Large production line optimisation using simulated annealing

21 Pages·2002·0.19 MB·English

Large production line optimisation using simulated annealing
(cid:1)y
DiomidisSpinellis
z
ChrissoleonPapadopoulos
x
J.MacGregorSmith
(cid:1)
November20,2000
Abstract 1 Introduction and literature review
Alargeamountofresearchhasbeendevotedtotheanaly-
sis and design of production lines. A lot of this research
Wepresentarobustgeneralisedqueuingnetworkalgorithm
concerns the design of manufacturing systems with con-
as an evaluative procedure for optimising production line
siderableinherentvariabilityintheprocessingtimesatthe
configurationsusingsimulatedannealing. Wecomparethe
various stations, a common situation with human opera-
resultsobtainedwithouralgorithmtothoseofotherstudies
tors/assemblers. The literature on the modellingand opti-
and findsome interestingsimilarities but also strikingdif-
misationof productionlines is vast, allowingus toreview
ferencesbetweenthemintheallocationofbuffers,numbers
only the most directly relevant studies. For a systematic
ofservers,andtheirservicerates.Whilecontextdependent,
classification ofthe relevantworkson thestochastic mod-
thesepatternsofallocationareoneofthemostimportantin-
elling of these and other types of manufacturing systems
sightswhichemergeinsolvingverylongproductionlines.
(e.g.,transferlines,flexiblemanufacturingsystems(FMS),
The patterns, however, are often counter-intuitive, which
andflexibleassemblysystems(FAS)),theinterestedreader
underscoresthedifficultyoftheproblemwe address. The
isreferredtoareviewpaperbyPapadopoulosandHeavey
mostinterestingfeatureofouroptimisationprocedureisits
(1996) and some recently published books, such as those
boundedexecutiontime,whichmakesitviableforoptimis-
byAskinandStandridge(1993),BuzacottandShanthiku-
ingverylongproductionlineconfigurations. Basedonthe
mar(1993),Gershwin(1994),Papadopoulosetal. (1993),
boundedexecutiontime property,we have optimisedcon-
ViswanadhamandNarahari(1992),andAltiok(1997).
figurationsofupto60stationswith120buffersandservers
Therearetwobasicproblemclasses:
inlessthanfivehoursofCPUtime.
1. the evaluation of production line performance mea-
Keywords:BufferAllocation,Non-linear,Stochastic,In-
suressuchasthroughput,meanflowtime,workstation
teger,NetworkDesign,SimulatedAnnealing
meanqueuelength,andsystemutilisation,and
2. the optimisation of the decision variables of these
lines.
InternationalJournalofProductionResearch,38(3):509–541,Febru- Examples of decision variables that have been considered
(cid:1)
ary2000.
are:
Thisisamachine-readablerenderingofaworkingpaperdraftthatled
y
toapublication. Thepublicationshouldalwaysbecitedinpreferenceto 1. the sizes of the buffers placed between successive
this draft using the reference inthe previous footnote. Thismaterial is
workstationsofthelines,
presentedtoensuretimelydisseminationofscholarlyandtechnicalwork.
Copyrightandallrightsthereinareretainedbyauthorsorbyothercopy-
2. the number of servers allocated to each workstation,
rightholders.Allpersonscopyingthisinformationareexpectedtoadhere
and
tothetermsandconstraintsinvokedbyeachauthor’scopyright. Inmost
cases,theseworksmaynotberepostedwithouttheexplicitpermissionof
3. theamountofworkloadallocatedtoeachworkstation.
thecopyrightholder.
DepartmentofInformationandCommunicationSystems,University
z The corresponding optimisation problems are named, re-
oftheAegean,83200KarlovasiSamos,Greece. (Correspondingauthor);
spectively,(1)thebufferallocationproblem,(2)theserver
contact address: Myrsinis 1, GR-145 62Kifisia, GREECE.Tel. +301
2719710,Fax[phone],email:[email protected] allocationproblem, and(3) the workloadallocationprob-
Department of Business Administration, University of the Aegean, lem,inaproductionline.
x
82100Chios,Greece.email:[email protected]
InPapadopoulosetal. (1993)bothevaluativeandgener-
Department of Mechanical and Industrial Engineering, Univer-
(cid:2) ative(optimisation)modelsaregivenformodellingthevar-
sity of Massachusetts, Amherst Massachusetts 01003, USA. email:
[email protected] ious typesof manufacturingsystems. This workfalls into
1
the second category. Evaluative and optimisation models Yamazakietal.,1992),amongothers,andtheserveralloca-
canbecombinedbyclosingtheloopbetweenthem;thatis, tionproblem(MagazineandStecke, 1996, Hillier andSo,
onecan use feedbackfroman evaluativemodelto modify 1989). HillierandSo(1995)studiedvariouscombinations
thedecisionstakenbytheoptimisationmodel. of these three design problems. Other references may be
Oneofthekeyquestionsthatthedesignersfaceinase- foundtherein(BuzacottandShanthikumar,1992,Buzacott
rialproductionlineisthebufferallocationproblem(BAP), andShanthikumar,1993),amongothers.
i.e., how muchbufferstorage to allow andwhere to place The present work deals with the same design problems
it within the line. This is an important question because (buffer allocation, server allocation, and workload alloca-
bufferscanhaveagreatimpactontheefficiencyofthepro- tion)butforlongproductionlineswithmulti-machinesta-
duction line. They compensate for the blocking and the tions.
starving of the line’s stations. For this reason, the buffer
As the problem being investigated is combinatorial in
allocation problem has received a lot more attention than
nature, traditional OperationsResearch techniquesare not
theothertwodesignproblems. Bufferstorageisexpensive
as practical for obtaining optimal solutions for long pro-
duebothtoitsdirectcost,andtotheincreaseofthework-
ductionlines. We proposea simulatedannealing(SA)ap-
in-process(WIP)inventories. Inaddition,therequirement
proachasthesearchmethodinconjunctionwiththeexpan-
tolimitthebufferstoragecanalsobearesultofspacelim-
sionmethoddevelopedbyKerbacheandMacGregorSmith
itationsintheshopfloor. TheliteratureontheBAP is ex-
(1987) as the evaluative tool. Simulated annealing is an
tensive. Asystematicclassificationoftheresearchworkin
adaptationofthesimulationofphysicalthermodynamican-
thisareaisgiveninSinghandMacGregorSmith(1997)and
nealingprinciplesdescribedbyMetropolisetal. (1953)to
Papadopoulosetal. (1993). Theworksaresplitaccording
thecombinatorialoptimisationproblems(Kirkpatricketal.,
to:
1983,Cerny,1985).Similartogeneticalgorithms(Holland,
1975,Goldberg,1989)andtabusearchtechniques(Glover,
Themethodusedtosolvethebufferallocationproblem
1990)itfollowsthe‘localimprovement’paradigmforhar-
searchmethods,(AltiokandStidham,1983,Smithand
nessingtheexponentialcomplexityofthesolutionspace.
Daskalaki, 1988, Seong et al., 1995, Hillier and So,
1991a,HillierandSo,1991b),dynamicprogramming Thealgorithmisbasedonrandomisationtechniques.An
methods, (Kubat and Sumita, 1985, Jafari and Shan- overview of algorithms based on such techniques can be
found in the survey by Gupta et al. (1994). A complete
thikumar,1989,YamashitaandAltiok,1997),among
others,andsimulationmethods,(Conwayetal.,1988, presentationofthemethodanditsapplicationsisdescribed
Hoetal.,1979). by Van Laarhoven and Aarts (1987) and accessible algo-
rithmsforitsimplementationarepresentedbyCoranaetal.
Thetypeofproductionline balanced/unbalanced,(Pow- (1987)andPressetal. (1988). Acriticalevaluationofdif-
ell(1994)presentsaliteraturereviewaccordingtothis ferentapproachestoannealingschedulesandothermethod
scheme), or reliable/unreliable; the majority of pa- optimisationsaregivenbyIngber(1993). Asatoolforop-
pers deal with reliable lines. Only a few algorithms erationalresearchSAispresentedbyEglese(1990),while
have been developed to calculate the performance Koulamas et al. (1994) provide a complete survey of SA
measures of unreliable productionlines (Glassey and applicationstooperationsresearchproblems.
Hong,1983,ChoongandGershwin,1987,Gershwin, The use of the Simulated Annealing algorithm appears
1989, Heavey et al., 1993). Hillier and So (1991a) tobeapromisingapproach. Webelievethatthisalgorithm
and Seong, Chang, and Hong (1995)have dealt with couldbeappliedin conjunctionwith a fast decomposition
thebufferallocationprobleminunreliableproduction algorithmtosolveefficientlyandaccuratelytheaforemen-
lines. tioned optimisation problems in much longer production
lines.
Thenumberofmachinesavailableateachworkstation
The remainderof the paperis organisedas follows: we
Similarly, few researchers have dealt with the buffer
firstdescribetheproductionlinemodelandtheproblemof
allocation problem in production lines with multiple
ourinterest followedby the methodologyof ourapproach
(parallel) machines at each workstation (Hillier and
namely: the performance model, the expansion method
So, 1995, Magazine and Stecke, 1996, Singh and
used for evaluating the line performance, an overview of
Smith,1997).
thecombinatorialoptimisationmethods,thesimulatedan-
Apart fromthe bufferallocationproblem,the othertwo nealingoptimisationmethod,andthecompleteenumeration
interesting design problems have also been considered by method; we then describe our experimental methodology
someresearchers,e.g.,theworkallocationproblem(Hillier andpresentanoverviewofthenumericalandperformance
andBoling,1979,HillierandBoling,1966,HillierandBol- resultsforshortandlongproductionlines.IntheAppendix
ing, 1977, Ding and Greenberg, 1991, Huang and Weiss, following the concluding section, we provide a full tabu-
1990,Shanthikumaretal.,1991,w.WanandWolff,1993, latedsetoftheexperimentalresults.
2
andinteger ,
M1 B2 M2 B3 … BN M(cid:1) qi (cid:2)(cid:6) (cid:4)i(cid:2)(cid:3)(cid:1)(cid:7)(cid:1)(cid:2)(cid:2)(cid:2)(cid:1)N(cid:5)
andinteger ,
si si (cid:2)(cid:1) (cid:4)i(cid:2)(cid:1)(cid:1)(cid:3)(cid:1)(cid:2)(cid:2)(cid:2)(cid:1)N(cid:5)
Mi : Station i consisting of si servers ,
wi (cid:3)(cid:6)(cid:4)i(cid:2)(cid:1)(cid:1)(cid:3)(cid:1)(cid:2)(cid:2)(cid:2)(cid:1)N(cid:5)
where and are fixed constants and , and are
Bi : Buffer i thedecQisi(cid:1)oSnvectoNrs. Thethirdconstraintindqicsatesthwatthe
sumoftheexpectedservicetimesisafixedconstant,which
bynormalisationcanbeequalto .
Figure1:An -workstationmulti-machineproductionline The objectivefunction of throNughput, , is not
with inNtermediatebuffers theonlyperformancemeasureofinterest.RTh(cid:4)eq(cid:1)asv(cid:1)ewra(cid:5)geWIP,
N (cid:1)(cid:1) the flow time, the cycle time, the system utilisation, the
2 The production line model and the averagequeuelengths andothermeasures are equallyim-
portantperformancemeasures. However,throughputisthe
optimal design problem
mostcommonlyusedperformancemeasureintheinterna-
tionalliterature.
Wedefineanasynchronouslineasoneinwhicheverywork-
station can pass parts on when its processing is complete
as longas bufferspace is available. Such a line is subject 3 Optimal allocation methodology
to manufacturing starving and blocking. We assume that
thefirststationisneverstarvedandthelaststationisnever 3.1 Performance models
blocked. Therefore we can say that the line operates in a
The queuing model that we use refers to a
pushmode: partsarealwaysavailablewhenneededatthe
queuingsystemwhereM: (cid:4)M(cid:4)C(cid:4)K
first workstation and space is always available at the last
workstationtodisposeofcompletedparts.
itisassumedthatthearrivalsaredistributedaccording
An -station line consists of workstations in series,
(cid:3) tothePoissondistribution(orequivalentlythatthein-
labelleNd and N locationsforbuffers,
termediate times between two successive arrivals are
labelledM(cid:1)(cid:1)M(cid:2)(cid:1)(cid:2)(cid:2)(cid:2)(cid:1)MN, is illNust(cid:1)ra(cid:1)ted in figure 1. Each
exponentiallydistributed),
station Bha(cid:2)s(cid:1)B(cid:3)(cid:1)s(cid:2)e(cid:2)r(cid:2)v(cid:1)eBrsNoperating in parallel. The buffer
capacitiies of sthie intermediate buffers , , the service (processing)times follow the exponential
are denoted by , whereas the mean sBerivicie(cid:2)tim(cid:3)e(cid:1)s(cid:2)(cid:2)o(cid:2)f(cid:1)tNhe (cid:3) distribution,
stations( qi )aredenotedby .
i The maiin(cid:2)(cid:1)p(cid:1)e(cid:2)rf(cid:2)o(cid:2)r(cid:1)mNance measure owf ithe production thereare parallelservers(machineswhichareiden-
line is the mean throughput, denoted by , (cid:3) ticalateaCchworkstation),and
where , R(cid:4)aqn(cid:1)sd(cid:1)w(cid:5)
the total capacity of the system is finite and equal to
q(cid:2) (cid:4)q(cid:2)(cid:1)q(cid:3)(cid:1).(cid:2)(cid:2)(cid:2)(cid:1)qN(cid:5) s(cid:2) (cid:4)s(cid:1)(cid:1)s(cid:2)(cid:1)(cid:2)(cid:2)(cid:2)(cid:1)sN(cid:5) w(cid:2)
(cid:3) .
(cid:4)wI(cid:1)f(cid:1)w(cid:2)d(cid:1)e(cid:2)n(cid:2)o(cid:2)t(cid:1)ewsNth(cid:5)etotalnumberofavailablebufferslotsto
K
beallQocatedtothe buffersand thetotalnumberof While our focus in this paper is on ap-
availableservers(mNac(cid:1)hi(cid:1)nes)to allocaStedto the stations proximationsfor openqueuingnetworksMof(cid:4)sMeri(cid:4)eCs-(cid:4)pKarallel
thenthegeneralversionoftheoptimisationmodNel(firstre- topologies, we also briefly discuss some of the available
portedbyHillierandSo,1995)is: approachesused for modelling systems since
mostoftheliteraturehasfocusedMo(cid:4)nM(cid:4)(cid:1)(cid:4)K systems.
(1) BothopenandclosedsystemshavMeb(cid:4)eMen(cid:4)(cid:1)st(cid:4)uKdiedbyex-
maxR(cid:4)q(cid:1)s(cid:1)w(cid:5)
subjectto: actanalysisalthoughresultshavebeenlimited.Exactanal-
ysesofopentwo,three,andfournode-servermodelswith
exponentialservicearelimited bytheexplosivegrowthof
N (2) theMarkovchainmodelsforanalysingthesesystems. The
analysisofverylargeMarkovchainmodelshasledtoeffec-
qi (cid:2)Q(cid:1)
i(cid:4)(cid:2) tive aggregation techniques for these models (Schweitzer
XN (3) and Altiok, 1989, Takahashi, 1989) but the computation
si (cid:2)S(cid:1) timeandpowerrequiredfortheseexactresultsleavesopen
i(cid:4)(cid:1) theneedforapproximationtechniques.
X
N (4) Van Dijk and his co-authors (1988, 1989) have devel-
wi (cid:2)N(cid:1) oped some bounding methodologies for both
i(cid:4)(cid:1) M(cid:4)M(cid:4)(cid:1)(cid:4)K
X
3
and systemsandhavedemonstratedtheiruse-
fulneMss(cid:4)iMnt(cid:4)hCe(cid:4)dKesignofsmallqueuingnetworks.Ofcourse, M/M/C/K M/M/C/K
whendoingoptimisationofmediumandlongqueuingnet- Λ i j R
works, bounds can be far off the optimum, so robust ap-
proximation techniques close to the optimal performance
measuresaremostdesirable.
Most approximation techniques appearing in the litera- P'K
turerelyondecomposition/aggregationmethodstoapprox-
imateperformancemeasures. Oneandtwonodedecompo-
M/M/C/K M/M/∞ M/M/C/K
sitionsofthenetworkhavebeencarriedout,allwithvary-
ingdegreeofsuccess. Λ i h j R
PK
Thefewapproximationapproachesavailableintheliter-
ature can be classified as follows: Isolation methods, Re-
peated Trials, Node-by-node decomposition, and Expan-
sionmethods. IntheIsolationmethod,thenetworkissub- 1 - PK
divided into smaller subnetworks and then studied in iso-
lation(LabetoulleandPujolle,1980,BoxmaandKonheim,
Figure2:TypeIBlockinginFiniteQueues
1981).ThismethodwasusedbyKuehn(1979)andGelenbe
andPujolle(1976)buttheyfailedtoconsidernetworkswith
finitecapacity.
Smith,1997).
Closely related to the Isolation method is the Repeated
Trials Method, a class of techniques based upon repeat-
3.2 Expansionmethod
edlyattemptingtosendblockedcustomerstoaqueuecaus-
ingtheblocking(CaseauandPujolle,1979,Fredericksand
TheExpansionMethodisarobustandeffectiveapproxima-
Reisner,1979,Fredericks,1980).
tion technique developed by Kerbache and Smith (1987).
In Node-by-nodedecomposition, the network is broken As described in previous papers, this method is charac-
down into single, pairs, and triplets of nodes with aug- terised as a combinationof Repeated Trials and Node-by-
mentedserviceandarrivalparameterswhicharethenstud- node Decomposition solution procedures. Methodologies
ied separately (Hillier and Boling, 1967, Takahashi et al., for computing performance measures for a finite queuing
1980, Altiok, 1982, Altiok and Perros, 1986, Perros and networkuseprimarilythefollowingtwokindsofblocking:
Altiok, 1986, Brandwajn and Jow, 1988). More general
service time approximations appear in the work by Gun TypeI: The upstream node gets blocked if the service
and Makowski (1989). The Expansion method is the ap- onacustomeriscompletiedbutitcannotmovedown-
proach argued for in this paper for computing the perfor- streamduetothequeueatthedownstreamnode be-
mance measures of finite queuing networks ingfull.ThisissometimesreferredtoasBlockingjAf-
(Kerbache,1984,KerMba(cid:4)cMhe(cid:4)aCnd(cid:4)KSmith,1987,Kerbacheand terService(BAS)(Onvural,1990).
Smith, 1988). It can be characterised conceptually as a
combinationofRepeatedTrialsandNode-by-NodeDecom- TypeII: The upstream node is blocked when the down-
positionwherethekeydifferenceisthata‘holding’nodeis stream node becomes saturated and service must be
added to the network to register blocked customers. The suspended on the upstream customer regardless of
addition of the holdingnode ‘expands’the network. This whetherserviceiscompletedornot.Thisissometimes
approach transforms the queuing network into an equiva- referredtoas BlockingBeforeService(BBS) (Onvu-
lent Jackson network which is then decomposed allowing ral,1990).
for each node to be solved independently. We have suc-
cessfullyusedtheExpansionMethodtomodel The Expansion Method uses Type I blocking, which
(KerbacheandSmith,1988), (JainMa(cid:4)nMdS(cid:4)m(cid:1)(cid:4)itKh, is prevalent in most production and manufacturing, trans-
1994, Han and Smith, 1992)M, (cid:4)M(cid:4)C(cid:4)K (Kerbache and portationandothersimilarsystems.
Smith, 1987), and most recentMly(cid:4)G(cid:4)(cid:1)(cid:4)K (Cheah and Considerasinglenodewithfinitecapacity (including
Smith,1994,Smith,1991,Smith,M19(cid:4)9G4)(cid:4)qCu(cid:4)eCuesandqueu- service).ThisnodeessentiallyoscillatesbetweKentwostates
ingnetworks.Inaddition,wehavealsousedourExpansion — the saturated phase and the unsaturated phase. In the
Methodologytomodelrouting(DaskalakiandSmith,1989, unsaturatedphase,node hasatmost customers(in
Daskalaki and Smith, 1986, Gosavi and Smith, 1990)and serviceorinthequeue).jOntheotherhKan(cid:1)d,(cid:1)whenthenode
optimalresourceallocationproblems(SmithandDaskalaki, issaturatednomorecustomerscanjointhequeue.Referto
1988, Smith, 1991, Smith and Chikhale, 1995, Singh and figure2foragraphicalrepresentationofthetwoscenarios.
4
The Expansion Method consists of the following three 3.2.2 StageII:parameterestimation
stages:
Thisstageessentiallyestimatestheparameters , and
(cid:1)
StageI:NetworkReconfiguration. utilisingknownresultsforthe pmKodpelK.
(cid:3) (cid:6)h M(cid:4)M(cid:4)C(cid:4)K
StageII:ParameterEstimation. : Analytical results from the model
(cid:3) (cid:3) pprKovidethefollowingexpressionfMor(cid:4)M:(cid:4)C(cid:4)K
StageIII:FeedbackElimination. pK
(cid:3)
The following notation defined by Kerbache and
K (5)
Smith (1987, 1988)shall be used in furtherdiscussion re- (cid:1) (cid:5)
gardingthismethodology: pK (cid:2) cK(cid:2)cc(cid:11) (cid:6) p(cid:5)
wherefor (cid:1) (cid:2)
(cid:4)(cid:5)(cid:4)c(cid:6)(cid:5)(cid:2)(cid:1)(cid:5)
The holding node established in the Expansion
h (cid:8)m(cid:2)ethod.
c(cid:2)(cid:1) n c K(cid:2)c(cid:6)(cid:1) (cid:2)(cid:1)
:=ExternalPoissonarrivalratetothenetwork. (cid:1) (cid:5) (cid:4)(cid:5)(cid:4)(cid:6)(cid:5) (cid:1)(cid:1)(cid:4)(cid:5)(cid:4)c(cid:6)(cid:5)
(cid:9) p(cid:5) (cid:2) (cid:12)
:=Poissonarrivalratetonode . (cid:3)n(cid:4)(cid:5)n(cid:11)(cid:1)(cid:6)(cid:2) c(cid:11) (cid:1)(cid:1)(cid:5)(cid:4)c(cid:6) (cid:4) (6)
X
(cid:5)j j andfor ,
:=Effectivearrivalratetonode . (cid:4)(cid:5)(cid:4)c(cid:6)(cid:2)(cid:1)(cid:5)
(cid:5)(cid:10)j j
:=Exponentialmeanservicerateatnode .
(cid:6)j j c(cid:2)(cid:1) n c (cid:2)(cid:1) (7)
(cid:1) (cid:5) (cid:4)(cid:5)(cid:4)(cid:6)(cid:5)
:=Effectiveservicerateatnode duetoblocking. p(cid:5) (cid:2) (cid:12) (cid:4)K(cid:1)c(cid:12)(cid:1)(cid:5)
(cid:6)(cid:10)j j (cid:3)n(cid:4)(cid:5)n(cid:11)(cid:1)(cid:6)(cid:2) c(cid:11) (cid:4)
:=Blockingprobabilityoffinitequeueofsize . : SXince there is no closed form solution for this
(cid:1)
pK K (cid:3) qpuKantityanapproximationisusedgivenbyLabetoulle
:=Feedback blocking probability in the Expansion
(cid:1) and Pujolle obtained using diffusion techniques (La-
pK method.
betoulleandPujolle,1980).
:=Unconditional probability that there is no customer
j
p(cid:5) intheservicechannelatnode (eitherbeingservedor
beingheldafterservice). j
K K K(cid:2)(cid:1) K(cid:2)(cid:1) (cid:2)(cid:1)
:=Meanthroughputrate. (cid:1) (cid:6)j (cid:12)(cid:6)h (cid:5)(cid:13)(cid:4)r(cid:2) (cid:1)r(cid:1) (cid:5)(cid:1)(cid:4)r(cid:2) (cid:1)r(cid:1) (cid:5)(cid:14)
R pK (cid:2)(cid:3) (cid:6)h (cid:1) (cid:6)h(cid:13)(cid:4)r(cid:2)K(cid:6)(cid:1)(cid:1)r(cid:1)K(cid:6)(cid:1)(cid:5)(cid:1)(cid:4)r(cid:2)K (cid:1)r(cid:1)K(cid:5)(cid:14)(cid:4)(8)
:=Servicecapacity(buffer)atnode ,i.e. fora
where and aretherootstothepolynomial:
Sj singlequeue. j S (cid:2)K
r(cid:1) r(cid:2)
(9)
(cid:2)
3.2.1 StageI:networkreconfiguration (cid:5)(cid:1)(cid:4)(cid:5)(cid:12)(cid:6)h(cid:12)(cid:6)j(cid:5)x(cid:12)(cid:6)hx (cid:2)(cid:6)
while, and and are the
(cid:1)
Usingtheconceptoftwophasesatnode ,anartificialnode actualarri(cid:5)val(cid:2)rat(cid:5)ejst(cid:1)ot(cid:5)hhe(cid:4)fi(cid:1)n(cid:1)itepaKn(cid:5)dartifi(cid:5)cijalhold(cid:5)ihngnodes
is added for each finite node in the njetwork to register respectively.
bhlocked customers. Figure 2 shows the additional delay, Infact, thearrivalratetothefinitenodeisgivenby:
causedtocustomerstryingtojointhequeueatnode when (cid:5)j
it is full,withprobability . Thecustomerssuccejssfully (10)
join queue with a probabpiKlity . Introductionof (cid:5)j (cid:2)(cid:5)(cid:10)i(cid:4)(cid:1)(cid:1)pK(cid:5)(cid:2)(cid:5)(cid:10)i(cid:1)(cid:5)h
anartificialjnodealsodictatesthe(cid:4)(cid:1)ad(cid:1)diptiKon(cid:5)ofnewarcswith Letusexaminethefollowingargumenttodeterminethe
and astheroutingprobabilities. service time at the artificial node. If an arrivingcustomer
pKThe b(cid:4)l(cid:1)oc(cid:1)kepdKc(cid:5)ustomerproceedsto the finite queuewith is blocked, the queue is full and thus a customer is being
probability onceagainafterincurringadelayatthe serviced, so thearrivingcustomertothe holdingnodehas
(cid:1)
artificialno(cid:4)d(cid:1)e(cid:1). IpfKth(cid:5)equeueisstillfull,itisre-routedwith to remain in service at the artificial holding node for the
probability totheartificialnodewhereitincursanother remainingservicetimeintervalofthecustomerinservice.
(cid:1)
delay.ThispprKocesscontinuestillitfindsaspaceinthefinite Thedelaydistributionofablockedcustomerattheholding
queue. A feedback arc is used to model the repeated de- nodehasthesamedistributionastheremainingservicetime
lays.Theartificialnodeismodelledasan queue. ofthecustomerbeingservicedatthenodedoingtheblock-
The infinite numberof servers is used simMpl(cid:4)yMto(cid:4)(cid:4)serve the ing.Usingrenewaltheory,onecanshowthattheremaining
blockedcustomeradelaytimewithoutqueuing. servicetimedistributionhasthefollowingrate :
(cid:6)h
5
forsolvingequation17with usedasadummyparameter
(11) forsimplicityofthesolution.zLastly,equation21givesthe
(cid:3)(cid:6)j
(cid:6)h (cid:2) (cid:2) (cid:2) approximationtotheblockingprobabilityderivedfromthe
(cid:1)(cid:12)(cid:7) (cid:6)j exactmodelforthe queue. Hence,weessen-
where, is the service time variance given by Klein-
rock(1975(cid:7))(cid:2). Notice that if theservice time distributionat tiallyhavefiveequatMio(cid:4)nMsto(cid:4)Cso(cid:4)lKve,viz.14to17and21. To
recapitulate, we first expandthe networkwith an artificial
thefinitequeuedoingtheblockingisexponentialwithrate
holdingnode;thisstageisthenfollowedbytheapproxima-
,then:
(cid:6)j tion of the routing probabilities, due to blocking, and the
(cid:6)h (cid:2)(cid:6)j service delay in the holding node; and, finally, the feed-
the service time at the artificial nodeis also exponentially
backarcattheholdingnodeiseliminated.Oncethesethree
distributedwithrate .
stages are complete, we have an expandednetworkwhich
(cid:6)j
canthenbeusedtocomputetheperformancemeasuresfor
3.2.3 StageIII:feedbackelimination the original network. As a decomposition technique this
approachallows successive additionof a holdingnodefor
Duetothefeedbacklooparoundtheholdingnode,thereare
everyfinite node, estimation of the parameters and subse-
strong dependencies in the arrival processes. Elimination
quenteliminationoftheholdingnode.
ofthesedependenciesrequiresreconfigurationofthehold-
ingnodewhichisaccomplishedbyrecomputingtheservice
time atthenodeandremovingthefeedbackarc. Thenew 3.3 Combinatorialoptimisationmodels
servicerateisgivenby:
TheBufferAllocationProblem(BAP) is perhapsbestfor-
(cid:1) (cid:1) (12) mulated as a non-linear multiple-objective programming
(cid:6)h (cid:2)(cid:4)(cid:1)(cid:1)pK(cid:5)(cid:6)h problemwherethedecisionvariablesaretheintegers. Not
Theprobabilitiesofbeinginanyofthetwophases(satu-
onlyistheBAPadifficult hardcombinatorialoptimi-
ratedorunsaturated)are and .Themeanservice
timeatanodeiprecedingpKthefini(cid:4)t(cid:1)e(cid:1)npodKe(cid:5)is wheninthe sationproblem,itismadeNalPlt(cid:1)hemoredifficultbythefact
Tunhsuast,uroanteadnpahvaesreaagned, t(cid:4)h(cid:6)e(cid:2)im(cid:1)e(cid:12)an(cid:6)(cid:1)hs(cid:2)e(cid:1)r(cid:5)viicnethtiem(cid:6)sea(cid:2)itau(cid:1)trattheednpohdaesei. tthoaitnttehrereolbajteectthiveeinftuengcetriodneciissinoontvoabrtiaaibnlaebsleianndcltohseepdefroformr-
mancemeasuressuchasthroughput ,wox(cid:15)rk-in-process ,
precedingafinitenodeisgivenby:
totalbuffersallocated ,andothRersystemperformanLc(cid:15)e
measuressuchassystemiuxtiilisation foranybutthemost
(13)
(cid:6)(cid:10)(cid:2)i (cid:1) (cid:2)(cid:6)(cid:2)i (cid:1)(cid:12)pK(cid:6)(cid:1)h(cid:2)(cid:1) trivial situations. ComPbinatorial Op(cid:8)(cid:15)timisation approaches
Similarequationscanbeestablishedwithrespecttoeach forsolvingproblemslike the BAP are generallyclassified
ofthefinitenodes. Ultimately,wehavesimultaneousnon- aseitherexactoptimalapproachesorheuristicones.
linearequationsinvariables , , alongwithaux-
(cid:1) (cid:2)(cid:1) Exactapproachesareappropriateforsolvingsmallprob-
iliaryvariablessuchas anpdK .pKSol(cid:6)vhingtheseequations leminstancesorforproblemswithspecialstructuree.g.the
simultaneously we can(cid:6)cojmput(cid:5)e(cid:10)iall the performancemea- Travelling Salesman Problem, which admit optimal solu-
suresofthenetwork. tions. Classical approachesforachievinganoptimalsolu-
tionincludeBranch-and-Bound,Branch-and-Cut,Dynamic
Programming,ExhaustiveSearch,andrelatedimplicitand
(14)
(cid:3)
(cid:1) (cid:1) (cid:1)j(cid:1)(cid:1)h(cid:2)(cid:3)(cid:1)pK(cid:4) explicitenumerationmethods. Thedifficultywithutilising
(15)
(cid:1)j (cid:1) (cid:1)(cid:5)i(cid:2)(cid:3)(cid:1)pK(cid:4) these exact approaches for the BAP such as Branch-and-
(16)
(cid:1)j (cid:1) (cid:1)(cid:5)i(cid:1)(cid:1)h Boundisthatthesubproblemsforwhichoneseekstocom-
p(cid:3)K (cid:1) (cid:2)j(cid:2)(cid:6)h(cid:2)h (cid:1) (cid:2)(cid:1)h(cid:7)(cid:2)(cid:7)(cid:2)rr(cid:1)K(cid:1)K(cid:3)(cid:1)(cid:2)r(cid:1)(cid:2)K(cid:4)r(cid:2)K(cid:1)(cid:3)(cid:2)(cid:2)r(cid:4)(cid:1)K(cid:1)(cid:4)(cid:2)(cid:2)r(cid:1)(cid:1)Kr(cid:1)(cid:2)K(cid:4)r(cid:2)K(cid:2)(cid:4)(cid:4)(cid:8)(cid:8)(1(cid:4)7(cid:2)) psdtuioftficehcuuapslttpiaces,rntahonend-ollroiniwgeieanrrabplorpourgonrbdalsmeommnisntohgeliptortbloejbelicestmigvaseiwnfuehdniccbhtyioantrheaearsees
(cid:5) (cid:1) (cid:6)(18)
z (cid:1) (cid:2)(cid:1)(cid:6)(cid:9)(cid:2)h(cid:4) (cid:1)(cid:10)(cid:1)(cid:2)h exactproblemdecompositionmethods.
(cid:1)
(cid:7)(cid:2)(cid:1)(cid:6)(cid:9)(cid:2)h(cid:4)(cid:1)z(cid:2)(cid:8) (19) This dilemma implies that heuristic approaches are the
r(cid:2) (cid:1)
(cid:9)(cid:2)h only reasonable methodology for large scale problem in-
(cid:1)(cid:2) (20) stances of the BAP problem. Heuristic approaches can
(cid:7)(cid:2)(cid:1)(cid:6)(cid:9)(cid:2)h(cid:4)(cid:6)z (cid:8)
r(cid:1) (cid:1) be classified as either classical Non-linear Programming
(cid:9)(cid:2)h
searchmethodsorMetaheuristics.
K
(21)
pK (cid:1) K(cid:4)(cid:3)c (cid:1) p(cid:4) Non-linear Programming (derivative-free)search (Him-
c c(cid:11) (cid:2) melblau, 1972) methods such as, to name a few, Hooke-
(cid:1) (cid:2)
Equations 14 to 17 are related to the arrivals and feed- Jeeves,Nelder-Meadsimplexmethods,PARTAN,Powell’s
backintheholdingnode. Theequations18to20areused ConjugateDirectionmetods,FlexibleTolerance,theCom-
6
plexMethodofBox,andotherrelatedtechniqueshavemet attemperaturesnearthefreezingpoint. Ifthisprocedureis
with variedlevels ofsuccess intheBAP literatureandare notfollowedtheresultingsubstancemayformaglasswith
viablemeansofdealingwiththeBAPbecauseofthenon- notcrystallineorderandonlymetastable,consistingoflo-
closed form nature of the non-linear objective function. cally optimalstructures(Kirkpatricket al., 1983). During
While many researchers feel that the objective functionis thecoolingprocessthesystemcanescapelocalminimaby
concave or pseudo-concave in the decision variables, the movingtoathermalequilibriumofahigherenergypoten-
discretenatureofthedecisionvariablesmakestheproblem tialbasedontheprobabilisticdistribution ofentropy :
discontinuousandsonoderivativeinformationisavailable. w S
(22)
Metaheuristic methods such as Simulated Annealing,
S (cid:2)klnw
Tabu Search, Genetic Algorithms, and related techniques
where isBoltzmann’sconstantand theprobabilitythat
havenothistoricallybeenutilisedtosolvetheBAP;inthis
thesysktemwillexistinthestateitiswinrelativetoallpos-
paperweshallexploretheuseofSimulatedAnnealing.
siblestatesitcouldbein. Thusgivenentropy’srelationto
energy andtemperature :
3.4 Simulatedannealing E T
(23)
Simulatedannealingisanoptimisationmethodsuitablefor d(cid:15)E
dS (cid:2)
combinatorial optimisation problems. Such problems ex- T
wearriveattheprobabilisticexpression ofenergydistri-
hibit a discrete, factorially large configuration space. In
butionforatemperature : w
commonwithallparadigmsbasedon‘localimprovements’
T
the simulated annealing method starts with a non-optimal
(24)
initialconfiguration(whichmaybechosenatrandom)and (cid:1)E
w (cid:6)exp(cid:4) (cid:5)
worksonimprovingitbyselectinganewconfigurationus- kT
This so-called Boltzmann probability distribution is illus-
ing a suitable mechanism (at randomin the simulated an-
tratedin figure3. The probabilistic‘uphill’energymove-
nealingcase)andcalculatingthecorrespondingcostdiffer-
mentthatismadepossibleavoidstheentrapmentinalocal
ential( ). Ifthecostisreduced,thenthenewconfigu-
minimumandcanprovideagloballyoptimalsolution.
rationis(cid:16)aRccKeptedandtheprocessrepeatsuntilatermination
Theapplicationoftheannealingoptimisationmethodto
criterionis satisfied. Unfortunately,such methodscan be-
otherprocessesworksbyrepeatedlychangingtheproblem
come‘trapped’inalocaloptimumthatisfarfromtheglobal
configurationandgraduallyloweringthetemperatureuntil
optimum. Simulated annealingavoids this problemby al-
aminimumisreached.
lowing ‘uphill’ moves based on a model of the annealing
The correspondence between annealing in the physical
processinthephysicalworld.
worldandsimulatedannealingas usedforproductionline
optimisationisoutlinedintable1.
3.5 Completeenumeration
Asabase-ratebenchmarkforourapproach,weusedcom-
Probability pleteenumeration(CE)ofallpossiblebufferandserveral-
1 locationpossibilities.Thenumberoffeasibleallocationsof
bufferslotsamongthe intermediatebufferlocations
0.5 Qand serversamongtheN(cid:1)s(cid:1)tationsincreasesdramatically
withS , ,and andisgNivenbytheformula:
0
Q S N
0.5
0 Energy0 .s5tate 0.1 Te0m.3perature QN(cid:12)N(cid:1)(cid:3)(cid:1)(cid:3) S(cid:12)NN(cid:1)(cid:1)(cid:1) (cid:1) (cid:2) (25)
1 (cid:1)(cid:7)Q(cid:6)(cid:1)(cid:8)(cid:7)Q(cid:6)(cid:2)(cid:8)(cid:3)(cid:3)(cid:3)(cid:7)Q(cid:6)(cid:2)N(cid:1)(cid:2)(cid:2)(cid:8)(cid:7)S(cid:6)(cid:1)(cid:8)(cid:7)S(cid:6)(cid:2)(cid:8)(cid:3)(cid:2)(cid:3)(cid:3)(cid:7)S(cid:6)N(cid:2)(cid:1)(cid:8)
(cid:7)N(cid:2)(cid:2)(cid:8)(cid:9) (cid:7)N(cid:2)(cid:1)(cid:8)(cid:9)
All bufferandservercombinationscanbe methodically
enumeratedbyconsideringavector denotingtheposition
within the productionline of each bpufferor server. Given
Figure3: Probabilitydistribution ofenergy
(cid:2)E the vector we can then easily map to or using the
statesaccordingtotemperature. w (cid:6) exp(cid:4)kT (cid:5)
followingepquation(forthecaseof ):p q s
q
Physical matter can be brought into a low-temperature
ground state by careful annealing. First the substance is Q if (26)
(cid:1) othpejrw(cid:2)isie
melted,thenitis graduallycooledwithalotoftimespent qi (cid:2)
(cid:6)
j(cid:4)(cid:1)(cid:7)
X
7
Thegeneralisedqueuingnetworkthroughputevaluation
PhysicalWorld Production Line
algorithm was initially ported from a VAX VMS operating
Optimisation
system,toaPC-basedIntelarchitecture. Mostchangesin-
Atomplacement Lineconfiguration volvedtheadjustmentofnumericalconstantsaccordingto
Random atom Buffer space, server, theIEEE488floatingpointrepresentationusedontheIntel
movements serviceratemovement platforms. Subsequently, the algorithmwas rewritten in a
Energy Throughput pure subroutine form so that it could be repeatedly called
EnergyEdifferential ConfiguratioRn through- from within a program and semi-automatically converted
(cid:16)E putdifferential fromFORTRANtoANSIC.
Energystateprobability Changes accor(cid:16)diRng to TheSA algorithmused fortheproductionline resource
distribution the Metropolis crite- optimisationisgiveninfigure4.
rion, (cid:2)(cid:10)E TheSAprocedurewasrunwiththefollowingcharacter-
exp(cid:4) ,T i(cid:5)mpl(cid:3)e- isticsbasedonthenumberofstations :
mraenndti(cid:4)n(cid:6)g(cid:2)(cid:2)th(cid:2)e(cid:1)(cid:5)Boltzmann N
Maximumtrialsatgiventemperature
probabilitydistribution
(cid:1)(cid:6)(cid:6)(cid:7)N
Temperature Variable for establish-
Maximumsuccessesatgiventemperature
ing configuration ac- (cid:1)(cid:6)(cid:7)N
ceptancetermination Initialtemperature
(cid:6)(cid:2)(cid:17)
Coolingschedule Exponential:
Table 1: Correspondencebetween annealing in the physi- TK(cid:6)(cid:1) (cid:2)(cid:6)(cid:2)(cid:18)TK
Initiallineconfiguration Equal division of buffers and
calworldandsimulatedannealingusedforproductionline
servers among stations with any remaining resources
optimisation.
placedonthestationinthemiddle.
Reportedtime Elapsedwallclocktimeinseconds.
Giventhenabufferorserverconfiguration weadvanceto
thenextconfiguration usingthefunctionpg:
(cid:1) Thefollowingfactsclarifytheuseoftheevaluationalgo-
p
rithm:
(27)
(cid:1)
p (cid:2)g(cid:4)p(cid:1)(cid:1)(cid:5)
thethroughputusedasthelinemetricandpresentedin
wheregisrecursivelydefined(for buffers)as:
(cid:3) the tabulated results is the throughputat the last sta-
Q
if tion,
(cid:2)p(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:3)pl(cid:6)(cid:3)(cid:3)(cid:4)(cid:4)(cid:4)(cid:3)pQ(cid:4) pl(cid:6)(cid:3)(cid:2)N
g(cid:2)p(cid:3)l(cid:4)(cid:1) (cid:2)wph(cid:3)(cid:2)e(cid:3)r(cid:4)e(cid:4)(cid:4)(cid:3)p(cid:3)l(cid:4)(cid:2)(cid:3)p(cid:3)l(cid:3)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:3)p(cid:3)l(cid:3)(cid:2)(cid:4) otherwise the line topology graph is a linear series of stations
(cid:8)(cid:9) p(cid:3) (cid:1)g(cid:2)p(cid:3)l(cid:6)(cid:3)(cid:4) (28) (cid:3) allowingforparallelservers,and
Essentially,gmapsthevectorofpositionstoanewonerep-
theinitialandtheeffectivearrivalrateatthefirstserver
resenting(cid:10)anotherlineconfiguration. Whentheposition (cid:3) aresetto .
ofbufferorserverresource ,asitisincremented,reachpesi (cid:1)(cid:2)(cid:17)
thelastplaceintheline (i )gisrecursivelyap- Threebatchesoftestswereplannedandexecuted. One,
pliedtothebufferorserNverpiin(cid:12)p(cid:1)os(cid:2)itioNn setting – presented in tables 2–13, was planned in order to com-
tothenewvalueof .Thecompletepeni(cid:6)um(cid:1) erationpteirmpiQ- pare the results of our approach with those of Hillier and
nateswhenallbuffepris(cid:6)o(cid:1)rserversreachthelineposition . So (1995). A second set, presented in tables 14–17, was
ToenumerateallbufferandservercombinationsonecoNm- plannedinordertocomparetheapproachtotheresultsob-
plete serverenumerationis performedforeach line buffer tainedusingadifferentevaluativeprocedure(Spinellisand
configuration. Papadopoulos,1997). Finally,thethirdbatchoftests, pre-
sentedintables18–27,aimedatestablishingourmethod’s
efficacyindeterminingoptimalconfigurationsoflargepro-
4 Experimental design
ductionlines. Tables14–27containanadditionalcolumn,
,presentingtheprogramexecutiontime( ).
ThemaingenerativeprocedureweusedwasSimulatedAn- T s
nealing (Kirkpatrick et al., 1983, Laarhoven and Aarts,
1987, Spinellis and Papadopoulos, 1997) with complete 5 Experimental results
enumeration used where practical in order to validate the
SAresults.Inordertotesttheapproach,339casesofsmall Thefollowingparagraphsoutlinetheresultsweobtainedby
andlargeproductionlinesfromawidecombinationofallo- utilising the described methodology on a variety of prob-
cationoptionsweresearchedusingSAorCE. lems. Where applicable we compare the obtained results
8
withotherpublisheddata. Inourdescriptionoftheresults
1. Set initial line configuration. Set , set wesometimesrefertothephenomenaofthe bowlandthe
(cid:13) (cid:14) q,ise(cid:8)t bQ(cid:4)Nc , -shape.
N
qseNt(cid:1)(cid:2) (cid:8) qN(cid:1)(cid:2)(cid:12)Q(cid:1) i(cid:4)(cid:2)bQ(cid:4)Nc ,sesti (cid:8) bS(cid:4)Nc. L Thebowlphenomenonoccurswhenfor ,
N
sN(cid:1)(cid:2) (cid:8)sN(cid:1)(cid:2)(cid:12)SP(cid:1) i(cid:4)(cid:1)bS(cid:4)Nc wi (cid:8)(cid:1)(cid:4)N j (cid:2)(cid:1)(cid:1)(cid:3)(cid:1)(cid:2)(cid:2)(cid:2)(cid:1)N
2. Setinitialtemperature . Set . (29)
(cid:4) (cid:4)
(cid:13) TP(cid:5)(cid:14) T (cid:8)(cid:6)(cid:2)(cid:17) wj (cid:2) wN(cid:6)(cid:1)(cid:2)j
3. Initializestepandsuccesscount. Set ,set
(30)
(cid:13). (cid:14) S (cid:8)(cid:6) U (cid:8) wj(cid:4) (cid:3) wj(cid:4)(cid:6)(cid:1) (cid:1)(cid:9)j (cid:9) N (cid:1)(cid:1)
(cid:6) (cid:3)
(cid:1) (cid:2)
4. Createnewlinewitharandomredistributionofbuffer (31)
(cid:4) (cid:4) N (cid:1)(cid:1)
(cid:13)space,servers,orservicerate. wj (cid:9) wj(cid:6)(cid:1) (cid:9)j (cid:9)N (cid:12)(cid:1)
(cid:3)
(cid:14) (cid:1) (cid:2)
Thenamearisesfromthefactthatthevalueof is
(a) Createa copyoftheconfigurationvectors. Set (cid:4) (cid:4)
(cid:13) ,set ,set . (cid:14) considerablylargerthantheother (cid:4),whereaswth(cid:1)eo(cid:2)thwerN (cid:4)
q(cid:1) (cid:8)q s(cid:1) (cid:8)s w(cid:1) (cid:8)w are relatively close, so that plottinwgjthe (cid:4) versus givwejs
(b) Determine which vectorto modify. Set theshapeofthecross-sectionoftheinterwiojrofabowjl.
(cid:13) . (cid:14) Rv (cid:8)
Correspondingly,the -phenomenonistheallocationof
brand(cid:13)(cid:6)(cid:2)(cid:2)(cid:2)(cid:3)(cid:14)c
(c) if Create new line with a random extraserverstothefirstLstationoftheproductionline. This
redRisvtrib(cid:2)utio(cid:6)n(cid:13)of buffer space. Move space leadstoalmostthemaximumthroughputoftheline,which
froma source buffer to a(cid:14)destinatiRonnbuffer is attained when the extra servers are allocated to the last
: set qRs , set stationoftheline,havingtheadvantagethatitleadstoless
qRd Rs (cid:8), sebtrand(cid:13)(cid:3)(cid:2)(cid:2)(cid:2)N(cid:5)c Rd (cid:8) work-in-processinventory.
bra,nsde(cid:13)t(cid:3)(cid:2)(cid:2)(cid:2)N(cid:5)c Rn ,(cid:8)set brand(cid:13)(cid:1)(cid:2)(cid:2)(cid:2)qRs (cid:12).
(cid:1)(cid:5)c qRs (cid:8)qRs (cid:1)Rn qRd (cid:8)qRd (cid:12)Rn
(d) if Createnewlinewitharandomredis- 5.1 Short lines
triRbuvtio(cid:2)n(cid:1)of(cid:13)serverallocation. Move servers
from source station to a(cid:14)destinatRionn station Tables 2–13 present results for the optimal allocation us-
: set sRs , set ing SA for lines similar to the ones examined by Hillier
sRd Rs (cid:8), sebtrand(cid:13)(cid:1)(cid:2)(cid:2)(cid:2)N(cid:5)c Rd (cid:8) andSo(1995).InallcasesCEresultshavebeencalculated
bra,nsde(cid:13)t(cid:1)(cid:2)(cid:2)(cid:2)N(cid:5)c Rn ,(cid:8)setbrand(cid:13)(cid:1)(cid:2)(cid:2)(cid:2)sRs (cid:12). forcomparisonpurposes.Incomparingtheresultsobtained
(cid:1)(cid:5)c sRs (cid:8)sRs(cid:1)Rn sRd (cid:8)sRd(cid:12)Rn withthosepresentedbyHillierandSothefollowingimpor-
(e) if Create new line with a random re-
tantobservationscanbemade:
disRtrvibu(cid:2)tio(cid:3)n(cid:13)of service rate. Move service
ratefromsourcestation (cid:14)toadestRinnationsta-
1. Theservicerateallocation(table2)doesnotfollowthe
tion : set sRs , set
bowlphenomenon,butdiminishestowardstheendof
sRd Rs,s(cid:8)et brand(cid:13)(cid:1)(cid:2)(cid:2)(cid:2)N(cid:5)c Rd,s(cid:8)et
theline.
brand(cid:13)(cid:1)(cid:2)(cid:2)(cid:2)N(cid:5)c ,sRetn (cid:8) rand(cid:4)(cid:6)(cid:2)(cid:2)(cid:2)wR.s(cid:14)
wRs (cid:8)wRs (cid:1)Rn wRd (cid:8)wRd (cid:12)Rn 2. Thebufferallocation(table3)doesnotfollowthebowl
5. Calculate energy differential. Set
phenomenon, but increases monotonically across the
(cid:13) . (cid:14)
(cid:1) (cid:1) (cid:1) line.
(cid:16)E (cid:8)R(cid:4)q(cid:1)s(cid:1)w(cid:5)(cid:1)R(cid:4)q (cid:1)s(cid:1)w (cid:5)
6. Decide acceptance of new configuration. Accept all
3. The server allocation (table 4) for a small numberof
(cid:13)new configurations that are more efficien(cid:14)t and, fol-
servers follows a pattern strikinglysimilar to the one
lowing the Boltzmann probability distribution, some
presented in Hillier and So (1995). However, as the
that are less efficient: if or
(cid:2)(cid:10)E numberof servers increases, servers tend to accumu-
,set (cid:1),(cid:16)seEt (cid:9) (cid:6)(cid:1),seetxp(cid:4) T (cid:1)(cid:5),s(cid:3)et
latetowardsthebeginningoftheline.
rand(cid:4)(cid:6)(cid:2)(cid:2)(cid:2)(cid:1).(cid:5) q (cid:8) q s(cid:8) s w (cid:8)w
U (cid:8)U (cid:12)(cid:1)
4. Theserverandservicerateallocation(table5)donot
7. Repeat forcurrenttemperature. Set . If
(cid:13) maximumnumberofsteps,(cid:14)gotosStep(cid:8)4.S (cid:12)(cid:1) exhibitthe -phenomenon.
L
S (cid:9)
8. Lowertheannealingtemperature. Set ( 5. Thebufferandservicerateallocation(table6)results
(cid:13) ). (cid:14) T (cid:8) cT (cid:6) (cid:9) areverysimilartotheresultspresentedinHillierand
c(cid:9)(cid:1) So(1995).Asfarasthebufferallocationisconcerned,
9. Check if progress has been made. If , go to bufferstendtoaccumulatetowardstheendoftheline.
(cid:13)step3;otherwisethealgorithmterm(cid:14)inatUes.(cid:3) (cid:6) The allocationof workhowever, does not exhibit the
(cid:1)
bowlphenomenonpresentedintheotherstudyandfol-
lowstheusualdescendingrateacrosstheline.
Figure4: Simulatedannealingalgorithmfordistributing
bufferspace, servers,and servicerateinan -statioQn 9
line. S N N
6. The buffer and server allocation (tables 7, 8) results
are roughly similar to those presented in Hillier and 18000
So (1995)inboththeserverandthebufferallocation 16000 q
vectors.Serverstendtoaccumulatetowardsthebegin- q, w
14000
q, s
ning and middle of the line, whereas buffers tend to
q, s, w
12000
accumulatetowardsthelineends. w
)
(s 10000 s
7. Finally, the buffer, server, and service rate allocation e s, w
m 8000
(table9)roughlyfollowstheshapeofservicerateal- Ti
locationpresentedinHillierandSo(1995),buttheal- 6000
locationofbuffersandserversisquitedissimilar. 4000
2000
ConcerningthebehaviourofSAcomparedtoCEthere-
sultsofthetwomethodsareasfollows: 0
0 10 20 30 40 50 60 70
1. Buffer allocation (tables 3, 10) is the same using SA Station number
andCE.
2. Theserverallocationofalargenumberofservers(ta- Figure5:Executiontimeforoptimalconfigurationcalcula-
bles4,11)usingSAdiffersfromtheallocationusing tionsusingSA.
CE but only in the last two experiments, and this is
probablyduetoincorrectresultsreturnedbytheeval-
uationalgorithm. 120servers,andtheservicerateconfigurationin hours
ona120MHzPentium-processorsystem. Thissol(cid:19)u(cid:2)t(cid:17)ionus-
3. For buffer and server allocation for up to 4 stations ing SA required the testing of 238,248 different configu-
(tables7,12)SAandCEgivethesameresults. rations. The dominant factor of the algorithm is the line
configurationevaluationtakingon oursystem about 70ms
4. Forbufferandserverallocationof5stations(tables8,
for every configuration test. Multiplying this time factor
13)SAandCEdifferintheallocatedvectorsofsome
by the number of all possible buffer and server allocation
configurations,butwithonlyaslightimpactonthere-
configurationsfortheabovecase—accordingtoequation
sultingthroughput.
25—givesusanapproximateevaluationtimeusingCEof
5.2 Longlines (cid:1)(cid:3)(cid:6)(cid:12)(cid:21)(cid:6)(cid:1)(cid:3) (cid:1)(cid:3)(cid:6)(cid:12)(cid:21)(cid:6)(cid:1)(cid:1) (cid:11)(cid:12) or
(cid:6)(cid:2)(cid:6)(cid:20)s (cid:2) (cid:1)(cid:6) s
y(cid:21)ea(cid:6)rs(cid:1). (cid:3) (cid:21)(cid:6)(cid:1)(cid:1)
Thetables(16–27)presentresultsforoptimalallocationus- (cid:1)(cid:11)(cid:2) (cid:2)(cid:1) (cid:2)
(cid:1)(cid:6)
ing SA for lines of stations with
buffers and N s(cid:2)erv(cid:19)er(cid:1)s. (cid:20)W(cid:6) here practicalQ(i.(cid:2)e. (cid:3)fNor
6 Summary and conclusions
small lines)SCE(cid:2)res(cid:3)uNlts have been calculated for compari-
sonpurposes. Theoptimalallocationofavariablenumber
of buffers to a fixed number of servers (9 servers in table Our approach of using the expansion method for evaluat-
14; 15 servers table 16) follows a regular pattern: buffers ing the production line configurations generated by simu-
areaddedtothelinefromtherighttotheleft. Theresults lated annealing optimisation allowed us to explore small
oftheCE(table15;table17)confirmtheSAresults.How- and large line configurations in bounded execution time.
ever, this placement strategy is different from the optimal Theresultsobtainedwithourapproach,comparedtothose
strategyobtainedusingthedecompositionmethod(Dallery of other studies exhibit some interesting similarities but
andFrein,1993)astheevaluativeprocedure(Spinellisand alsostrikingdifferencesbetweentheminthe allocationof
Papadopoulos,1997). buffers,numbersofservers,andtheirservicerates. While
context dependent, these patterns of allocation are one of
the most important insights which emerge in solving very
5.3 Performanceanalysis
long production lines. The patterns, however, are often
The execution time of the SA optimisation algorithm in- counter-intuitive, which underscores the difficulty of the
creasedexponentiallydependingonthenumberofstations problemweaddressed.
(figure 5). This increase was a lot better than the combi- Havingdemonstratedtheviabilityofusinganalgorithm
natorial explosionof CE, and allowedus to producenear- based on randomisation techniques for optimising large
optimalconfigurationsforrelativelylargeproductionlines productionlineswenowplantoexploreotheroptimisation
inreasonabletime. Attheextremeendfora60stationline methods based on similar principles such as genetic algo-
SA produced a solution for the allocation of 120 buffers, rithmstofindhowtheycomparetoourapproach.Thesim-
10