffeasiblesolutionsanditeratively replacesthecurrentpopulationbyanewpopulationuntilcertain stoppingconditionisreached.Itrequiresasuitableencodingfor
theproblemandafitnessfunctionthatrepresentsameasureof
thequalityofeachencodedsolution(chromosome).The
reproduc-tionmechanismselectstheparentsandrecombinesthemusinga
crossoveroperatortogenerateoffspringwhicharesubmittedtoa mutationoperatorinordertoalterthemlocallytoavoidpremature
convergence.ThecomponentsoftheGAappliedtoourproblemare
asfollows. 4.1. Encoding
Inthisstudy,weadopttherandomnumberencodingmethod
[39].Foraproblemofnjobs,wegenerateachromosomewithn
uniformrandomrealnumbersbetween0and1torepresentthe
genes,whereeachgenecorrespondstoajob.Theorderofthese
randomnumbersrepresentsthejobsequence.For instance,the
chromosomeofa5-jobproblem(0.33,0.78,0.13,0.94,0.26)would standforthesequence(3,5,1,2,4).
4.2. Populationsize
Thepopulationsizeisanimportantfactorintheperformanceof GA.Foralargepopulationsize,itiseasiertoobtainabettersolution, butitconsumesmoretime.Afterapreliminarytrial,thepopulation sizeNissetat500inourcomputationalexperiment.
4.3. Fitnessfunction
Inordertomimicthenaturalprocessofthesurvivalofthefittest,
thefitnessfunctionassignstoeachmemberofthepopulationa
valuereflectingtheirrelativesuperiority.Inthispaper,weadopt theideabyHomaifaretal.[40]ofaddingapenaltyfunctiontothe
Parent1 0.45 0.32 0.15 0.78 0.53 0.36
Offspring 0.45 0.32 0.27 0.49 0.18 0.72
Parent2 0.18 0.87 0.27 0.49 0.18 0.72 Fig.1.Onecut-pointcrossover.
infeasiblesolution.Thus,theobjectivefunctionofchromosomek is
objk=
j∈AG1
Tj+˛maxj∈ AG2max{Tj−M,0}, where ˛ is set at
5000inthisstudy.Inaddition,weusethereciprocaloftheobjective valueasthefitnessvalueforeachchromosome,andtheprobability thatachromosomeisselectedastheparentisproportionaltoits fitnessvalue.Thatis,theprobabilityofselectingchromosomeiis fi=hi/
Nj=1hj,wherehi=1/obji,i=1,...,N,isthereciprocalofthe objectivevalueofchromosomeiinapopulationofsizeN.Thisisto ensurethattheprobabilityofselectionforasequencewithlower valueoftheobjectivefunctionishigher.4.4. Crossover
Crossoverisanoperationtogeneratenewoffspringfromtwo
parents.ItisthemainoperatorinGA.Inthisstudy,weusetheone cut-pointcrossoverasshowninFig.1andtheratePcwaschosen at95%aftersomepretests.
4.5. Mutation
Mutationisanothermainoperatortopreventpremature
con-vergenceandfallintolocaloptimum.Suchanoperationcanbe
viewedasatransitionfromacurrentsolutiontoitsneighborhood solutioninalocalsearchalgorithm.Inthisstudy,weusethe one-pointmutationasshowninFig.2andthemutationratePmisset
at80%basedonourpreliminaryexperiment.
4.6. Selection
Itis aproceduretoselectoffspringfromparentstothenext generation.Inourstudy,thepopulationsizeisfixedat500from
generation togeneration. In our study, we choose thebest 50
chromosomes(10%)fromtheparentpopulationandthebest450
chromosomes(90%)fromtheoffspringtoformthenextgeneration. 4.7. Termination
Aftersomepretests,weterminatetheproposedGAafter20n
generations,wherenisthenumberofjobs. 4.8. Initialsequences
Agoodinitialsequencemightbeusefultofacilitatethe conver-genceoftheprocessortoobtainabetterapproximatesolution.In thispaper,threemethodsareimplemented.InthefirstGA(GA1),
Offspring 0.45 0.32
0.1
5
0.78 0.53 0.36
Offspring 0.45 0.32
0.38
0.78 0.53 0.36
Fig.2.One-pointmutation.Find a job j from Uwith a minimal release time.
Can job j be scheduled in position? th k Set 1, , k j j k r p k C = + = + \ {j}, U S S S = ∪ =Ω
Remove job j from U.
Is k larger than n?
SetV={rj≤Ck j∈AG}1
Is Vempty?
Find a job jfrom Vwith a minimal due date.
Can job j be scheduled in position? th k Set 1,r} , 1, max{C \ {j}, j j k k p k k C S U S S − + = + = Ω = ∪ = SetW={rj≤Ck j∈AG2} Is Wempty?
Find a job j from Wwith a minimal due date.
Can job j be scheduled in
position? th
k
Output the job sequence
Yes No Yes No Yes No No Yes Yes No Yes No
Fig.3.BlockdiagramforHA.
thefirstgenerationconsistsof500randomsequences.Inthe sec-ondGA(GA2),thefirstgenerationconsistsof499randomsequences
andonedesignatedsequencefromtheheuristicalgorithm(HA)as
describedbelow.InthethirdGA(GA3),thefirstgenerationconsists
of51randomsequencesand441designatedsequences.The
des-ignatedsequencessortjobsaccordingtothenon-decreasingorder
ofw1rj+w2pj+(1−w1−w2)dj,wherew1=0,1/60,...,20/60and w2=0,1/60,...,20/60.Thealgorithmisgivenbelowandshownin
Fig.3.
Heuristicalgorithm(HA)
Table1
Theperformanceofthebranch-and-boundalgorithmwithn=12,P=50%,=1,and M=30n.
R Numberofnodes CPUtime
Mean Max Mean Max
0.25 0.25 52.32 474 0.001 0.016 0.50 71.59 1314 0.001 0.016 0.75 46.37 328 0.001 0.016 0.50 0.25 117.22 1889 0.001 0.016 0.50 121.95 776 0.001 0.016 0.75 156.68 1870 0.002 0.016 0.75 0.25 107.12 583 0.001 0.016 0.50 27.37 698 0.001 0.016
Step2.FindajobjfromUwithaminimalreleasetime.
Step3.Ifjobjcanbescheduledinthekthpositionwithoutcausing
theviolationoftheconstraint,putjobjinthekthposition,set
Ck=rj+pj,k=k+1,S=S∪{j},U=˝\S.Otherwise,delete{j}from
UandgotoStep2.
Step 4. If k>n, go to Step 8. Otherwise, form the set
V={rj≤Ckandj∈AG1},andifVisempty,gotoStep6.
Step5.FindajobjfromVwithaminimalduedate.Ifjobjcanbe
scheduledinthekthpositionwithoutcausingtheviolationofthe
constraint,setjobjinthekthposition,setCk=max{Ck−1,rj}+pj,
k=k+1,S=S∪{j},U=˝\S,andgotoStep4.
Step6.FormthesetW={rj≤Ckandj∈AG2}.IfWisempty,goto
Step2.
Step7.FindajobjfromWwithaminimalduedate.Ifjobjcan
bescheduledinthekthpositionwithoutcausingtheviolationof
theconstraint,setjobjinthekthposition,Ck=max{Ck−1,rj}+pj,
k=k+1,S=S∪{j},U=˝\S,andgotoStep4.Otherwise,gotoStep
2.
Step8.Outputthejobsequence.
4.9. Computationalexperiments
Acomputationalexperimentisconductedinthissectionto
eval-uatetheperformanceofthebranch-and-boundandtheGAs.Allthe
algorithmsarecodedinFortran90andrunonapersonalcomputer
withAMDAthlon(tm)64Processor3500+,2.21GHzand1GBRAM
underWindowsXP.Theprocessingtimesaregeneratedfroma
uni-formdistributionovertheintegers1–100.Thejobreleasetimesare
generatedfromuniformdistributionsbetween0and50.5nwhere
nisthenumberofjobsandisacontrolvariable,assuggestedin
[41].Theduedateofjobjisgeneratedfromauniformdistribution overtheintegersbetweenrj+T(1−−R/2)andrj+T(1−+R/2), whererjistheduedateofjobj,Tisthetotaljobprocessingtimes, isthetardinessfactor,andRistheduedaterange.Toensurethe feasibilityoftheinstance,jobsfromagentAG2areplacedbasedon theEDDrule,anditisregeneratedifthemaximumtardinessofjobs
fromagentAG2exceedstheupperboundM.
Thecomputationalexperimentsaredividedintofourparts.The firstpartistotesttheimpactoftheduedatefactorsandRtothe
performanceofthebranch-and-boundalgorithm.Thenumberof
jobsis12,andP,theproportionofjobsfromagentAG1,is50%.The
releasetimefactoris1andtheupperboundofmaximum
tardi-nessis30n,wherenisthenumberofjobs.Eightcombinationsof (,R)valuesareused,i.e.(0.25,0.25),(0.25,0.50),(0.25,0.75),(0.5, 0.25),(0.5,0.50),(0.5,0.75),(0.75,0.25),and(0.75,0.50).Themean
andmaximumnumbersofnodesandthemeanandmaximumCPU
times(inseconds)arereportedfor thebranch-and-bound
algo-rithm. 100instances arerandomlygenerated foreach case and
theresultsarepresentedinTable1andFig.4.Itisseenthatthe
tardinessfactor ismore significantthantherange factorRto
theperformanceofthebranch-and-boundalgorithm.Problemsare
Fig.4.Themeannumbersofnodesofthebranch-and-boundalgorithmwithn=12, P=50%,=1,andM=30n.
moredifficulttosolvewhen=0.5.Moreover,thecase(,R)=(0.5,
0.75)hasthemostmeannumberofnodesamongthe8cases.
Thesecondpartoftheexperimentistotesttheimpactsofthejob releasetime(),theproportionofjobsfromagentAG1(P),andthe
upperboundofthemaximumtardiness(M)totheperformanceof
thebranch-and-boundalgorithm.Thenumberofjobsis12,andthe valueof(,R)is(0.5,0.5).Threevaluesof(0.2,1.0,3.0),ofP(0.25, 0.50,0.75)andofM(10n,30n,50n,wherenisthenumberofjobs) aretested.Asaresult,27casesareconsideredand100instances
arerandomlygeneratedforeachcase.Theresultsarepresented
inTable2andFigs.5–7.Itisseenthatthejobreleasetime()is themostsignificantfactoramongthesethreefactors.Inaddition, problemsaremoredifficulttosolvewhenthevalueofissmaller. TheproportionofjobsfromagentAG1(P)isthesecondmost
sig-nificantfactor,andproblemstendtobeharderwhenthevalueof
Pissmaller.Ontheotherhand,theupperboundofthemaximum
tardiness(M)seemstohavelittleinfluenceontheperformanceof
thebranch-and-boundalgorithm.
Thethirdpartoftheexperimentistostudytheperformance
ofthebranch-and-boundalgorithmandtheaccuracyofthethree
proposedgeneticalgorithmswhenthenumberofjobsis16.Wefix
=0.5and=0.2sinceproblemsarethemostdifficulttosolveas shownintheresultsofthefirstandthesecondpartsofthe exper-iments.Inaddition,threedifferentvaluesofR(0.25,0.5,0.75),ofP (0.25,0.5,0.75),andofM(10n,30n,50n)arechosen.Themeanand
themaximumnumbersofnodesandthemeanandthemaximum
Fig.5.Themeannumbersofnodesofthebranch-and-boundalgorithmwithn=12, =1,=0.5,andR=0.5.
Table2
Theperformanceofthebranch-and-boundalgorithmwithn=12,=0.5,andR=0.5.
P M Numberofnodes CPUtime
Mean Max Mean Max
0.20 0.25 10n 7181.83 60,051 0.060 0.469 30n 19,012.42 166,562 0.155 1.172 50n 14,122.12 159,389 0.118 1.000 0.50 10n 3253.14 21,118 0.025 0.156 30n 1210.51 27,201 0.010 0.172 50n 971.36 9635 0.008 0.063 0.75 10n 932.56 9003 0.008 0.063 30n 852.61 7611 0.005 0.063 50n 889.96 12,724 0.005 0.078 1.00 0.25 10n 339.41 4462 0.002 0.031 30n 154.73 2084 0.001 0.016 50n 230.49 3733 0.001 0.016 0.50 10n 114.85 1012 0.001 0.016 30n 121.95 776 0.001 0.016 50n 152.39 1790 0.001 0.016 0.75 10n 79.14 531 0.001 0.016 30n 89.12 736 0.001 0.016 50n 74.93 350 0.001 0.016 3.00 0.25 10n 15.71 51 0.000 0.000 30n 17.01 181 0.000 0.016 50n 14.33 28 0.001 0.016 0.50 10n 15.19 43 0.000 0.016 30n 14.42 45 0.001 0.016 50n 14.43 46 0.000 0.016 0.75 10n 14.67 37 0.001 0.016 30n 15.26 54 0.000 0.016 50n 13.94 41 0.000 0.016
CPU times(in seconds) arereported for thebranch-and-bound
algorithm,whileonlythemeanandthemaximumerror
percent-agesoftheGAsaregiven.Forinstance,theerrorpercentageofthe
solutionproducedbyGA1iscalculatedas
(V−V∗)
V∗ ×100%
whereVistheobjectivefunctionofthesequencegeneratedbyGA1
andV*istheobjectivefunctionoftheoptimalsequencefromthe
branch-and-boundalgorithm.Foreachcase,100randominstances
aregeneratedandtheresultsaregiveninTable3.Notethatthe
branch-and-boundalgorithmisterminatedifthenumberofnodes
exploredisover108,whichwasapproximately0.5hintermsof theexecutiontime.Theinstancewithnumberofnodesover108is
Fig.6.Themeannumbersofnodesofthebranch-and-boundalgorithmwithn=12, P=50%,=0.5,andR=0.5.
denotedasanasteriskinTable3.Itisobservedthatthe
branch-and-boundalgorithmcansolvemostoftheproblemswith16jobsina
reasonableamountoftime.Amongthe2700problems,thereare
only4unsolvableproblems.Acloserlookrevealsthat,amongthe threefactorsconsidered,theproportionofjobsfromagentAG1(P)
isthemostsignificantone,andproblemstendtobeharderwhen
Pissmaller.Theduedaterange(R)isthesecondsignificant,and
problemsaremoredifficultwhenRissmaller.Astothe
perfor-manceofGAs,itisnoticedthattheperformanceofallthethreeGAs isquitegood.Inaddition,itisseenthatGAwithmoredesignated initialsequencestendstohavebetteroverallsolutions.However, thereisaninstanceinwhichGA1yieldsanobjectivevalueof3but thetotaltardinessis0fortheoptimalsequence.Thus,wewould
Fig.7.Themeannumbersofnodesofthebranch-and-boundalgorithmwithn=12, M=30n,=0.5,andR=0.5.
W.-C. Lee et al. / Applied Soft Computing 12 (2012) 3580–3589 3587
Theperformanceoftheproposedalgorithmswithn=16,=0.2,and=0.5.
R P M Branch-and-boundalgorithm Errorpercentages
Numberofnodes CPUtime GA1 GA2 GA3 GA*
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
0.25 0.25 10n 7,733,551.02 88,667,901** 104.54 1256.59 0.10 4.41 0.06 2.54 0.06 2.54 0.06 2.54 30n 7,419,256.32 86,013,946 103.43 1255.38 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 7,355,622.14 83,886,484* 109.69 1246.44 0.05 5.30 0.00 0.00 0.00 0.00 0.00 0.00 0.50 10n 689,226.16 9,501,342 7.79 113.78 0.38 20.18 0.29 16.94 0.15 7.44 0.01 1.18 30n 0.00 0 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 10,060.48 594,503 0.15 8.70 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.75 10n 137,298.25 1,636,596 1.63 18.66 0.84 83.66 0.84 83.66 0.13 12.99 0.00 0.00 30n 0.00 0 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 0.00 0 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.25 10n 2,760,314.24 37,327,750 35.86 449.00 0.27 15.29 0.08 5.61 0.19 7.44 0.00 0.00 30n 1,492,637.44 33,707,790 19.67 361.89 0.02 1.56 0.02 1.56 0.02 1.56 0.02 1.56 50n 737,745.28 11,652,452 10.35 153.08 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 10n 281,446.60 4,129,379 3.44 51.52 0.53 36.40 0.55 36.40 0.21 16.07 0.16 16.07 30n 3015.02 301,502 0.04 3.94 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 3230.40 197,842 0.04 2.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.75 10n 37,923.48 513,825 0.46 6.39 0.00 0.00 0.00 0.00 0.08 8.08 0.00 0.00 30n 0.00 0 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 0.00 0 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.75 0.25 10n 906,273.05 32,881,824* 11.56 388.05 0.22 13.88 10.23 1000.00 0.18 8.47 0.05 4.55 30n 233,616.08 7,809,830 2.90 104.58 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 187,185.72 11,482,136 2.34 137.83 0.00 0.00 0.00 0.00 0.50 50.00 0.00 0.00 0.50 10n 52,821.20 696,483 0.68 9.20 0.37 35.22 2.61 200.00 0.01 0.68 0.01 0.68 30n 1254.95 94,666 0.02 1.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 43.32 3826 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.75 10n 11,824.61 282,636 0.15 3.31 0.61 56.25 0.05 5.19 0.14 8.33 0.05 5.19 30n 70.48 5561 0.00 0.09 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50n 0.00 0 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Table4
Theperformanceofthegeneticalgorithmswithn=50,=0.2,and=0.5.
R P M GA1 GA2 GA3
RDP CPUtime RDP CPUtime RDP CPUtime
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
0.25 0.25 10n 1.05 10.60 11.53 11.92 1.55 14.73 11.65 12.00 0.67 10.44 11.52 11.91 30n 0.07 5.87 5.53 12.06 0.00 0.00 5.42 12.11 0.15 8.74 5.56 12.25 50n 0.22 11.74 6.79 11.98 0.13 11.74 6.66 12.17 0.09 5.18 6.81 12.13 0.50 10n 2.66 38.94 11.39 11.80 2.73 51.34 11.50 12.02 1.65 51.34 11.33 11.81 30n 0.00 0.00 0.16 0.25 0.00 0.00 0.00 0.02 0.00 0.00 0.15 0.25 50n 0.00 0.00 0.15 0.23 0.00 0.00 0.00 0.00 0.00 0.00 0.15 0.27 0.75 10n 1.99 56.07 10.88 11.83 1.57 57.10 10.90 11.81 1.21 28.87 10.72 11.69 30n 0.00 0.00 0.05 0.11 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.16 50n 0.00 0.00 0.04 0.08 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.11 0.50 0.25 10n 3.35 79.15 10.95 12.06 3.48 110.54 11.05 12.22 0.50 12.44 10.89 12.06 30n 0.00 0.00 0.88 12.05 0.00 0.00 0.63 12.06 0.00 0.00 0.76 12.05 50n 0.00 0.00 1.18 11.98 0.00 0.00 0.85 11.98 0.00 0.00 1.08 11.84 0.50 10n 2.17 108.71 6.29 11.69 2.46 96.83 6.27 11.89 0.90 46.24 6.00 11.61 30n 0.00 0.00 0.18 0.34 0.00 0.00 0.00 0.03 0.00 0.00 0.12 0.27 50n 0.00 0.00 0.15 0.25 0.00 0.00 0.00 0.02 0.00 0.00 0.11 0.22 0.75 10n 0.52 27.97 2.58 11.53 0.00 0.00 2.29 11.61 1.01 54.09 2.12 11.41 30n 0.00 0.00 0.06 0.16 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.13 50n 0.00 0.00 0.05 0.09 0.00 0.00 0.00 0.02 0.00 0.00 0.06 0.11 0.75 0.25 10n 1.61 127.27 4.53 12.16 0.78 47.91 4.53 12.53 0.26 22.71 4.24 12.16 30n 0.00 0.00 0.53 12.00 0.00 0.00 0.34 11.91 0.00 0.00 0.27 11.89 50n 0.04 4.00 0.49 12.13 0.00 0.00 0.14 12.00 0.04 4.00 0.26 12.05 0.50 10n 0.08 7.52 1.31 11.44 0.03 2.62 1.21 11.63 0.00 0.00 0.80 11.41 30n 0.00 0.00 0.23 0.48 0.00 0.00 0.01 0.27 0.00 0.00 0.07 0.22 50n 0.00 0.00 0.16 0.31 0.00 0.00 0.00 0.03 0.00 0.00 0.06 0.14 0.75 10n 0.00 0.00 0.49 1.38 0.00 0.00 0.26 4.13 0.00 0.00 0.06 0.36 30n 0.00 0.00 0.09 0.25 0.00 0.00 0.01 0.11 0.00 0.00 0.04 0.08 50n 0.00 0.00 0.06 0.11 0.00 0.00 0.00 0.02 0.00 0.00 0.04 0.06
recommend to use the best sequence from three GAs,
GA*=min{GA1,GA2,GA3},astheapproximatesolution,sincethey
areallfinishedwithinasecondanditsmeanerrorpercentagesare
lessthan0.2%forallthetestedcases.
Thelastpartofthecomputationalexperimentsistotestthe
performanceofGAswhenthenumberofjobsislarge.The
num-berofjobsissetat50and100.Asetof100instancesistested,
andtheresultsarepresentedinTables4and5.Themeanandthe
Table5
Theperformanceofthegeneticalgorithmswithn=100,=0.2,and=0.5.
R P M GA1 GA2 GA3
RDP CPUtime RDP CPUtime RDP CPUtime
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
0.25 0.25 10n 2.72 14.55 73.68 76.53 4.13 44.99 73.76 75.97 1.24 9.41 73.69 76.33 30n 0.19 15.79 24.28 75.70 0.25 12.31 22.96 75.31 0.22 15.79 23.69 75.92 50n 3.89 216.67 20.89 76.47 0.00 0.00 18.08 74.33 0.00 0.00 20.46 75.95 0.50 10n 3.44 40.27 72.93 75.45 2.80 40.27 72.50 75.55 1.77 31.56 72.11 75.17 30n 0.00 0.00 1.26 1.73 0.00 0.00 0.00 0.05 0.00 0.00 1.18 1.84 50n 0.00 0.00 1.13 1.55 0.00 0.00 0.00 0.02 0.00 0.00 1.05 1.59 0.75 10n 2.59 36.81 70.56 75.08 1.61 48.99 69.80 74.70 1.95 36.81 69.63 74.55 30n 0.00 0.00 0.56 1.23 0.00 0.00 0.00 0.02 0.00 0.00 0.53 0.95 50n 0.00 0.00 0.45 0.69 0.00 0.00 0.00 0.02 0.00 0.00 0.48 0.77 0.50 0.25 10n 13.64 813.85 71.68 76.97 13.00 726.15 71.54 76.33 5.14 97.06 71.29 76.48 30n 0.00 0.00 3.08 4.66 0.00 0.00 1.59 4.39 0.00 0.00 1.88 3.48 50n 0.00 0.00 2.76 5.02 0.00 0.00 0.08 0.23 0.00 0.00 1.75 3.44 0.50 10n 2.52 90.38 26.04 74.42 3.72 90.38 23.81 73.97 7.68 432.82 22.28 73.95 30n 0.00 0.00 1.53 2.22 0.00 0.00 0.00 0.00 0.00 0.00 0.87 1.52 50n 0.00 0.00 1.19 1.63 0.00 0.00 0.00 0.02 0.00 0.00 0.74 1.19 0.75 10n 0.02 1.54 5.26 70.39 0.00 0.00 2.51 69.06 0.00 0.00 1.34 70.03 30n 0.00 0.00 0.80 1.50 0.00 0.00 0.00 0.00 0.00 0.00 0.39 0.92 50n 0.00 0.00 0.48 0.78 0.00 0.00 0.00 0.02 0.00 0.00 0.36 0.77 0.75 0.25 10n 2.13 124.38 21.55 77.30 1.90 107.72 21.00 75.75 0.19 7.38 17.48 76.13 30n 0.00 0.00 3.07 5.25 0.00 0.00 2.31 5.98 0.00 0.00 0.82 2.00 50n 0.00 0.00 2.68 7.19 0.00 0.00 0.06 0.23 0.00 0.00 0.74 3.72 0.50 10n 0.00 0.00 4.77 17.44 0.00 0.00 2.45 7.20 0.00 0.00 0.58 2.78 30n 0.00 0.00 1.95 2.89 0.00 0.00 0.05 2.86 0.00 0.00 0.43 1.48 50n 0.00 0.00 1.28 2.02 0.00 0.00 0.01 0.05 0.00 0.00 0.36 0.78 0.75 10n 0.00 0.00 3.41 5.69 0.00 0.00 0.98 4.66 0.00 0.00 0.22 0.97 30n 0.00 0.00 1.13 2.06 0.00 0.00 0.51 27.25 0.00 0.00 0.18 0.50 50n 0.00 0.00 0.60 1.25 0.00 0.00 0.01 0.08 0.00 0.00 0.18 0.39
maximumrelativedeviationpercentages,andthemeanandthe
maximumCPUtime(insecond)arerecorded.Therelative
devia-tionpercentage(RDP)ofthesolutionproducedbyGAiiscalculated
as
(Vi−min{V1,V2,V3})
min{V1,V2,V3} ×
100%
fori=1,2,3whereViistheobjectivevalueofthesequencefrom
GAi.ItisobservedthattheexecutiontimesoftheGAsareaboutthe
same.ItisseenthatGA3hasthebestoverallperformanceandthe
trendbecomesmoresignificantwhenthenumberofjobsincreases.
TheexecutiontimesoftheGAsareaboutthesame.Ittakesabout
12sforaninstanceof50jobsand75sforaninstanceof100jobs.
5. Conclusion
Inthispaper,westudiedatwo-agentschedulingproblemon
asinglemachinewithreleasetime.Theobjectiveistominimize
thetotaltardinessofjobsfromthefirstagentgiventhatthe
max-imumtardiness ofjobsfromthesecondagentcannot exceeda
givenupperbound.Computationalresultsshowthatthe
branch-and-boundalgorithmcouldsolvemostoftheproblemswith16
jobswithinareasonableamountoftime.Inaddition,itshowsthat
theperformanceofthecombinedgeneticalgorithmisverygood
withmean errorpercentagesof lessthan0.2%for allthecases.
Consideringobjectivefunctions,otherthanthetwoexaminedin
thispaperorextendingthesingle-machinecasetoothermachine
environmentswouldbeaninterestingtopicforfutureresearch.
Acknowledgements
Theauthorsaregratefultotheeditorandthereferees,whose
constructivecommentshaveledtoasubstantialimprovementin
thepresentationofthepaper.ThisworkwassupportedbytheNSC
ofTaiwan,ROC,underNSC100-2221-E-035-029-MY3.
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