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A multi-start Dynasearch algorithm for the time dependent single-machine total weighted tardiness
scheduling problem
Eric Angel, Evripidis Bampis
To cite this version:
Eric Angel, Evripidis Bampis. A multi-start Dynasearch algorithm for the time dependent single- machine total weighted tardiness scheduling problem. European Journal of Operations Research, 2005, 162 (1), pp.281–289. �hal-00341340�
single-mahine total weighted tardiness sheduling problem
E. Angel, E.Bampis
Universite d'
EvryVal-d'Essonne
LaMI
91025 Evry, Frane
email: fangel, bampisglami.univ-evry.fr
June 11, 2001
Abstrat
Weextendthedynasearhtehnique,reentlyprop osedbyCongrametal,intheontext
of time-dep endent ombinatorialoptimization problems. As an appliation we onsider a
general time-dep endent (idleness) version of thewell known single-mahine total weighted
tardinessshedulingproblem,inwhihthepro essing timeofajobdep ends onitsstarting
timeofexeution. Wedevelopamulti-startlo alsearhalgorithmandpresentexp erimental
resultsonseveraltyp esofinstanesshowingthesup eriorityofthedynasearhneighb orho o d
overthetraditionalone.
1 Introdution
Inthis pap erwe evaluatethep erformane ofdynasearh intheontext of time-dep endent
sheduling. Dynasearhisareentlyprop osedneighb orho o dsearhtehnique[5℄thatallows
aseries ofmoves tob e p erformedat eah iteration of alo alsearh algorithm,generating
inthat wayan exp onentialsize neighb orho o d. Inorder to eÆiently exploresuh aneigh-
b orho o d,dynasearhusesdynamiprogramming. Congrametal. applieddynasearhtothe
lassialsinglemahinetotalweightedtardinessproblem(1jj P
w
j T
j
)[5℄andomparedthe
qualityoftheobtainedsolutionswithtraditionalmulti-startanditerateddesentalgorithms.
Theobtainedomputationalresultswereveryenouragingintheasewheredynasearhwas
appliedinsideaniteratedlo alsearhalgorithmwhenomparedtolassialiterateddesent
algorithms. However, the appliationof dynasearh in the ase of a multi-start algorithm
gave marginallyb etter results than the lassial multi-start metho ds. It is then natural
to ask if dynasearh is notso appropriate for multi-start lo alsearh algorithms. In this
work,we show that this is not true. More preisely, we study thetimedep endent version
ofthesinglemahinetotalweightedtardinessproblem,and we presentomputationresults
showingthatmulti-startdynasearh learlydominatesthelassialmulti-startlo alsearh
algorithms.
In sheduling theory there have b een an inreasing interest, in the last few years, for
shedulingproblemswithtime-dep endentpro essing times[1℄. Inthispap erwe onsidera
general time-dep endent version of the well known single-mahine total weighted tardiness
shedulingproblem. The probleman b e stated as follows. We are given aset of n jobs,
eahjobjhasaduedated
j
andap ositiveweightw
j
. Thepro essingtimef
j
(t)ofeahjob
j dep ends onits startingtimeof exeution t and isgivenby afuntionf
j
. We shall note
p t
j for f
j
(t). Soif ajobj immediatelystarts after ajobi, itsduration is p C
i
j
with C
i the
valuesareinteger ones.
Sine we make no assumptions on the funtions f
j
, this mo del aptures a wide range
of pratial appliations. A rst example is when the availability of the resoures (e.g.
pro essingp ower) vary (e.g. inamonotoneor yli way)over time;thinkfor exampleat
theloadof aomputer network. A seond example is whenany delayin theexeution of
ajobmayleads to aninrease (resp. derease) ofthe diÆultyof thejoband therefore to
amo diedduration; think for exampleto re ghting(resp. destroying atarget whih is
gettingloser).
WedenotebyC
j
theompletiontimeandbyT
j
=maxfC
j d
j
;0gthetardinessofjobj.
Theobjetiveistondashedulewhihminimizesthetotalweightedtardiness P
n
j=1 w
j T
j .
Adoptingthethree-eldstandard notationofGrahametal. [7℄we willdenotethisproblem
by1jp t
j
;idl enessj P
j w
j T
j .
ThisproblemisstronglyNP-hardsine itisageneralizationofthesingle-mahinetotal
weightedtardiness problem1jj P
j w
j T
j
[11℄. Indeed, thereexistsadynamiprogramming
algorithmwitharunningtimeO (n 3
P
n
j=1 p
j
)fortheproblem1jj P
j w
j T
j
,but onlywhen
weightsare agreeable,that isp
j p
k )w
j w
k
for alljobs j andk [11℄. There existsa
branh and b ound algorithmforthe 1jj P
j w
j T
j
problem[14℄, but as it is rep orted in[5℄
itannot b eused inpratie oninstanes withmorethan 50jobs. Moreoverthe designof
approximationalgorithmsseems very hard, sine theonly known results onern onlythe
farless general 1jj P
j T
j
problemwith aFPTAS due to Lawler [12℄ in O (n 7
=) time,and
slightlyimprovedbyKovalyov[10℄inO (n 6
logn+n 6
=)time.Sinetheproblemweonsider
isabroad generalizationofthe1jj P
j w
j T
j
problem,theseresults stress theimp ortaneof
themetaheuristi approahifonewantstopratiallydealwithinstanesofthisproblem.
2 Dynasearh neighborhood
Lo al searh algorithms, and their generalizations suh as simulated annealing and tabu
searh (also alled metaheuristis), are often used to obtain near optimal solutions for a
widerangeof NP-hardombinatorialoptimizationproblems[15, 3,4℄. Inthese metho dsa
neighborhood is dened, usually by givinga set of transformations that an b e appliedon
theurrent solution. In the simplestlo al searh metho d, at eah iteration the algorithm
searhes theneighb orho o d oftheurrent solutionfor ab etter one. Ifsuh asolutionexists
it b eomes thenew urrent solutionand the pro ess go eson, otherwise thealgorithmhas
foundaloaloptimalsolutionandstops.
Ingeneral, themorelarge isthe neighb orho o d,theless there arelo aloptima andthe
b etterinqualitytheyare. Reentlyexp onentialsizeneighb orho o ds,whihanb e neverthe-
lesssearhed inp olynomialtime,haveb eenprop osed forthetravelingsalesmanproblem[8℄
(see also the pioneer works [16℄ and [17℄), the one mahine bathing problem[9℄,and the
singlemahinetotalweightedtardinessshedulingproblem[5℄.
Thedynasearhneighb orho o dweuseisbasedontheswapneighb orho o dwhihgivesthe
b estresults ompared to other ones for the 1jj P
j w
j T
j
problem[6℄ [2℄,and probablyfor
thegeneralized problemthatwe onsider.
Weshallrepresentasolutionbyap ermutation=( (1);:::; (n))ofthesetf1;2;:::;ng,
meaningthat job (j)isthej-thjobtob e sheduled.
Givenap ermutation=( (1);:::; (i);:::; (j)::: (n))theswapneighb orho o d on-
sistsofalln(n 1)=2p ermutations 0
=( (1);:::; (j);:::; (i)::: (n)),with1i<j
n,thatanb eobtainedfrombyswappingtwojobs. Amovethatswapsjob (i)with (j),
andamovethatswapsjob (k )with (l )aresaidtob eindependentifmaxfi;jg<minfk ;l g
ormaxfk ;l g<minfi;jg.
Thedynasearhswapneighb orho o d,intro duedin[5℄,onsistsofallsolutionsthatanb e
obtainedbyaseriesofpairwiseindep endentswapmoves.Forexample,giventhep ermutation
=(134625810 9712141311),we anapplythe3indep endent swapmovesgured
to obtainthe neighb oringp ermutation 0
= (1 64 325 710 9 812 14 11 13). It isnot
diÆultto seethatthisneighb orho o dhassize2 1.
Tosearh this exp onential neighb orho o d inaneÆient way,i.e. to ndthe b estneigh-
b oringp ermutationamongthe2 n 1
1andidatep ermutations(i.e. weusesteep estdesent
lo alsearh), adynamiprogrammingalgorithmisused. We use abakwardenumeration
shemeinwhihjobsareapp endedto theb eginningoftheurrent partialsequeneandare
p ossiblyswapp ed withjobsalreadysheduled inthepartialsequene.
We note(x) +
=maxfx;0gforanyintegerx. Let=( (1);:::; (i);:::; (j)::: (n)),
b eap ermutation. Wenote(
i
;t)theb estp ossiblewaytoshedulejobs (i); (i+1);:::; (n)
byapplyingaseriesofindep endentswapsonthesub-p ermutation( (i); (i+1);:::; (n)),
assuming that the rst jobsheduled inthat sub-p ermutation is sheduled at timet. We
takeonlyintoaountthetotalweightedtardinessofjobs (i), (i+1),::: , (n)andforget
jobs (1); (2);:::; (i 1)when dealing with(
i
;t). We note F(
i
;t) theorresp onding
totalweightedtardiness of jobs (i), (i+1), ::: , (n)in the state (
i
;t). We shall put
(
n+1
;t)=;andF(
n+1
;t)=0foranytimettosimplifythedesription ofthealgorithm
b elow.
Now thestate (
i
;t) must b eobtainedeither by app endingthejob (i)infront ofthe
state(
i+1
;t+p t
(i)
)orbyapp endingthesequene( (j); (i+1);:::; (j 1); (i)),obtained
byswapping jobs (i)and (j), infrontof thestate (
j+1
;t 0
)for somejobi+1<j n
andtimet 0
(tob e determinedlater).
We havefortherstaseF(
i
;t)=w
(i) (t+p
t
(i) d
(i) )
+
+F(
i+1
;t+p t
(i)
):Forthe
seondase,lett
k
b ethestartingtimeof thek -thsheduled jobforikj after having
swapp ed (i)and (j). By denitionofF(
i
;t),t
i
=t. Then sinejobs (i)and (j)have
b eenswapp ed,t
i+1
=t
i +p
ti
(j)
. Finally,t
k
=t
k 1 +p
t
k 1
(k 1)
fori+1<kj. Thuswehave
F(
i
;t)= w
(j) (t
i +p
ti
(j) d
(j) )
+
+ X
i<k <j w
(k ) (t
k +p
tk
(k ) d
(k ) )
+
+
w
(i) (t
j +p
t
j
(i) d
(i) )
+
+F(
j+1
;t
j +p
t
j
(i) ):
Ifj =i+1,thesum P
i<k <j
isempty.
Finallythedynamiprogrammingalgorithmanb e statedas:
F(
n
;t)=w
(n) (t+p
t
(n) d
(n) )
+
and
F(
i
;t)=minfw
(i) (t
i +p
ti
(i) d
(i) )
+
+F(
i+1
;t
i +p
ti
(i) );
min
j;i<jn w
(j) (t
i +p
t
i
(j) d
(j) )
+
+ X
i<k <j w
(k ) (t
k +p
t
k
(k ) d
(k ) )
+
+
w
(i) (t
j +p
tj
(i) d
(i) )
+
+F(
j+1
;t
j +p
tj
(i)
)g; for1i<n
We wantto alulate F(
1
;0). Toimplementthe dynamiprogrammingalgorithmwe
haveused abasi memoizetehnique: anarraystores thevalues ofF alreadyomputedin
ordertoreduethenumb erofreursive alls. Theoptimalsetofindep endentswapsanb e
retrieved by examiningan array whih stores, for eah job j and eah timet for whih a
valueF(
j
;t)wasomputed,thep ositionof (j)inthestate(
j
;t).
We shall assume that the pro essing times are b ounded up by a onstant p
max , i.e.
0p t
(j) p
max
,8j;t. Under thisassumption,thetimetat whihthelastjobanstartis
b oundedupby(n 1)p
max
. Therearen(n 1)p
max
states(
i
;t),eahoneanb eomputed
inO (n 2
) time(assuming that the previous required states have b een already omputed),
leadingtoatotalO (n 4
p
max
)timeomplexity. ThespaeomplexityisO (n 2
p
max ).
3 Experimental results
Thereisertainlyatradeo b etweentheb enee ofusingalargeneighb orho o dintermsof
thequalityoflo aloptimaandtheindued timeinrease,relative toasmallneighb orho o d,
to sp end moretimeexploring a larger neighb orho o d. Toompare the p erformaneof the
standard and the dynasearh swap neighb orho o ds we have used multi-start lo al searh
(MLS).
In MLS with the dynasearh swap neighb orho o d, a xed numb er of lo al searh are
p erformed,10 inourase,and we retain theb est lo aloptima foundout ofthese 10 lo al
searh. InMLS withthe standardswap neighb orho o d,the numb erk of lo alsearh isnot
knownapriorianditisafuntionofthetimesp ent,sayT,bytheMLSwiththedynasearh
swapneighb orho o d,namelywe p erformlo alsearh untilthetotaltimesp entis greater or
equal to T. In this way we an fairlyompare the two algorithms. In the sequel, MLS
means multi-start lo al searh with the swap neighb orho o d, and MDS means multi-start
dynasearh,thatis multi-startlo alsearhwiththedynasearh swapneighb orho o d.
Tosp eed-up the lo alsearh algorithmwith the standard swap neighb orho o d, whena
swapinvolvingjobs (i)and (j),with j>i, isevaluated,onlythep ortionintherightof
thejob (j)istakenintoaounttoalulatetheostofthenewsolution(weuseadditional
arraystoreord foreah jobitsstartingtime,andforeahp ositionthepartialostindued
bytheurrentsolutionuptothis p osition).
Aninstaneoftheproblem1jp t
j
;idl enessj P
j w
j T
j
withnjobsisompletelydesrib ed
bygivingthe(1+(n 1)p
max
)npro essingtimesp t
j
with1jnand0t(n 1)p
max ,
withthenweightsand thendue datesforeahjob. Wehavehosenp
max
=20.
We have generated random instanes of various typ es with sizes 30 and 40. For eah
jobj anintegerweightw
j
israndomlyuniformlygeneratedin[1;10℄,whatever thetyp e of
theinstane. Inthe rst typ e of instanes, eah pro essing timep t
j
is indep endent of the
others,andishosenrandomlyin[1;p
max
℄.Thismetho dertainlyleadstothemostdiÆult
instanes, but from apratial p oint of viewthey are notrealisti. Inthe seond typ e of
instanes, eah jobj has an ideal starting timeb
j
and is p enalized if its starts either to o
earlyorto olate. Itspro essing timeisgivenbyp t
j
=1+minfp
max 1;b
2jt bjj
n 1
g. Inthe
third(resp. fourth)typ eofinstanes,thepro essingtimesaredereasing(resp. inreasing)
with the time, i.e. the latter a job starts, the shorter (resp. longer) its duration. More
preisely,thepro essingtimeofajobj giventhatit startsattimetismaxf1;bp
max r
j
t
pmax 1
(n 1)p
max
g for the third typ e of instane, with r
j
a real random numb er in[0,1[, t in
f0;1;:::;(n 1)p
max
g,andminfp
max
;b1+r
j t
pmax 1
(n 1)pmax
gforthefourthtyp eofinstane.
The integer due dates were generated in a similar way than what it is usually done
forthe lassial 1jj P
j w
j T
j
problem: For eah job j, an integer due date d
j
is randomly
generated in the interval [P(1 TF RD D =2);P(1 TF +RD D =2)℄ using a uniform
distribution, with P = P
n
j=1 p
j
, RDD 2 f0:2;0:4;0:6;0:8;1:0g the relative range of due
dates, and TF 2 f0:2;0:4;0:6;0:8;1:0gthe tardiness fator. When the tardiness fator is
loseto 1 it means that the majorityof jobs willb e ompleted after their due dates and
theproblem is very onstrained. The relative range of due dates indiates the variability
of the due dates around their mean value. We adopted the same pro edure, with P =
P
n
j=1 P
(n 1)pmax
t=0
p t
j
=((n 1)p
max
+1)totake into aountthetimedep endentpro essing
times.
Wehavegeneratedveinstanesforeahofthe25pairsofvaluesofRDDandTF,whih
yieldsatotalof125instanesforeah sizeandeahtyp eofinstane. Theexp erimentswere
p erformedonaPentiumI I Ibipro essor-500Mhz. Averageresults(over10exeutionsofMLS
andMDS)areshowninTables1to8.
InTables 1and 2,for eah TF and RDDvalues we have three entries. The rstentry
is theost of theb est lo aloptimumfound using MLS (orMDS), theseond entry is the
exeutiontime inseonds, and the last entry is the numb er of lo alsearh p erformedfor
MLSinordertohavethesameexeutiontimethanMDS.Table3indiatestheimprovement
inp erentage ontheostofthesolutionfoundbyMDSversusMLS.
Weanseethandespiteamuhlessnumb eroflo alsearh(10againstseveralhundreds),
MDSp erformsb etter thanMLS.Theimprovementanb e verysigniant,forexamplefor
instaneswith RDD=0.2and TF=0.6,we obtaintheoptimalsolutionwithMDS, whereas
MLSgivessolutionswithaostof20.1.
0.2 0.0;2.3;366 0.0;2.4;122 20.1;8.7;217 1966.1;23.2;422 12939.8;24.4;390
0.4 0.0;2.3;271 0.0;3.1;130 2.3;10.8;231 2237.4;23.9;405 14287.1;24.9;397
0.6 0.0;2.4;231 0.0;4.9;187 0.9;13.5;265 3297.3;24.6;404 11723.3;25.4;399
0.8 0.0;2.7;186 0.0;7.2;235 65.6;16.5;306 4356.7;23.9;378 17836.4;25.4;391
1 0.0;3.6;208 0.0;10.3;280 1035.2;20.0;345 6970.6;25.0;380 18160.4;26.6;398
Table 1: size: 40,typ e: 1,multi-startlo alsearh
rddntf 0.2 0.4 0.6 0.8 1.0
0.2 0.0;2.3;10 0.0;2.4;10 0.0;8.7;10 880.3;23.1;10 11769.6;24.3;10
0.4 0.0;2.3;10 0.0;3.1;10 0.0;10.8;10 1245.7;23.9;10 13064.3;24.9;10
0.6 0.0;2.4;10 0.0;4.9;10 0.5;13.5;10 2507.8;24.6;10 10820.6;25.4;10
0.8 0.0;2.7;10 0.0;7.2;10 49.8;16.5;10 3895.1;23.9;10 17010.8;25.4;10
1 0.0;3.6;10 0.0;10.3;10 994.0;19.9;10 6567.3;24.9;10 17361.0;26.5;10
Table 2: size: 40, typ e: 1,multi-startdynasearh
rddntf 0.2 0.4 0.6 0.8 1.0
0.2 0 0 -100.0 -55.2 -9.0
0.4 0 0 -100.0 -44.3 -8.6
0.6 0 0 -44.4 -23.9 -7.7
0.8 0 0 -24.1 -10.6 -4.6
1 0 0 -4.0 -5.8 -4.4
Table 3: size: 40, typ e: 1
Due to spae limitation we present only the p erentage of improvementfor the other
typ esofinstaneswithsize30and40inTables4to8. Weanseethatintheoverwhelming
ofases, multi-startdynasearhgivesb etterresults. Thereisanexeptionhoweverfortyp e
4instanes. For them we obtain exatly the same ost with multi-start lo al searh and
dynasearh. Thisprobablyindiatesthatthese instanesaresolvedto optimality.
In the pap er of Congram et al. [5℄, MDS gave only little improvement over MLS for
the 1jj P
w
j T
j
problem. The improvementwas however signiant for a more elab orated
metho dalled iterated lo alsearh (in whih anewlo alsearh is started froma solution
\lose"tothepreviouslyfoundlo aloptimum,insteadofarandomlygeneratedsolutionasit
istheaseforMLSandMDS).TheexplanationwegivefortheeÆienyofMDSoverMLS
for the time-dep endent sheduling problem we onsider is the followingone. Inour ase,
startingfromasolution , theswapoftwojobs (i)and (j)(i<j)an leadtoasolution
0
inwhihtheswap oftwojobs (k ) and (l ) (i<j <k<l or k<l <i<j)leads toa
solution 00
withalowerostthansolution ,whereasp erformingrsttheswapofjobs (k )
and (l ) onsolution isnot protable,i.e. itinreases the ost of thesolution obtained.
Thissituationisnotenounteredinthestatishedulingproblem1jj P
w
j T
j
. Therefore,we
take theb enee of alo okahead apabilitywhih is absent fromthestandard lo alsearh
algorithmswhiharetraditionallymyopiinnature.
4 Conlusion and extensions
Ourwork is intheontinuityof Congramet al. [5℄ whih have intro dued the dynasearh
swapneighb orho o dforthe1jj P
j w
j T
j
problem. Byintro duingthetimeparameterinside
thedynamiprogrammingalgorithmwe obtainapseudop olynomialalgorithmintimeand
spae,whereas theiralgorithmneeded O (n 3
)timeand O (n)spae, but we enlarge onsid-
erably thelass of problems whih an nowb e treated. We need notanymoreto onsider
