Large production line optimization using simulated annealing

33 Pages·2000·0.37 MB·English

int.j.prod.res.,2000, vol.38, no.3, 509± 541
Large production line optimization using simulated annealing
D. SPINELLIS *, C. PAPADOPOULOS and
y z
J. MACGREGOR SMITH§
We present a robust generalized queuing network algorithm as an evaluative
procedure foroptimizing production line con® gurations usingsimulated anneal-
ing.Wecomparetheresultsobtainedwithouralgorithmtothoseofotherstudies
and® ndsomeinterestingsimilaritiesbutalsostrikingdi(cid:128)erencesbetweenthemin
theallocationofbu(cid:128)ers,numbersofservers,andtheirservicerates.Whilecontext
dependent, these patterns of allocation are one of the most important insights
which emerge in solving very long production lines. The patterns, however, are
often counter-intuitive, which underscores the di(cid:129) culty of the problem we
address. The most interesting feature of our optimization procedure is its
bounded executiontime, whichmakesitviable foroptimizing verylong produc-
tionlinecon® gurations. Basedontheboundedexecutiontimeproperty, wehave
optimized con® gurations of upto60stations with120bu(cid:128)ersandserversinless
than® vehours of CPUtime.
1. Introduction and literature review
A large amount of research has been devoted to the analysis and design of
productionlines. Alotof thisresearch concernsthedesignof manufacturingsystems
withconsiderableinherent variability intheprocessing timesatthevarious stations,
acommonsituation withhumanoperators/assemblers. The literature onthemodel-
lingandoptimizationof productionlinesisvast, allowing ustoreviewonlythemost
directly relevant studies. For asystematicclassi® cationof therelevant worksonthe
stochasticmodellingof theseandothertypesof manufacturingsystems(e.g.,transfer
lines, ¯exible manufacturing systems (FMS), and ¯exible assembly systems (FAS)),
the interested reader is referred to a review paper by Papadopoulos and Heavey
(1996) and some recently published books, such as those by Askin and Standridge
(1993), Buzacott and Shanthikumar (1993), Gershwin (1994), Papadopoulos et al.
(1993), Viswanadham and Narahari (1992), and Altiok (1997).
There are two basic problem classes:
(1) theevaluation of productionlineperformance measuressuchasthroughput,
mean¯owtime,workstationmeanqueuelength, andsystemutilization, and
(2) the optimization of the decision variables of these lines.
Revision receivedApril1999.
{Department of Information and Communication Systems, University of the Aegean,
83200 Karlovasi Samos, Greece.
{DepartmentofBusinessAdministration,UniversityoftheAegean,82100Chios,Greece.
e-mail: [e-mail]
}Department of Mechanical and Industrial Engineering, University of Massachusetts,
AmherstMA 01003, USA, e-mail: [e-mail]
*Towhom correspondenceshould beaddressed. e-mail: [e-mail]
International JournalofProductionResearchISSN0020± 7543print/ISSN1366± 588Xonline# 2000Taylor&FrancisLtd
http://www.tandf.co.uk/journals/tf/00207543.html
510 D. Spinellis et al.
Examples of decision variables that have been considered are:
(1) the sizes of the buffers placed between successive workstations of the lines,
(2) the number of servers allocated toeach workstation, and
(3) the amount of workload allocated to each workstation.
Thecorrespondingoptimization problems are named, respectively, (1) thebu(cid:128)er
allocation problem, (2) the server allocation problem, and (3) the workload allocation
problem, in a production line.
In Papadopoulos et al. (1993) both evaluative and generative (optimization)
models are given for modelling the various types of manufacturing systems. This
work falls into the second category. Evaluative and optimization models can be
combined by closing the loop between them; that is, one can use feedback from
an evaluative model to modify the decisions taken by the optimization model.
Oneof thekey questions thatthedesigners face in aserial production lineisthe
bu(cid:128)erallocation problem (BAP), i.e. howmuchbu(cid:128)erstoragetoallowandwhereto
place it within the line. This is an important question because bu(cid:128)ers can have a
great impact on the e(cid:129) ciency of the production line. They compensate for the
blocking andthestarving of theline’sstations. Forthisreason, thebu(cid:128)erallocation
problemhasreceivedalotmoreattentionthantheothertwodesignproblems. Bu(cid:128)er
storage is expensive due both to its direct cost, and to the increase of the work-in-
process (WIP) inventories. In addition, the requirement to limit the bu(cid:128)er storage
canalso bearesult of space limitations intheshop¯oor. Theliterature ontheBAP
is extensive. A systematic classi® cation of the research work in this area is given in
Singh andSmith (1997)andPapadopoulosetal. (1993). Theworksaresplitaccord-
ing to:
. Themethodused tosolve thebu(cid:128)erallocation problem: search methods (Altiok
andStidham1983, SmithandDaskalaki1988, Seongetal.1995, HillierandSo
1991a, 1991b), dynamic programming methods (Kubat and Sumita, 1985,
Jafari and Shanthikumar 1989, Yamashita and Altiok 1997), among others,
and simulation methods (Conway et al. 1988, Ho, et al. 1979).
. The type of production line. Balanced/unbalanced (Powell 1994 presents a
literature review according tothis scheme), or reliable/unreliable; themajority
of papers deal withreliable lines. Only afew algorithms have been developed
tocalculate theperformance measures of unreliable production lines (Glassey
and Hong 1983, Choong and Gershwin 1987, Gershwin 1989, Heavey et al.
1993). HillierandSo (1991a)andSeongetal. (1995)havedealtwiththebu(cid:128)er
allocation problem in unreliable production lines.
. The number of machines available ateachworkstation. Similarly, few research-
ers have dealt with the bu(cid:128)er allocation problem in production lines with
multiple (parallel) machines at each workstation (Hillier and So 1995,
Magazine and Stecke 1996, Singh and Smith 1997).
Apart from thebu(cid:128)er allocation problem, theothertwointeresting design prob-
lems havealso been considered by someresearchers, e.g., theworkallocation prob-
lem (HillierandBoling 1979, Hillier andBoling 1966, Hillier andBoling 1977, Ding
and Greenberg 1991, Hyang and Weiss 1990, Shanthikumar et al. 1991, Wan and
Wol(cid:128) 1993, Yamazaki etal. 1992), amongothers, andtheserver allocation problem
(Magazine and Stecke 1996, Hillier and So 1989). Hillier and So (1995) studied
various combinations of these three design problems. Other references may be
Production line optimization by SA 511
foundtherein(BuzacottandShanthikumar1992, BuzacottandShanthikumar1993),
among others.
The present work deals withthesame design problems (bu(cid:128)er allocation, server
allocation, and workload allocation) but for long production lines with multi-
machine stations.
As the problem being investigated is combinatorial in nature, traditional
Operations Research techniques arenotaspractical for obtainingoptimalsolutions
for long production lines. We propose a simulated annealing (SA) approach as the
search method in conjunction with the expansion method developed by Kerbache
andSmith (1987) as theevaluative tool. Simulatedannealing is anadaptation of the
simulationof physical thermodynamicannealing principles described byMetropolis
et al. (1953) to the combinatorial optimization problems (Kirkpatricket al. 1983,
Cerny1985). Similartogeneticalgorithms (Holland1975, Goldberg1989) andtabu
search techniques (Glover 1990) it follows the `local improvement’ paradigm for
harnessing the exponential complexity of the solution space.
Thealgorithm isbasedonrandomizationtechniques. Anoverview of algorithms
based on such techniques can be found in the survey by Gupta et al. (1994). A
complete presentation of the method and its applications is described by Van
Laarhoven and Aarts (1987) and accessible algorithms for its implementation are
presented by Corana et al. (1987) and Press et al. (1988). A critical evaluation of
di(cid:128)erent approaches to annealing schedules and other method optimizations are
given by Ingber (1993). As a tool for operational research SA is presented by
Eglese (1990), while Koulamas et al. (1994) provide acomplete survey of SA appli-
cations tooperations research problems.
The use of the simulated annealing algorithm appears to be a promising
approach. We believe that this algorithm could be applied in conjunction with a
fast decompositionalgorithm tosolve e(cid:129) ciently and accurately theaforementioned
optimization problems in much longer production lines.
Theremainder of thepaper isorganized asfollows: we® rst describe theproduc-
tionlinemodel andtheproblem of ourinterestfollowed bythemethodology of our
approach namely: theperformance model, theexpansionmethodusedforevaluating
the line performance, an overview of the combinatorial optimization methods, the
simulated annealing optimization method, and the complete enumeration method;
we then describe our experimental methodology and present an overview of the
numerical and performance results for short and long production lines. At the end
of the paper we provide a full tabulated set of the experimental results.
2. The production line model and the optimal design problem
Wede® neanasynchronouslineasoneinwhicheveryworkstationcanpassparts
onwhenitsprocessing iscomplete aslongasbu(cid:128)er space isavailable. Suchalineis
subject to manufacturing starving and blocking. We assume that the ® rst station is
neverstarvedandthelaststationisneverblocked. Therefore wecansaythattheline
operates in a push mode: parts are always available when needed at the ® rst work-
stationandspaceisalways available atthelast workstation todispose of completed
parts.
An N-station line consists of N workstations in series, labelled M ;M ;...;M
1 2 N
and N 1 locations for bu(cid:128)ers, labelled B ;B ;...;B , is illustrated in ® gure 1.
¡ 2 3 N
Each station i has s servers operating in parallel. The bu(cid:128)er capacities of theinter-
i
512 D. Spinellis et al.
Figure 1. AnN-workstationmulti-machineproductionlinewithN 1intermediatebu(cid:128)ers.
¡
mediatebu(cid:128)ersBi, i ˆ 2;...;N,aredenotedbyqi, whereasthemeanservicetimesof
the i stations (i 1;...;N) are denoted by w.
ˆ i
The main performance measure of the production line is the mean throughput,
denoted by R…q;s;w†, where q ˆ …q2;q3;...;qN†, s ˆ …s1;s2;...;sN† and
w ˆ …w1;w2;...;wN†.
If Q denotes the total number of available bu(cid:128)er slots to be allocated to the
N 1bu(cid:128)ersandS thetotalnumberof available servers (machines)toallocated to
¡
theN stations thenthegeneral version of the optimization model (® rst reported by
Hillier and So, 1995) is
maxR q;s;w 1
… † … †
subject to
N
qi ˆ Q; …2†
i 2
Xˆ
N
s S; 3
i ˆ … †
i 1
Xˆ
N
w N; 4
i ˆ … †
i 1
Xˆ
q 0and integer; i 2;3;...;N ;
i … ˆ †
s 1 and integer; i 1;2;...;N ;
i … ˆ †
wi > 0; …i ˆ 1;2;...;N†;
where Q;S and N are ® xed constants and q, s and w are the decision vectors. The
third constraint indicates that the sum of the expected service times is a ® xed con-
stant, which by normalization can be equal to N.
The objective function of throughput, R q;s;w , is not the only performance
… †
measure of interest. The average WIP, the ¯ow time, the cycle time, the system
utilization, the average queue lengths and other measures are equally important
performance measures. However, throughput is the most commonly used perform-
ance measure in the international literature.
Production line optimization by SA 513
3. Optimal allocation methodology
3.1. Performance models
The queuing model M=M=C=Kthat we use refers to aqueuing system where:
. it is assumed that the arrivals are distributed according to the Poisson distri-
bution (or equivalently that the intermediate times between two successive
arrivals are exponentially distributed),
. the service (processing) times follow the exponential distribution,
. thereareCparallel servers (machineswhichareidenticalateachworkstation),
and
. the total capacity of the system is ® nite and equal to K.
WhileourfocusinthispaperisonM=M=C=Kapproximationsforopenqueuing
networks of series-parallel topologies, we also brie¯y discuss some of the available
approaches used for modelling M=M=1=Ksystems since most of the literature has
focused on M=M=1=Ksystems.
Both open and closed systems have been studied by exact analysis although
results have been limited. Exact analyses of open two, three, and four node-server
models with exponential service are limited by the explosive growth of the Markov
chain models for analysing these systems. The analysis of very large Markov chain
models has led toe(cid:128)ective aggregation techniques for these models (Schweitzer and
Altiok 1989, Takahashi 1989) but the computation time and power required for
these exact results leaves open the need for approximationtechniques.
Van Dijk and hisco-authors (1988, 1989) have developed somebounding meth-
odologies for both M=M=1=K and M=M=C=K systems and have demonstrated
their usefulness in the design of small queuing networks. Of course, when doing
optimization of medium and long queuing networks, bounds can be far o(cid:128) the
optimum, so robust approximation techniques close to the optimal performance
measures are most desirable.
Most approximation techniques appearing in the literature rely on decomposi-
tion/aggregationmethodstoapproximateperformance measures. Oneandtwonode
decompositions of the network have been carried out, all with varying degree of
success.
Thefewapproximationapproaches available intheliterature canbeclassi® edas
follows: isolationmethods,repeatedtrials,node-by-nodedecomposition,andexpansion
methods. In the isolation method, the network is subdivided into smaller subnet-
works and then studied in isolation (Labetoulle and Pujolle 1980, Boxma and
Konheim 1981). This method was used by Kuehn (1979) and by Gelenbe and
Pujolle (1976), but they failed to consider networks with ® nite capacity.
Closely related to the isolation method is the repeated trials method, a class of
techniques based upon repeatedly attempting tosendblocked customers toaqueue
causing the blocking (Caseau and Pujolle 1979, Fredericks and Reisner 1979,
Fredericks 1980).
In node-by-node decomposition, the network is broken down into single, pairs,
andtriplets of nodeswithaugmented service and arrival parameters which are then
studied separately (Hillier and Boling 1967, Takahashi et al. 1980, Altiok 1982,
Alriok and Perros 1986, Perros and Alriok 1986, Brandwajn and Jow 1988). More
general service time approximations appear in the work by Gun and Makowski
(1989). The expansion methodistheapproach argued for in thispaper forcomput-
ing the performance measures of M=M=C=K ® nite queuing networks (Kerbache
514 D. Spinellis et al.
1984, Kerbache and Smith 1987, 1988). It can be characterized conceptually as a
combination of repeated trials and node-by-node decomposition where the key dif-
ferenceisthata`holding’ nodeisaddedtothenetworktoregisterblocked customers.
The addition of the holding node `expands’ the network. This approach transforms
thequeuing networkintoanequivalent Jackson networkwhichis thendecomposed
allowing for each node to be solved independently. We have successfully used the
Expansion Method to model M=M=1=K (Kerbache and Smith 1988), M=M=C=K
(JainandSmith1994, HanandSmith1992), M=G=1=K(KerbacheandSmith1987),
andmostrecentlyM=G=C=C(CheahandSmith1994, Smith1991, 1994)queuesand
queuing networks. In addition, we have also used our expansion methodology to
model routing (Daskalaki and Smith 1989, 1986, Gosavi and Smith 1990) and
optimal resource allocation problems (Smith and Daskalaki 1988, Smith 1991,
Smith and Chikhale 1995, Singh and Smith 1997).
3.2. The expansion method
The expansion method is a robust and e(cid:128)ective approximation technique devel-
opedbyKerbacheandSmith(1987). Asdescribedinprevious papers, thismethodis
characterized as a combination of repeated trials and node-by-node decomposition
solution procedures. Methodologies for computing performance measures for a
® nite queuing network use primarily the following two kinds of blocking:
. Type I. The upstream node i gets blocked if the service on a customer is
completedbutitcannotmovedownstreamduetothequeueatthedownstream
nodej beingfull. Thisissometimesreferred toasblocking afterservice (BAS)
(Onvural 1990).
. Type II. The upstream node is blocked when the downstream node becomes
saturated and service mustbesuspended ontheupstreamcustomerregardless
of whether service iscompletedornot. Thisis sometimes referred toasblock-
ing before service (BBS) (Onvural 1990).
The expansion method uses type Iblocking, which is prevalent in most produc-
tion and manufacturing, transportation and other similar systems.
Consider a single node with ® nite capacity K (including service). This node
essentially oscillates between two statesÐ the saturated phase and the unsaturated
phase. Intheunsaturatedphase, nodejhasatmostK 1customers (inserviceorin
¡
the queue). On the other hand, when the node is saturated no more customers can
join thequeue. Refer to® gure 2foragraphical representationof thetwoscenarios.
The expansion method consists of the following three stages:
. Stage I: network recon® guration.
. Stage II: parameter estimation.
. Stage III: feedback elimination.
Thefollowing notationde® nedbyKerbache andSmith (1987, 1988)willbeused
in further discussion regarding this methodology:
h The holding node established in the expansion method,
External Poisson arrival rate to the network,
¶ Poisson arrival rate to node j;
j
¶~j E(cid:128)ective arrival rate to node j,
·j Exponential mean service rate at node j,
Production line optimization by SA 515
·~j E(cid:128)ective service rate at node j due to blocking,
p Blocking probability of ® nite queue of size K,
K
pK0 Feedback blocking probability in the expansion method,
pj Unconditional probability that there is no customer in the service channel
0
at node j (either being served or being held after service),
R Mean throughput rate,
S Service capacity (bu(cid:128)er) at node j, i.e. S Kfor a single queue.
j ˆ
3.2.1. Stage I: network recon® guration
Usingtheconcept of twophases atnode j, anarti® cial node his added for each
® nite node in the network to register blocked customers. Figure 2 shows the
additional delay, caused to customers trying to join the queue at node j when it is
full, withprobability p . The customers successfully join queue j withaprobability
K
1 p . Introduction of an arti® cial node also dictates the addition of new arcs
… ¡ K†
with p and 1 p as the routing probabilities.
K … ¡ K†
Theblockedcustomerproceedstothe® nitequeuewithprobability…1¡ p0K† once
again after incurring a delay at the arti® cial node. If the queue is still full, it is re-
routed withprobability p0K tothearti® cial node where it incurs another delay. This
process continues till it ® nds a space in the ® nite queue. A feedback arc is used to
modeltherepeateddelays. Thearti® cialnodeismodelledasanM=M= queue. The
1
in® nitenumber of servers is used simply toserve theblocked customer adelay time
without queuing.
3.2.2. Stage II: parameter estimation
This stage essentially estimates the parameters pK, p0K and ·h utilizing known
results for the M=M=C=Kmodel.
. pK: Analytical results from the M=M=C=K model provide the following
expression for p :
K
Figure 2. Type Iblocking in® nite queues.
516 D. Spinellis et al.
1 ¶ K
p p ; 5
K ˆ cK¡cc! · 0 … †
where for ¶=c· 1
… ˆ6 † ¡ ¢
1
c¡1 1 ¶ n ¶=· c1 ¶=c· K¡c‡1 ¡
p … † ¡ … † 6
0 ˆ " nXˆ0 n! · ‡ c! 1¡ ¶=c· # … †
and for ¶=c· 1 , ¡ ¢
… ˆ †
c¡1 1 ¶ n ¶=· c ¡1
p0 ˆ n! · ‡ … c! † …K¡ c‡ 1† : …7†
" n 0 #
Xˆ
. p0K: Sincethereisnoclosedform¡ s¢olutionforthisquantityanapproximationis
used given by Labetoulle and Pujolle obtained using di(cid:128)usion techniques
(Labetoulle and Pujolle 1980).
1
p0K ˆ " ·j·‡h·h¡ ·¶h‰‰……rrK2K2‡¡1r¡K1†rK1¡‡1…r†K2¡¡1…r¡K2 r¡K1¡r1K1††ŠŠ# ¡ ; …8†
where r and r are the roots to the polynomial:
1 2
¶ ¡ …¶‡ ·h‡ ·j†x‡ ·hx2 ˆ 0 …9†
while, ¶ ˆ ¶j¡ ¶h…1¡ p0K† and ¶j and ¶h are the actual arrival rates to the
® nite and arti® cial holding nodes respectively.
In fact, ¶j the arrival rate to the ® nite node is given by:
¶j ˆ ¶~i…1¡ pK† ˆ ¶~i¡ ¶h …10†
Let us examine the following argument to determine the service time at the
arti® cial node. If an arriving customer is blocked, the queue is full and thus a
customer is being serviced, so the arriving customer to the holding node has to
remaininserviceatthearti® cialholdingnodefortheremainingservice timeinterval
of the customer in service. The delay distribution of a blocked customer at the
holdingnodehasthesamedistribution astheremainingservicetimeof thecustomer
being serviced at thenode doing theblocking. Using renewal theory, one can show
that the remaining service time distribution has the following rate · :
h
2·
j
· ; 11
h ˆ 1 ¼2·2 … †
‡ j
where ¼2 is the service time variance given by Kleinrock (1975). Notice that if the
service time distribution at the ® nite queue doing the blocking is exponential with
rate ·j, then
· ·
h ˆ j
the service time at the arti® cial node is also exponentially distributed with rate ·j.
3.2.3. Stage III: feedback elimination
Duetothefeedbacklooparoundtheholdingnode, therearestrongdependencies
in the arrival processes. Elimination of these dependencies requires recon® guration
Production line optimization by SA 517
of the holding node which is accomplished by recomputing the service time at the
node and removing the feedback arc. The new service rate is given by
·0h ˆ …1¡ p0K†·h: …12†
Theprobabilitiesof beinginanyof thetwophases (saturatedorunsaturated)are
pwKheanndin…t1h¡eupnKs†a.tTurhaetemdepahnasseervaincde …ti·m¡i e1‡at·a0h¡n1o†dine ithperescaetduirnagtedthpeh® ansiet.eTnhoudse, iosn·a¡in1
average, the mean service time at the node i preceding a ® nite node is given by
·~¡i 1 ˆ ·¡i 1‡ pK·0h¡1: …13†
Similar equations can be established with respect to each of the ® nite nodes.
Ultimately, we have simultaneous nonlinear equations in variables pK, p0K,
·¡h1 along withauxiliary variables suchas ·j and ¶~i. Solving these equations simul-
taneously we can compute all the performance measures of the network:
¶ ˆ ¶j¡ ¶h…1¡ p0K† …14†
¶j ˆ ¶~i…1¡ pK† …15†
¶j ˆ ¶~i¡ ¶h …16†
1
p0K ˆ " ·j·‡h·h¡ ·¶h‰‰……rrK2K2‡¡1r¡K1†rK1¡‡…1r†K2¡¡1…r¡K2 r¡K1¡r1K1††ŠŠ# ¡ …17†
2
z ˆ …¶‡ 2·h† ¡ 4¶·h …18†
1
r ‰…¶‡ 2·h† ¡ z2Š 19
1 ˆ 2·h … †
1
r ‰…¶‡ 2·h† ‡ z2Š 20
2 ˆ 2·h … †
K
1 ¶
pK ˆ cK¡cc! · p0: …21†
Equations (14) to (17) ar¡e r¢elated to the arrivals and feedback in the holding
node. Equations (18) to (20) are used for solving equation (17) with z used as a
dummy parameter for simplicity of the solution. Lastly, equation (21) gives the
approximation to the blocking probability derived from the exact model for the
M=M=C=Kqueue. Hence, weessentially have® veequations tosolve, namelyequa-
tions (14)± (17) and (21). To recapitulate, we ® rst expand the network with an arti-
® cial holding node; this stage is then followed by the approximation of the routing
probabilities, duetoblocking, andtheservicedelayintheholdingnode; and, ® nally,
the feedback arc at the holding node is eliminated. Once these three stages are
complete, we have an expanded network which can then be used to compute the
performance measures for the original network. As a decomposition technique this
approach allows successive addition of aholding node for every ® nitenode, estima-
tion of the parameters and subsequent elimination of the holding node.
518 D. Spinellis et al.
3.3. Combinatorial optimization models
The bu(cid:128)er allocation problem (BAP) is perhaps best formulated as a nonlinear
multiple-objective programming problem where the decision variables are the inte-
gers. NotonlyistheBAPadi(cid:129) cult hardcombinatorialoptimizationproblem,
NP¡
it is made all themore di(cid:129) cult by thefact that theobjective function is not obtain-
able in closed form to interrelate the integer decision variables x and the perform-
ance measures such as throughput R, work-in-process L, total bu(cid:128)ers allocated
x, and other system performance measures such as system utilization « for
i i
anybutthemosttrivialsituations. Combinatorialoptimizationapproaches forsolv-
P
ingproblemsliketheBAParegenerally classi® edaseitherexactoptimalapproaches
or heuristic ones.
Exact approaches are appropriate for solving small problem instances or for
problems with special structure, e.g., the travelling salesman problem, which admit
optimal solutions. Classical approaches for achieving an optimal solution include
branch-and-bound, branch-and-cut, dynamic programming, exhaustive search, and
related implicit and explicit enumeration methods. Thedi(cid:129) cultywithutilizing these
exactapproachesfortheBAPsuchasbranch-and-boundisthatthesubproblems for
which one seeks tocompute upper and lower bounds on the objective function are
stochastic, nonlinear programming problems which are as di(cid:129) cult as the original
problem so little is gained by these exact problem decomposition methods.
Thisdilemmaimplies that heuristic approaches are theonlyreasonable method-
ology for large scale problem instances of the BAPproblem. Heuristic approaches
canbeclassi® edaseitherclassical nonlinear programming search methods ormeta-
heuristics.
Nonlinear programming (derivative-free) search (Himmelblau 1972) methods
such as, to name a few, Hooke± Jeeves, Nelder± Mead simplex methods, PARTAN,
Powell’sconjugatedirectionmethods,¯exibletolerance, thecomplexmethodof Box,
and other related techniques have met with varied levels of success in the BAP
literature and are viable means of dealing with the BAPbecause of the non-closed
formnatureof thenonlinear objectivefunction. Whilemanyresearchers feelthatthe
objective function is concave or pseudo-concave in the decision variables, the dis-
crete nature of the decision variables makes the problem discontinuous and so no
derivative information is available.
Metaheuristic methods such as simulated annealing, Tabu search, genetic algor-
ithmsandrelated techniques havenothistorically beenutilized tosolve theBAP; in
this paper we shall explore the use of simulated annealing.
3.4. Simulated annealing
Simulated annealing is an optimization method suitable for combinatorial
optimization problems. Suchproblems exhibit adiscrete, factorially large con® gura-
tion space. In common with all paradigms based on `local improvements’ thesimu-
latedannealingmethodstartswithanon-optimalinitialcon® guration (whichmaybe
chosenatrandom)andworksonimprovingitbyselectinganewcon® gurationusing
a suitable mechanism (at random in the simulated annealing case) and calculating
the corresponding cost di(cid:128)erential ( R ). If the cost is reduced, then the newcon-
K
® gurationisacceptedandtheprocessrepeatsuntilaterminationcriterionissatis® ed.
Unfortunately, such methods can become `trapped’ in a local optimum that is far
from the global optimum. Simulated annealing avoids this problem by allowing
`uphill’ moves based on a model of the annealing process in the physical world.