Job Shop Scheduling by Simulated Annealing

14 Pages·2007·0.48 MB·English

Job Shop Scheduling by Simulated Annealing
Author(s): Peter J. M. van Laarhoven, Emile H. L. Aarts, Jan Karel Lenstra
Source: Operations Research, Vol. 40, No. 1 (Jan. - Feb., 1992), pp. 113-125
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JOB SHOP SCHEDULINGB Y SIMULATEDA NNEALING
PETER J. M. VAN LAARHOVEN
NederlandseP hilipsB edrijvenE, indhoven,T he Netherlands
EMILEH . L. AARTS
PhilipsR esearchL aboratoriesE, indhoven,T he Netherlands
and
EindhovenU niversityo f TechnologyE, indhoven,T heN etherlands
JAN KARELL ENSTRA
EindhovenU niversityo f TechnologyE, indhoven,T heN etherlandsa nd CWI,A msterdam,T heN etherlands
(ReceivedJ uly 1988;r evisionsr eceivedM ay 1989,J anuary1 990;a cceptedJ une 1990)
We describe an approximationa lgorithmf or the problem of finding the minimum makespani n a job shop. The
algorithmi s based on simulated annealing,a generalizationo f the well known iterativei mprovementa pproacht o
combinatorialo ptimizationp roblems.T he generalizationin volves the acceptanceo f cost-increasingtr ansitionsw ith a
nonzero probabilityt o avoid getting stuck in local minima. We prove that our algorithma symptoticallyc onvergesi n
probabilityt o a globally minimal solution, despite the fact that the Markov chains generatedb y the algorithma re
generallyn ot irreducible.C omputationale xperimentss how that our algorithmc an find shorterm akespanst han two
recenta pproximationa pproachest hat are more tailoredt o the job shop schedulingp roblem.T his is, however,a t the cost
of larger unningt imes.
W e are concernedw ith a problemi n machine tions. Adams, Balas and Zawack( 1988) developed
schedulingk nown as the job shop scheduling a shifting bottleneckp rocedure,w hich employs an
problem( Coffman 1976, French 1982). Informally, ingeniousc ombinationo f schedulec onstructiona nd
the problemc an be describeda s follows.W e areg iven iterative improvement, guided by solutions to
a set of jobs and a set of machines.E achj ob consists single-machinep roblems.T he approachp ursuedb y
of a chain of operations,e ach of which needs to be Adams,B alasa nd Zawacki s stronglyt ailoredt o the
processedd uringa n uninterruptedti me period of a problema t hand. It is based on a fair amount of
given lengtho n a given machine.E achm achinec an combinatoriailn sight into the job shop scheduling
processa t most one operationa t a time. A schedulei s problem,a nd its implementationr equiresa certain
an allocationo f the operationst o time intervalso n level of programmesro phistication.
the machines.T he problemi s to find a scheduleo f In this paper,w e investigateth e potentialo f a more
minimuml ength. generala pproachb asedo n the easilyi mplementable
The job shop schedulingp roblem is among the simulateda nnealinga lgorithm( KirkpatrickG, elatt
hardestc ombinatorialo ptimizationp roblems.N ot and Vecchi 1983, and Cerny 1985). A similar
only is it NP-hard,b ut even among the memberso f approachw as independentlyin vestigatedb y Matsuo,
the latterc lassi t appearsto belongt o the mored ifficult Suh and Sullivan (1988). Their controlleds earch
ones (LawlerL, enstraa nd RinnooyK an 1982).O pti- simulated annealing method is less general than
mizationa lgorithmsf or job shop schedulingp roceed ours in the sense that it uses more problems pecific
by branch and bound, see, for instance, Lageweg, neighborhoods.
Lenstraa nd Rinnooy Kan (1977), and Carliera nd The organizationo f this paper is as follows. In
Pinson( 1989). Most approximational gorithmsu se a Section 1 we give a formal problemd efinition.I n
priorityr ule, i.e., a rule for choosing an operation Section2 the basice lementso f the simulateda nneal-
from a specifieds ubseto f as yet unscheduledo pera- ing algorithma re reviewed.I n Section3 we describe
Subjectc lassifications:A nalysiso f algorithms,s uboptimala lgorithmss: imulateda nnealing.P roduction/schedulings,e quencing,d eterministicm, ultiple
machine:j ob shop scheduling.
Area of review: MANUFACTURING, PRODUCTION AND SCHEDULING.
OperationsR esearch 0030-364X/92/4001-01 13 $01.25
Vol. 40, No. 1, January-Februar1y9 92 113 ? 1992 OperationsR esearchS ocietyo f America
114 / VAN LAARHOVENA, ARTS AND LENSTRA
the applicationo f simulateda nnealingt o job shop
schedulingW. e provea symptoticc onvergenceo f the
algorithmt o a globallym inimals olutionb y showing 1 2 3~~~~~~~~~~~
that the neighborhoods tructurei s such that each
ergodics et contains at least one global minimum. O~ ~~ a~' " 10
Section4 containst he resultso f a computationaslt udy
in whichs imulateda nnealingi s used to find approxi-
mate solutionst o a larges et of instanceso f the job
shop schedulingp roblem.W e compareo ur simulated
7 8 9
annealingm ethod with three other approachesi,. e.,
time-equivalentit erativei mprovement,t he shifting Figure1 . The disjunctiveg raph G of a 3-job 3-
bottleneckp rocedureo f Adams, Balas and Zawack, machinei nstance.O perations1 , 5 and 9 are
andt he controlleds earchs imulateda nnealingm ethod processedb y machine1 , operation2s , 4 and
of Matsuo, Suh and Sullivan.I n Section 5 we end 8 by machine2 , and operations3 , 6 and 7
with some concludingr emarks. by machine3 ; 0 and 10 are the fictitious
initial and final operations,r espectively.
Thicka rrowsd enotea rcsi n A, dottedl ines
1. THE PROBLEM
edgesi n E.
We areg ivena set -f of n jobs, a set X of m machines,
and a set a of N operations.F or each operation
v E a there is a job J, E /o to which it belongs,a Figure1 illustratest he disjunctiveg raphf or a 3-job,
machineM , E 4 on which it requiresp rocessing, 3-machinei nstance,w heree achj ob consistso f three
anda processingti me t, E NJT. herei s a binaryr elation operations.
-> on a that decomposesa into chainsc orresponding For each pair of operationsv , w E a with v - ,
taon dth teh jeorbes i;s mnoo rxe s4p {evc,i fwic}a slluyc, hif tvh a->t vw ->, thxe onr J x, -=> JwW. , cSoimndiliatirolnyf, ( o3r) eisa crhe ppraeirse onf toepdbe yr aatnio anrsvc , (wv, Ew a) inw iAth.
The problemi s to find a startt ime s, for each opera- M, = MWt,h e disjunctivec onstraint( 4) is represented
tion v E a sucht hat by an edge {v, w} in E, and the two wayst o settlet he
disjunctionc orrespondt o the two possibleo rienta-
max s, + t,, (1) tions of the edge. There is an obvious one-to-one
vE&
correspondencbee tweena set of choicesi n (4) that is
is minimizeds ubjectt o overallf easiblea nd an orientationo f all the edgesi n
E for which the resultingd igraphi s acyclic. The
s >- O for all v E a (2)
objectivev alue (the makespan)o f the corresponding
SW- S >, tv if v ->W, v, w E a (3) solutioni s givenb y the lengtho f a longestp athi n this
digraphS. ucha set of orientationds ecomposesa into
SW- SV; tv V sV- S tw chainsc orrespondintgo the machines,i .e., it defines
if MV = Mw, v, w Ec. (4) for each machinea n orderingo r permuationo f the
operationst o be processedb y that machine. Con-
It is usefult o representth e problemb y the disjunctive verselya, set of machinep ermutationds efinesa set of
graphm odel of Roy and Sussmann( 1964). The dis-
orientationso f the edgesi n E, thoughn ot necessarily
junctiveg raphG = (V, A, E) is defineda s follows: one which results in an acyclic digraph.S ince the
* V = A U {0, N + I}, where 0 and N + 1 are two longestp athi n a cyclicd igraphh as infinitel ength,w e
fictitiouso perationst;h e weighto f a vertexv is given can now rephraseth e problema s that of findinga set
by the processingti me tv (to = tN+1 = 0). of machinep ermutationtsh at minimizest he longest
* A = 1(v, w) Iv , w E ,- v-> w} U 1(0, w) IwE , pathi n the resultingd igraphI. n Section3 we use this
Av E a: v -> w} U {(v, N + 1) I v E A, Aw E a formulationo f the problemt o find approximatseo lu-
v -> WI.T hus, A contains arcs connecting consecu- tions by simulateda nnealing.
tive operationso f the samej ob, as well as arcsf rom
0 to the firsto perationo f eachj ob and fromt he last
2. SIMULATEDA NNEALING
operationo f eachj ob to N + 1.
* E -= v, w} IM v = MW}.T hus, edges in E connect Ever since its introduction, independently by
operationst o be processedb y the samem achine. KirkpatrickG, elatt and Vecchi (1983) and Cerny
Job ShopS chedulingb yS imulatedAnnealing/ 115
(1985),s imulateda nnealingh asb eena ppliedt o many tioned probabilityd ecreasesf or increasingv alues of
combinatoriaol ptimizationp roblemsi n such diverse C(j) - C(i) and for decreasingv alueso f c, and cost-
areasa s computer-aidedde signo f integratedc ircuits, decreasingtr ansitionsa rea lwaysa ccepted.
image processing,c ode design and neural network For a fixed value of c, the configurationtsh at are
theory;f or a review the readeri s referredt o Van consecutivelyv isitedb y the algorithmc an be seen as
Laarhovena nd Aarts( 1987). The algorithmi s based a Markovc hainw itht ransitionm atrixP = P(c) given
on an intriguingc ombinationo f ideas from at first by (Aartsa ndV anL aarhoven1 985a,L undya ndM ees
sightc ompletelyu nrelatedf ieldso f science,v iz. com- 1986,a nd Romeo and Sangiovanni-Vincente1ll9i 85)
binatoriaol ptimizationa nd statisticapl hysics.O n the
[GjAij(c) if 5$4 i
one hand, the algorithmc an be considereda s a gen-
eralizationo f the well known iterativei mprovement Pij (c) IS} (5)
approachto combinatoriaolp timizationp roblemso, n l- GjkAik(c) if = i,
k=1
the otherh and,i t can be vieweda s an analogueo f an
algorithmu sedi n statisticapl hysicsf orc omputers im- wheret he generationp robabilitieGs ija reg ivenb y
ulationo f the annealingo f a solid to its grounds tate,
i.e., the statew ith minimume nergy.I n this paper,w e Iv (z) I if j E X(i)
mainly restricto urselvest o the first point of view; G1j(c)= (6)
thus,w e firstb rieflyr eviewi terativei mprovement. 0O otherwise,
Generallya, combinatoriaol ptimizationp roblemi s and the acceptancep robabilitieAs ijb y
a tuple( I, W),w hereR is the set of configurationosr
solutionso f the problem, and C: S -11 R the cost -
Ai(c) = min{1, exp ((Cj) C(i)))}. (7)
function (Papadimitrioua nd Steiglitz 1982). To be
able to use iterativei mprovementw e need a neigh-
The stationaryd istributiono f this Markovc haine xists
borhoods tructureY : R -- 2'; thus, for each config-
and is given by (Aartsa nd Van Laarhoven1 985a,
urationi , XA(i)i s a subseto f configurationsc,a lledt he
Lundya nd Mees 1986,a nd Romeoa nd Sangiovanni-
neighborhooodf i. Neighborhoodasr eu suallyd efined
Vincentelli1 985):
by firstc hoosinga simplet ype of transitiont o obtain
a new configurationfr oma giveno ne and then defin- i(c) = IX l(i) IA i0(c) (8
ing the neighborhooda s the set of configurationtsh at Ej)E=- I, /-(j) IA =4(c)
can be obtainedf rom a given configurationin one
transition. for some io E _opt where Ropt is the set of globally
minimalc onfigurationsp, rovidedt he neighborhoods
Givent he set of configurationsa, cost functiona nd
ares ucht hatf ore achp airo f configuration(si ,j ) there
a neighborhoodst ructurew, e can definet he iterative
is a finite sequenceo f transitionsle adingf rom i to j.
improvementa lgorithma s follows. The algorithm
The latterc onditioni s equivalentt o the requirement
consistso f a numbero f iterationsA. t the starto f each
that the matrix G be irreducibleI. t can readilyb e
iterationa, configurationi is givena nd a transitiont o
shownt hat
a configurationj E XA(i) is generated. If C(j) < C(i),
the start configurationi n the next iteration is j, [ Mo m1 if i E Ropt
otherwiseit is i. If R is finitea nd if the transitionsa re lim qi(c) = (9)
generatedin some exhaustivee numerativew ay, then CI lo otherwise.
the algorithmte rminatesb y definitioni n a local min-
We recall that the stationaryd istributiono f the
imum. Unfortunatelya, local minimum may differ
Markovc haini s defineda s the probabilitdy istribution
considerablyin cost from a globalm inimum.S imu-
of the configurationasf tera n infiniten umbero f tran-
lated annealingc an be vieweda s an attemptt o find
sitions.T hus,w e concludef rom( 9) thatt he simulated
near-optimalol cal minimab y allowingt he acceptance
annealinga lgorithmc onvergesw ith probability1 to a
of cost-increasintgr ansitionsM. orep reciselyi, f i and
globallym inimalc onfigurationif the followingc on-
j E X(i) are the two configurationtso choose from,
ditionsa re satisfied:
thent he algorithmco ntinuesw ithc onfiguratiojn w ith
a probabilityg ivenb y mint1 , exp(-(C(j) - C(i))Ic)} * the sequence of values of the control parameter
wherec is a positivec ontropl arameterw, hichi s grad- convergest o 0;
uallyd ecreasedd uringt he executiono f the algorithm. * the Markovc hainsg enerateda t each value of c are
Thus, c is the analogueo f the temperaturein the of infinitel ength;a nd
physicala nnealingp rocess.N ote that the aforemen- * the matrixG is irreducible.
116 / VAN LAARHOVENA, ARTS AND LENSTRA
Unfortunately, the neighborhood structurec hosen for jE5'9X
job shop schedulingi n Section 3 is such that the
correspondingm atrix G is not irreducibleI. n that 0 < lim PrIX(k) =j =E PrIX(O)= i}* qj
k-~~~~~~ooi
case, we can still provea symptoticc onvergencep ro-
videdt he neighborhoodas re such that for each con-
= ( E Pr{X(O) = ix xi, + E PrIX(O) =i
figurationi there is a finite sequenceo f transitions
leading from i to some configuration io E 4op(, Van
Laarhoven1 988). To do so, we use the fact that in A10j(c)
every chain the recurrentc onfigurationsc an be A101(c)
divided uniquely into irreduciblee rgodic sets Sl,
Y2, ..., 5T. In additiont o the ergodics ets there Z5A i,,j(c) ( 12)
is a set S9 of transientc onfigurationsf rom which
configurationisn the ergodics ets can be reached( but Using (7) we find
not vice versa).N ote that if the neighborhoodssa tisfy
the aforementionedco ndition,t hen each 5Y, contains lim Aioj(c) 0 (13)
-
at leasto ne globallym inimalc onfiguration.
Now considert he sequenceo f configurationcso n- ifj E Yt, jI Mopt. Consequently,
stitutingt he Markovc haina ssociatedw ithP (c). There
aret wo possibilitiese:i thert he Markovc hains tartsi n limcjO(limk,0P- r{X(k)= j) = 0
a transientc onfigurationo r it does not. In the latter for any transiento r nongloballym inimal recurrent
case,t he configurationcso nstitutingth e Markovc hain configurationj . In other words
all belongt o the same irreduciblee rgodics et 5t and
we can provea symptoticc onvergencea s before,w ith lim(lim Pr{X(k)E =opt = 1 (14)
M replaced by St. On the other hand, if the Markov c4O kk--oo
chain startsi n a transientc onfigurationi,t will even- whereM ' ptd enotest he nonemptys et of globallym in-
tually "land" (Feller 1950) in an ergodic set 5t, t E imal recurrenct onfigurations.
fl, ..., T), thoughi t is not a priorik nownw hicho ne. Someo f the conditionsf ora symptoticc onvergence,
The line of reasoningd escribeda bove can then be as, for instance,t he infinitel engtho f Markovc hains,
applieda gain. cannot be met in practice.I n any finite-timei mple-
We can maket he precedinga rgumentms orep recise mentation,w e thereforeh ave to make a choice with
by introducingt he notion of a stationary matrix Q, respectt o each of the followingp arameters:
whosee lementsq jja red efinedb y
* the lengtho f the Markovc hains,
* the initialv alueo f the controlp arameter,
qij = lim Pr{X(k)= j I X(0) = i}. (10)
* the decrementr uleo f the controlp arameter,
k-+oo
* the finalv alueo f the controlp arameter.
Usingt he resultsi n Chapter1 5, Sections6 -8 of Feller,
we obtain Sucha choicei s usuallyr eferredto as a coolings ched-
ule or an annealings cheduleT. he resultsi n this paper
have been obtainedb y an implementationo f simu-
0 ifj E or i E tj Yt
lated annealingt hat employs the cooling schedule
for some t E{,.. ., T,
derivedb y Aartsa nd Van Laarhoven(1 985a,b ). This
is a three-parametesrc hedule:T he parameterXs oa nd
q-j= Aio(c) if i, jE Yt( E.d eterminet he initiala nd final valueso f the control
alEl(=-9,,Ai,,I(c) fors omet E I1 ,. . .. , T), parameterr, espectivelyw, hereast he decrementr ule
dependso n a parameter6 in the followingw ay:
| Awo(c) if i E-- jEE t
X"Et s>tA1(c) for some tE {1,.. . T, Ck+ I = ~Ck (5
1 + [Ck * ln(1 + 6)/3ck] (15)
where xit is the probabilityt hat the Markovc hain, wherec k is the valueo f the controlp arameterfo r the
startingf romt he transientc onfigurationi, eventually kth Markovc haina nd Sk is the standardd eviationo f
reaches the ergodic set Yt. the cost valueso f the configurationosb tainedb y gen-
From ( 11) we obtain,f or a recurrenct onfiguration eratingt he kth Markovc hain.
Job ShopS chedulingb y SimulatedAnnealing/ 117
The decremento f Ck is such that the stationary annealingt o the job shop schedulingp roblemi s dis-
distributions of two succeeding Markov chains cussedi n mored etail.
approximatelys atisfy (Aarts and Van Laarhoven
1985a)
3. SIMULATEDA NNEALINGA ND JOB SHOP
SCHEDULING
I < qi(ck) < +
1+
<qi(ck+1) We recallf rom the previouss ectiont hat in ordert o
applys imulateda nnealingt o any combinatoriaol pti-
k 1,2,... foralliEM. (16)
mization problem,w e need a precise definitiono f
configurationsa, cost function and a neighborhood
Thus,f ors mallv alueso f 6, the stationaryd istributions
structureF. urthermoret,o prove asymptoticc onver-
of succeedingM arkovc hainsa re" close"t o eacho ther
gencew e must show that the neighborhoodst ructure
andw e thereforea rguet hat,a fterd ecreasingC kt o Ck+l, is sucht hatf or an arbitrarcyo nfigurationi t heree xists
a smalln umbero f transitionss ufficest o let the prob-
abilityd istributiono f the configurationasp proachth e at least one globallym inimalc onfigurationio E Ropt
that can be reachedf rom i in a finite number of
new stationary distribution q(ck+1 ). Note that small
transitions.H ereinafterw, e discuss these items in
values of 6 correspondt o a slow decremento f the
mored etail.
controlp arameter.
Finally,w e chooset he lengtho f eachM arkovc hain, 3.1. Configurations
Lk, equalt o the size of the largestn eighborhoodi,. e.,
We recallf rom Section 1 that we can solve the job
Lk =max IX (i)I, k= 1, 2, . . . . (17) shop scheduling problem by considering sets of
iE R machinep ermutationsa nd by determiningf,o r such
a set of permutationst,h e longestp ath in the digraph
We have applieds imulateda nnealingb asedo n this
whichr esultsf romg ivingt he edgesi n the disjunctive
coolings chedulet o many problemsi n combinatorial
grapht he orientationsd eterminedb y the permuta-
optimization( see, for example,V an Laarhovena nd
tions. We therefored efine a configurationi of the
Aarts)a nd have foundi t extremelyr obusti n that the
final results are typicallyw ithin 2% of the global problem as a set Hli = [7ri, ..., 7rim}oI f machine
minimum,w hen 6 is chosen sufficientlylo w (0.1 or permutationsl.r ik is to be interpreteda s the orderi n
whicht he operationso n machinek are processedI: f
smaller).
Unders omem ilda ssumptionsi,t is possiblet o show Mv = k for some v E &, then lrik(v) denotes the
operationf ollowingv on machinek . Consequently,
that the aforementionedc ooling schedulel eads to a
time-complexitoyf the simulateda nnealinga lgorithm the numbero f configurationiss given by HlkmI Mk!,
whereM ki s the numbero f operationsto be processed
givenb y &(,rLi n I M I), wherer is the time involved
in the generationa nd (possible)a cceptanceo f a tran- by machine k(mk =I{v E &a Mv = k 1).
sition and L is the size of the largestn eighborhood
3.2. Cost Function
(the length of the Markovc hains)( Aaartsa nd Van
For each configurationi we definet he followingt wo
Laarhoven1 985a).I f one workso ut this bound for a
digraphs:
particularc ombinatoriaol ptimizationp roblem,i t is
usuallyp olynomiailn the size of the problemI. n those Di = (V, A U Ei), where
cases,w e havea polynomial-timaep proximational go-
rithm. Such a resultw ith respectt o the efficiency of Ei= {(v, w) I{ v, w) E E and rik(v) = w
the algorithmis only worthwhilein combinationw ith for some k E X1. (18)
resultso n its effectiveness, viz. on the differencei n
D,= (V, A UE i), where
cost betweens olutionsr eturnedb y the algorithma nd
globallym inimal ones. From a theoreticalp oint of i= {(v, w) I {v, w) E E and Xrik(v) = w
view, very little is known about the effectivenesso f
simulateda nnealing,b ut there are many empirical for some k EX,/I 1<l mk - l}. (19)
results;s ee for instancet he extensivec omputational In other words,D i is the digrapho btainedf rom the
experimentos f Johnsone t al. (1989).F ort hej ob shop disjunctiveg raphb y givingt he edgesi n E the orien-
schedulingp roblem,w e presenta n empiricala nalysis tations resultingf rom Hli;t he digraphD i can be
of the effectivenesas nde fficiencyo f simulateda nneal- obtainedf rom D, by takingo nly those arcs from E,
ing in Section4 , but firstt he applicationo f simulated that connect successive operations on the same
118 / VAN LAARHOVEANA, RTSA ND LENSTRA
machine.I t is well knownt hat the longestp athsi n Di to some globallym inimalc onfigurationI.n ordert o
and Di have equall ength.T hus, the cost of a config- do so, we need two lemmas.
urationi can be found by determiningth e lengtho f
a longestp ath from 0 to N + 1 in Di. To compute Lemma 1. Considera n arbitraryc onfigurationi and
such a cost, we use a simple labeling algorithm, an arbitrargyl obalm inimumio E Mopt. If i 4 Mopt,
based on Bellman'se quations( Bellman 1958), for thent he set Ki(io)d efinedb y
solving the longest-pathp roblemi n a digraph.T he
Ki(io) = {e = (v, w) E Es I e
time-complexityo f this algorithmi s proportionatlo
the number of arcs in the graph. In our case, this is criticalA (w, v) E E11 (20)
number equals IA I + I Eij = (N + n) + (N - m); is not empty.
accordinglyt,h e labelinga lgorithmt akes 6(N) time
to computet he cost of a configuration. Proof. The proofc onsistso f two parts:F irst,w e show
that Ei always contains critical arcs, unless i E Mop,;
3.3. NeighborhoodS tructure next that there are alwaysc riticala rcs in Ei that do
A transition is generated by choosing vertices v and not belongt o EiOun lessa gaini E8
opt.
w, such that:
1. Supposet hat Ei containsn o criticala rcs,t hen all
1. v and w ares uccessiveo perationso n somem achine criticala rcs belongt o A. Consequentlya, longest
k; path consists of arcs connectingv ertices corre-
2. (v, w) E E, is a criticala rc, i.e., (v, w) is on a spondingt o operationso f the same job; accord-
longest path in Di; ingly,i ts lengthi s givenb y the totalp rocessingti me
of thatj ob. But this is a lowerb oundt o the length
andr eversingth e orderi n whichv and w arep rocessed
of a longest path in any digraphD a, hence i E
on machinek . Thus, in the digraphD i such a transi- Mpt.
2. Supposet hat for all criticala rcs e in Ei, we have
tion resultsi n reversingt he arc connectingv and w
e E EiP.W e then know that any longestp athp in
and replacingth e arcs( u, v) and (w, x) by (u, w) and
Di is also a path q in Di,. The lengtho f a longest
(v, x), respectively, where u = 1rki1ri(kv()W a)n. d x = path r in DBios also the lengtho f a longestp athi n
Our choice is motivated by two facts:
Dio and because io E Ropt, we have length(r) <
* Reversing a critical arc in a diagraph Di can never length(p). But by definition length(r) > length
lead to a cyclicd iagraphD j (see Lemma2 ). (q) = length(p). Consequently, length(p) =
* If the reversal of a noncritical arc in Di leads to an length(r)a nd i E Ropt.
acyclic graph Dj, a longest path q in Dj cannot be
shorter than a longest path p in Di (because Dj still Lemma 2. Supposet hat e = (v, w) E Ei is a critical
containst he pathp ). arc of an acyclic digraphD i. Let Dj be the digraph
obtainedfromD i by reversingth e arc e in Ei. ThenD j
Thus, we exclude beforehand some noncost- is also acyclic.
decreasing transitions and, in addition, all transitions
that might resulti n a cyclic digraph.C onsequently, Proof. Supposet hatD j is cyclic.B ecauseD i is acyclic,
the neighborhoods tructurei s such that the algo- the arc( w, v) is parto f the cyclei n D>.C onsequently,
rithm visits only digraphsc orrespondingto feasible there is a path (v, x, y, . . ., w) in D>.B ut this path
solutions. can also be found in Di and is clearlya longerp ath
The neighborhoodo f a configurationi is thusg iven from v to w than the arc (v, w). This contradictsth e
by the set of acyclicd iagraphtsh at can be obtainedb y assumptiont hat (v, w) is on a longest path in Di.
reversinga criticala rcb elongingt o Ej in the graphD i. Hence,D j is acyclic.
ConsequentlyI,X (i) I < k=1 (Mk - 1) = N- m-
Givena configurationio E we definet wo sets
Mopt,
for an arbitraryco nfigurationi:
3.4. AsymptoticC onvergence
It is not difficult to construct a problem instance Mi(io)= {e = (v, w) E Ei l(w, v) EG io} (21)
containing pairs of configurations( i, j) for which there
is no finite sequence of transitions leading from i to j Mi(io)= {e = (v, w) E Ei I (w, v) e EioJI (22)
(Van Laarhoven)T. hus,t o provea symptoticc onver- In view of Section 2, the followingt heorem now
gence, we must show that for each configurationi ensuresa symptoticc onvergencei n probabilityt o a
therei s a finites equenceo f transitionsle adingf rom i globallym inimalc onfiguration.
JobS hopS chedulingb y SimulatedAnnealing/ 119
Theorem 1. For each configurationi ( it is operations.F or the Lawrencei nstances,p rocessing
4opt
possible to constructa finite sequenceo f transitions times are drawnf rom a uniformd istributiono n the
leadingf rom i to a globallym inimalc onfiguration. interval[ 5, 99];t he sequenceo f machinesf or eachj ob
is random.
Proof. We choosea n arbitrarcyo nfigurationio E Mopt The performance of simulated annealing on
and construct a sequence of configurations Xo, these instancesi s reportedi n Table I for the Fisher-
XI, . . .} as follows: Thompson instances, and in Table II for the
1. X0=i. Lawrencein stances.T he averagesi n these tablesa re
2. Xk+I is obtained from Xk by reversinga n arc computedf rom five solutions,o btainedb y running
e E Kx k(o) in EXk. According to Lemma 2, this can the algorithm,c ontrolledb y the cooling schedule
be done withoutc reatinga cyclei n Further- describedi n Section 2, five times on each instance
Dxk+,.
more,t his operationi s of the aforementionedty pe andr ecordingth e bestc onfigurationen counteredd ur-
of transition. ing each run (this need not necessarilyb e the final
configuration)T. he probabilisticn atureo f the algo-
It is easyt o see that if IJ 1k(io) I > 0 then rithm makesi t necessaryt o carryo ut multipler uns
|MI.k+l(iO)=I |I k(iO) I 1. (23) on the same problemi nstancei n ordert o get mean-
-
ingfulr esults.
Hence, for k = IM 1(iol), IM xk(io) I = 0. Using All resultsa re obtainedw ith the parameterXs oa nd
Ki(io)5 Mi(io)C M1(io)w, e find KXk(io)= 0 for k = e, set to 0.95 and 10-6, respectivelya, nd for different
Mi i(io)1 . According to Lemma 1, this implies valueso f the distancep aramete8r . Runningt imesa re
Xk E Ropt* CPU times on a VAX-785.
From Tables I and II we can make the following
observations:
4. COMPUTATIONALR ESULTS
1. The qualityo f the averageb est solutionr eturned
We have analyzedt he finite-timeb ehavioro f the by the algorithmi mprovesc onsiderablyw hen 8 is
simulateda nnealinga lgorithme mpiricallyb y running decreasedT. his is in accordancew ith the theory
the algorithmo n a numbero f instanceso f the job underlyingth e employedc oolings chedules: maller
shops chedulingp roblemv, aryingin size froms ixj obs values of 8 correspondt o a better approxima-
on six machinest o thirtyj obs on ten machines.F or tion of the asymptoticb ehavior( Aartsa nd Van
all instances,t he numbero f operationso f each job Laarhoven 19 85a). Furthermoret, he difference
equals the number of machines and each job has betweent he averageb est solution and a globally
preciselyo ne operationo n eachm achine.I n thatc ase, minimalo ne doesn ot deterioratsei gnificantlwy ith
the numbero f configurationosf eachi nstancei s given increasingp roblems ize.F ort he FIS2 instance,t he
by (n!)m,t he labelinga lgorithmt akes6 (nm) time to fiveb ests olutionso btainedw ith8 = 10-' havec ost
computet he cost of a configurationa, nd the size of values of 930 (twice),9 34, 935, and 938, respec-
the neighborhoodo f a configurationis boundedb y tively.T hus, a globallym inimals olutioni s found
m(n- 1). 2 out of 5 times,w hichi s quitea remarkablree sult,
FIS1, FIS2 and FIS3 (Table I) are three problem consideringth e notorietyo f this instance.
instancesd ue to Fisher and Thompson( 1963), the 2. As for runningt imes, the bound for the running
fortyi nstancesin TableI I ared ue to Lawrence( 1 984). time given in Section 2 is ((nm)3 In n) (L =
FIS2 is a notorious1 0-job,1 0-machinein stancet hat 6(nm), I II = 6((n!)m) andr = 6(nm)). Thus, for
hasd efieds olutiont o optimalityf orm oret hant wenty fixed m the boundi s &(n3I n n), and for fixedn it
years.A coupleo f yearsa go, a solutionw ith cost 930 is (m'3).F or the A, B and C instancesi n TableI I,
was found after 1 hour 47 minuteso f runningt ime, for whichm is constant,t he averager unningt ime
and no improvemenwt as founda fter9 hours6 min- T for 8 = 0.01 is approximatelyg iven by t =
utes( Lageweg1 984).T hisc ost valuew aso nly recently to * n2215 In n, for some constantt o (X2 = 1.00);
provedt o be globallym inimalb y Carliera nd Pinson fort he G, B andI instancesf, orw hichn is constant,
(1989).F orF IS1, FIS2 andF IS3,t he processingti mes the averager unningt ime for 8 = 0.01 is approxi-
of the operationsa rer andomlyd rawna nd rangef rom matelyg ivenb y T= t mr 2.406f,o r some constant
1 to 10 (FIS1)o r to 99 (FIS2 and FIS3) unitso f time. t1 (x 2 = 1.00). Thus, the observedr unningt imes
The sequenceo f machinesf or each job is such that are in good accordancew ith the bound given in
lower-numberemd achinest end to be used for earlier Section2 .
120 / VAN LAARHOVENA, ARTS AND LENSTRA
Table I
Results for ProblemI nstanceso f Fishera nd Thompson (1963)"
SimulatedA nnealing ABZ MSS
Problem 6 C tC % T at Cbt CA t4A CM tM
6 machines,6 jobs
FISi 10'1 56.0 1.3 1.82 8 1 55* 55* 1 -
10-2 55.0* 0.0 0.00 52 8 55*
10 machines, 10 jobs
FIS2 10-1 1,039.6 15.1 11.78 113 13 1,028 930* 426 946 494
10-2 985.8 22.1 6.00 779 61 951
10-3 942.4 4.5 1.33 5,945 180 937
10-4 933.4 3.1 0.37 57,772 2,364 930*
20 machines,5 jobs
FIS3 10-1 1,354.2 26.5 16.24 123 13 1,325 1,178 40
10-2 1,229.0 33.6 5.49 848 93 1,184
10-3 1,187.0 18.7 1.89 6,840 389 1,173
104 1,173.8 5.2 0.76 62,759 7,805 1,165*
IterativeI mprovement
#C UrC % t a-t Gaist
FISi 803.2 55.4 0.8 0.73 52 0 55*
FIS2 9,441.2 1,018.2 9.1 9.48 5,945 0 1,006
FIS3 5,221.0 1,331.4 9.5 14.28 6,841 0 1,319
a Simulateda nnealing: tion over five macroruns;
6 : the distancep arameter( smaller1 -valuesi mply slower t, oS the averager unningt ime and standardd eviationo ver
cooling); five macroruns( in seconds);
C, uc: the averagec ost and standardd eviationo f best solu- Cbes, the best cost found over five macroruns;
tion over five runs; S the percentageo f C over optimalc ost.
t, o? the averager unningt ime and standardd eviationo ver Adams,B alasa nd Zawack( 1988):
five runs (in seconds); CA : the best cost found;
Cbes, : the best cost found over five runs; tA runningt ime (in seconds).
So : the percentageo f C over optimalc ost. Matsuo,S uh and Sullivan( 1988):
Iterativei mprovement: CM the best cost found;
: the averagen umbero f local minima over five macro- tM runningt ime (in seconds);
runs (see text); * provablyo ptimalc ost.
C, rc: the averagec ost and standardd eviationo f best solu-
In the remaindero f this section we comparet he ment areo btainedf romf ive macrorunsE. achm acro-
resultso btainedw itho ur methodw ithr esultso btained run consists of repeatede xecution of the iterative
with threeo therm ethods,v iz. improvementa lgorithmf or a large numbero f ran-
domlyg eneratedin itialc onfigurationasn d thusy ields
* time-equivalenitt erativei mprovement, a largen umbero f local minima. Executiono f each
* the shiftingb ottleneckp rocedure(A dams,B alasa nd macroruni s terminateda s soon as the runningt ime
Zawack)a, nd is equal to the runningt ime of an averager un of
* controlleds earchs imulateda nnealing( Matsuo,S uh simulateda nnealinga pplied to the same problem
and Sullivan). instancew ith the distancep arameter6 set to i0-'
(10-2f or FIS1);C is the averageo f the best cost value
foundd uringe ach macrorun.
4.1. Time-Equivalent Iterative Improvement
We observe that repeatede xecution of iterative
Table I also contains results obtained by repeated improvementi s easily outperformedb y simulated
executiono f a time-equivalenitt erativei mprovement annealingf or the two largerp roblemsT. he difference
algorithmb asedo n the same neighborhoodst ructure is significantf:o r FIS3, for instance,t he averageb est
as our simulateda nnealinga lgorithm.I t is basedo n solution obtainedb y simulateda nnealingi s almost
repeatede xecution of iterative improvement.T he 11 %b etteri n cost than the one obtainedb y repeated
averagesf or the time-equivalentit erativei mprove- executiono f iterativei mprovement.
Job ShopS chedulingb y SimulatedAnnealing/ 121
Table II
Results for ProblemI nstanceso f Lawrence( 1984)a
SimulatedA nnealing ABZ MSS
Problem 6 CYc t St Ctst CA tA CM tM
10 machines, 10 jobs
Al 1.0 1023.4 30.6 26 1.5 991 978 120 959 78
0.1 981.0 17.3 110 17.1 956
0.01 966.2 10.1 686 83.3 956
A2 1.0 861.0 41.2 23 3.7 797 787 96 784 47
0.1 792.4 6.2 112 7.0 784
0.01 787.8 1.6 720 109.0 785
A3 1.0 902.6 30.9 23 1.6 870 859 112 848 53
0.1 872.2 12.4 112 22.1 861
0.01 861.2 0.4 673 69.0 861
A4 1.0 950.0 54.5 24 5.3 904 860 120 842 58
0.1 881.4 6.9 97 20.4 874
0.01 853.4 4.6 830 85.4 848
A5 1.0 1021.6 26.2 30 1.9 994 914 144 907 50
0.1 927.6 18.9 86 7.9 907
0.01 908.4 4.2 667 126.9 902
10 machines, 15 jobs
BI 1.0 1176.2 37.8 69 6.7 1133 1084 181 1071 103
0.1 1115.2 23.9 299 50.9 1085
0.01 1067.6 3.7 1991 341.1 1063
B2 1.0 1125.6 35.6 65 3.6 1094 944 210 927 92
0.1 977.4 19.5 307 36.5 963
0.01 944.2 4.7 2163 154.6 938
B3 1.0 1155.8 64.2 63 5.6 1056 1032* 113 1032* 10
0.1 1051.0 24.6 275 35.8 1032*
0.01 1032.0* 0.0 2093 89.7 1032*
B4 1.0 1101.0 53.5 71 5.0 1032 976 217 973 100
0.1 977.6 8.1 252 28.5 968
0.01 966.6 8.7 2098 406.0 952
B5 1.0 1114.6 9.1 77 16.9 1103 1017 215 991 90
0.1 1035.4 10.6 283 44.3 1017
0.01 1004.4 14.4 2133 374.5 992
10 machines,2 0 jobs
Cl 1.0 1397.0 69.1 139 16.0 1311 1224 372 1218* 27
0.1 1268.0 9.7 555 81.7 1252
0.01 1219.0 2.0 4342 597.8 1218*
C2 1.0 1434.2 40.0 139 6.4 1390 1291 419 1274 143
0.1 1311.6 12.7 651 82.9 1295
0.01 1273.6 5.2 4535 392.0 1269
C3 1.0 1414.6 57.8 135 7.4 1335 1250 451 1216 153
0.1 1280.2 23.6 614 83.3 1246
0.01 1244.8 15.4 4354 349.8 1224
C4 1.0 1387.4 47.0 138 14.1 1307 1239 446 1196 134
0.1 1260.4 35.4 581 24.0 1203
0.01 1226.4 6.5 4408 450.9 1218
C5 1.0 1539.2 44.2 145 20.6 1492 1355* 276 1355* 4
0.1 1393.6 9.6 605 84.4 1381
0.01 1355.0* 0.0 3956 428.2 1355*