DANTE – The combina ion be ween an An Colony Op imiza ion algo i hm and
a Dep h Sea ch me hod
Ped o Ca doso – M´a io Jesus
EST – Uni e sidade do Alga e
8005-139 Fa o, Po ugal
[email p o ec ed]
Albe o Ma quez
Dep . Ma em´a ica Aplicada I
ETSII – Uni e sidad de Se illa
A da. Reina Me cedes
s/n 41012 Se illa, Spain
[email p o ec ed]
Abs ac
The -DANTE me hod is an hyb id me a-heu is ic. In
combines he e olu iona y An Colony Op imiza ion (ACO)
algo i hms wi h a limi ed Dep h Sea ch. This Dep h Sea ch
is based in he phe omone ails used by he ACO, which
allows i o be o ien ed o he mo e p omising a eas o he
sea ch space. Some esul s a e p esen ed o he mul iple
objec i e k-Deg ee Spanning T ees p oblem, p o ing he
e ec i eness o he me hod when compa ed wi h o he al-
eady es ed e olu iona y me hods.
1. In oduc ion
I is well known ha , e y o en, he use o pu e E o-
lu iona y Algo i hms has a lack o pe o mance, namely in
he op imiza ion o la ge combina o ial p oblems ins ances.
Mo eo e , he majo i y o he e olu iona y me a-heu is ics
ha e some common cha ac e is ics. Fo example, hey usu-
ally each good app oxima ions o he op imum solu ions
(despi e he di icul ies o e ine hose app oxima ions, i.e.,
ge ing o he eal op imum). O en only he bes solu ion(s)
a e kep . The o he solu ions a e disca ded wi hou u he
explo a ion, which o ge s he expensi e compu a ional e -
o necessa y o build hem. Examples o op pe o m-
ing me a-heu is ics ha employ his me hodology a e he
Simula ed Annealing, he Tabu Sea ch, o he An Colony
Op imiza ion algo i hms [11]. The e o e, his s a egy does
no allow a p ope local explo a ion o he e en ually mo e
p omising egions o he sea ch space. In o he wo ds, he
neighbo hoods o he ob ained solu ions do no go h ough
an exhaus i e explo a ion.
Algo i hmic hyb idiza ions appea ed as an e o o sol e
some o he abo e men ioned p oblems. The imp o emen
o he me hods by hose combina ions can gi e us he bes
o se e al algo i hmic s a egies. In he implemen ed cases,
di e en possible hyb idiza ions can be hough , like he use
o ini ial solu ions ia ano he me hod han he main one
(used many imes o ins ance in he Gene ic Algo i hms o
ob ain he ini ial popula ion[9, 11]) o a local imp o emen
o he ob ained solu ions. Usually, hyb id me hods a e con-
side ed as he ones ha use wo o mo e me hods in he
ollowing sense: he p ima y me hods a e used o gene a e
mo e o less ough app oxima ions o he p oblems solu-
ions, ollowed by o he (s) me hod(s) ha e ine he ea lie
solu ions. Some examples o he applica ion o he second
phase o he solu ion ob ained by some me a-heu is ic, a e
he use o he 2-OPT o 3-OPT o he T a elling Salesman
Pe son [5], he SOP-3-exchange o he Sequen ial O de -
ing P oblem [6], o he I e a ed Local Sea ch o he Bin
Packing P oblem [10]. In [1] is p esen ed he Beam–ACO,
which is a combina ion o a Beam Sea ch heu is ic wi h
an ACO algo i hm. In his case, he solu ion cons uc ion
mechanism o s anda d ACO algo i hms is eplaced by he
solu ion cons uc ion mechanism in which each a i icial an
pe o ms a p obabilis ic Beam Sea ch. This is done by e-
placing he de e minis ic choice o a solu ion componen a
each cons uc ion s ep by a p obabilis ic choice based on
ansi ion p obabili ies.
In his pape , we p opose a hyb id me aheu is ic called
–Dep h ANT Explo e (-DANTE), ha uses an e icien
local sea ch (adap ed o a phe omone-o ien ed p ocedu e).
Mo e p ecisely, in each one o he -DANTE cycles, se s o
solu ions a e compu ed using a manda o y i s phase ha ,
in some cases, is ollowed by an elec i e second phase ha
depends on he quali y o he o me solu ion. Mo e speci -
ically,
•In he i s phase, simila o mos ACO algo i hm, a
solu ion is gene a ed based in a cons uc i e p ocedu e
ha successi ely adds selec ed componen s acco ding
o a pseudo- andom o mula.
Eigh h In e na ional Con e ence on Hyb id In elligen Sys ems
978-0-7695-3326-1/08 $25.00 © 2008 IEEE
DOI 10.1109/HIS.2008.115
36
•In he second phase, a limi ed dep h sea ch p ocedu e
based on he bes i ing solu ions is made. In o he
wo ds, i he i s phase ou come is wi hin  ange o
he app oxima ion se ( he se o he bes known pe -
o ming solu ions) o imp o es his se , hen a lim-
i ed dep h sea ch p ocedu e is applied. The same
phe omone ails, ha we e used in he i s phase,
a e also used he e. This enhances he limi a ions o
he dep h sea ch me hods by leading he p ocedu e o
include he mo e p omising componen s in he con-
s uc ed solu ions.
The e o e, his pape is s uc u ed as ollows. The
-DANTE is desc ibed in he nex sec ion. In he hi d sec-
ion a e p esen ed some esul s and, in he las one, a e
d awn some conclusions.
2. –DANTE – Dep h ANT Explo e
The second phase local sea ches men ioned abo e (e.g.,
2-OPT,3-OPT,...) a eused o e ine he solu ions, bu do
no use he in o ma ion acqui ed by he e olu iona y me a-
heu is ic, excep o he base solu ion i sel . Mo eo e , mos
o he imes he local sea ch ope a o a e applied o all solu-
ions e en in he cases whe e hey a e no much p omising.
To y o explo e some o hese weaknesses we de el-
oped he –Dep h ANT Explo e (–DANTE) which is a
hyb id me hod. In ac , –DANTE is li le mo e since, i is a
usion be ween an An Colony Op imiza ion me hod [4] and
an Dep h Sea ch me hod [8]. F om his combina ion esul s
mo e han a me a-heu is ic ollowed by he local sea ch, as
we will see nex .
The ke nel o he me hod, simila ly o he An Colony
Op imiza ion me hod, has a se o cycles. In each cycle a se
o solu ions is buil using he phe omone ails. Those so-
lu ions a e hemsel es used o upda e he phe omone ails
and o upda e he se o (bes ) app oxima ions o he p ob-
lem solu ion: a single se o he single objec i e p oblems
o he Pa e o se o he mul iple objec i e case [3] (see Sec-
ion 3). Al e na i ely, he upda ing o he phe omone ails
can be done in a mo e g eedy me hod using only he bes
solu ions ha whe e ob ained du ing he en i e p ocess [2].
One o he main ideas o he -DANTE, is o make a local
explo a ion o he bes solu ions. In o he wo ds, a solu ion
is a ac i e whene e ce ain quali y pa e n is achie ed.
This can be quan i ied has he solu ion being wi hin an 
ange om he bes known solu ions. In his case, he Dep h
Sea ch based on he ea lie buil solu ion is applied. This
s a egy ies o a oid he, p obably useless, compu a ional
e o associa ed wi h he explo a ion o neighbo hoods o
he wo s solu ions. The Dep h Sea ch is limi ed bo h in
he dep h i sel and in he numbe o possible b anches o
he sea ch ee. Fu he mo e, he dep h sea ch is o ien ed
1. Ini ialize he phe omone ail.
2. While s opping c i e ion is no me do
(a) Fo all an s do
•Cons uc a new solu ion, S, using he
cu en phe omone ail.
•I he dis ance o S o he app oxima-
ion se is in e io o o S imp o es
he app oxima ion se hen apply a lim-
i ed dep h sea ch p ocedu e, based on he
phe omone ails, om ha solu ion.
(b) Upda e he phe omone ail
Figu e 1. -DANTE Algo i hm
by he phe omone ails and he local heu is ics used in he
An Colony me hod (Sec ion 2.1 desc ibes in mo e de ail
he Dep h Sea ch phase).
The main -DANTE algo i hm is ske ched in Figu e 1.
2.1 Dep h Sea ch phase
As e e ed, he i ness o each gene a ed solu ion, S,is
compa ed wi h he i ness o he elemen s in he app ox-
ima ion se , P(Pis he app oxima ion se : in he single
objec i e case i will be a singula se and in he mul iple
objec i e case i will ha e he non-domina ed solu ions [3]
– see sec ion 3.1). Then he limi ed dep h sea ch goes o
Le el D–I Simp o es P, ha is,Sis no wo s han any
elemen o P;o
Le el d–I Sis wo s han some elemen o P, bu i s el-
a i e dis ance o he elemen s o he app oxima ion se
is smalle han .
He e dand Da e algo i hm pa ame e s and should e i y
d>D. Fu he mo e, he numbe o b anches used in each
le el o he Dep h Sea ch, M, is also an algo i hm pa ame-
e .
Figu e 2 p esen s a high le el desc ip ion o he p ocess
whe e Func ion -DANTE Solu ion equi es some
pa ame e s besides he solu ion in cons uc ion T, namely
•–Thepa ame e will in luence he numbe o imes
ha he p ocess en e s in he Dep h Sea ch mode.
Small alues will gua an ee ha only he solu ions
wi h objec i e alue nea he bes known solu ions will
go in o he o ien ed Dep h Sea ch. La ge alues will
do he opposi e.
•The algo i hm con ains a abu lis o each le el, abul,
ha is ini ialized as emp y se . This abu lis a oids
37
Func ion -DANTE Solu ion(T)
l←|T|/*de ines he le el by he numbe o added
componen s*/
i Tis a solu ion hen
Se Δas he ela i e dis ance om T o P
i Δ<0 hen /*Timp o es P*/
Upda e he app oxima ion se wi h T
Upda e Δk(k=1,2,...,m) wi h T†
e u n D
else
i Δ< hen
Upda e Δk(k=1,2,...,m) wi h T†
e u n d
else e u n 0
else
sle el ←0
NB anches ←M
o k=1 o NB anches
i ∃e∈E− abul:T∪{e}is admissible hen
Choose an edge, e, om E− abul‡
Tabu
l←Tabu
l∪{e}†
T←T∪{e}
/*Recu si e call*/
L←-DANTE SOLUTIONS(T)
i L>0 hen
NB anches ←NB anches +M
sle el ←max{sle el, L}
T←T−{e}
else
b eak /* o */
Tabu
l←∅†/*clean he abu lis */
e u n max{sle el −1,0}
Figu e 2. -DANTE’s solu ions explo a ion al-
go i hm (whe e he s eps ma ked wi h †a e
op ional and he ones wi h ‡a e p oblem spe-
ci ic).
ha he same solu ions a e ebuil in he Dep h Sea ch,
by es ic ing he addi ion o he same compone s in
he same le el. The use o he abu lis is implemen a-
ion dependen and should be hough acco ding o he
cleaning abuls ep, since i also es ic s he cons uc-
ion o new solu ions con aining hose componen s.
•The compu a ion o he dis ance om T o he app ox-
ima ion se P e u ns a non-posi i e alue i Tim-
p o es P;
•The upda e p ocedu e consis s in inse ing Tin Pand
emo ing he elemen s o P ha a e wo s han T.
•The upda e o Δkis op ional since, i all solu ions con-
ibu e o he a ia ion o he phe omones, he esul
is a noisy ail wi h consequen los o pe o mance.
Al e na i e s a egies include a g eedy upda e whe e
only he bes solu ions con ibu e o he a ia ion o
he phe omone ails, o e en a mo e es ic ed selec-
ion o e he app oxima ion se (see sec ion 3.4).
A mo e de ailed desc ip ion o he me hod can be ound
in [2].
3 Some compu a ional es s
3.1 Tes p oblems - Mul iple Objec i e k-
Deg ee Minimum Spanning p oblem
To es he -DANTE me hod i was conside ed he Mul-
iple Objec i e k-Deg ee Minimum Spanning p oblem. A
k-Deg ee Minimum Spanning T ee is a minimal weigh
spanning ee such ha he maximum deg ee o any node
is k. The ee is buil o e a ne wo k (V,E,W),whe e
Vis he nodes se , Eis he edges se , W:E→
IR mis he weigh ec o - unc ion de ined as W(e)=
(w1(e),w
2(e),...,w
m(e)),andmis he numbe o objec-
i es.
He e, since we ha e a mul iple objec i e p oblem, min-
imal weigh is conside ed in he Pa e o op imali y con ex .
Tha is, gi en wo solu ions Xand Yo he easible se
S,i issaid ha Xdomina es Y,X≺Y,i o all
i∈{1,2,...,m}we ha e wi(X)≤wi(Y), and exis s
j∈{1,2,...,m}such ha wj(X)<w
j(Y). A solu ion
Xis Pa e o (op imal) i i is no domina ed by any o he so-
lu ions o S, ha is, o allY∈S−{X}i is e i ied ha
Y≺ X. The se o all Pa e o solu ions is called Pa e o se
o e icien se .
3.2 Me ics
To e i y he pe o mance ou me hod we used wo me -
ics: R1and R3[7]. The R1me ic measu es he p obabil-
i y ha an app oxima ion se P1is be e han ano he P2
o e a amily o u ili y unc ions U.I R1(P1,P2,U)>1
2
hen, acco ding o his measu e, P1is be e han P2and i
will be no wo se i R1(P1,P2,U,p)≥1
2.TheR3me ic
measu es he expec ed p opo ion o supe io i y o one se
o e ano he . The la ge R3(P1,P2,U)is, he wo s is P1
when compa ed wi h P2. The opposi e is alid i R3is neg-
a i e. I R3(P1,P2,U)≈0 hen P1and P2ha e simila
quali y o e U.
38
3.3 k-Deg ee Spanning T ees p oblem
cons uc ion
In he p oposed p ocess, o build he k-deg ee spanning
ee, easible edges a e successi ely added un il a solu ion
is comple e. Be o e he addi ion o any edge, i mus be
e i ied ha such addi ion will no o m a cycle nei he he
nodes maximum deg ee condi ion is iola ed.
Mo e p ecisely, he p ocess s a s by andomly se-
lec ing a node om V,s, and se ing TN={s}
(TNis he se o he nodes al eady included i he
ee). Then n−1admissible edges om A=
{eu ∈E:u∈TN∧δ(u)<k∧ ∈V−TN},a e se-
quen ially added, whe e δis he deg ee o node uin he sub-
ee ha is being cons uc ed, and k he maximum deg ee al-
lowed. The selec ion o he edges is pseudo- andomly made
using o mula
es =a g max ∈A m
j=1 τj( )αjwj( )βji q≤q0
ei q>q
0,
(1)
whe e
•τj(e)is he phe omone alue associa ed o he jweigh
in edge e;
•wj(e)is he j-weigh o edge e;
•αjis an algo i hm pa ame e associa ed o he ele-
ance o weigh j;
•βjis an algo i hm pa ame e associa ed o he local
heu is ic ha a ou s edges wi h lowe j-weigh ;
•e∈Ais an edge pseudo- andomly chosen wi h p ob-
abili y
p(e)= m
j=1 τj(e)αjwj(e)−βj
 ∈A m
j=1 τj( )αjwj( )−βj.(2)
•qis a uni o m andom alue in [0,1];and
•q0∈[0,1] is a pa ame e ha in luences which b anch
o (1) is used mo e o en: a smalle alue o q0p o-
duces a mo e explo a o y sea ch, since i implies he
use o he pseudo- andom o mula (2) wi h highe
p obabili y. When q0is nea 1, he easible edge
wi h la ge p obabili y o en e ing he ee is used wi h
g ea e equency, which sugges an exploi ing sea ch.
I he solu ion sa is ies he dis ance cons ain hen he
p ocess en e s Dep h Sea ch me hod desc ibed in Algo i hm
2. The selec ion o he edges o en e he sea ch ee in his
phase also ollows he abo e o mulas, ha is, (1) and (2).
3.4 Phe omone upda e
To upda e he phe omones ma ices i was used he
Angle-Phe omone Upda e s a egy [2]. This s a egy can
be conside ed g eedy in he sense ha i only uses elemen s
o he app oxima ion se . The objec i e is o explo e small
egions o he sea ch space by using, in he phe omone up-
da ing o mula, only a subse o he solu ions con ained in
ha se . This idea is mo i a ed by he ac ha , in mos o
he cases, he numbe o elemen s in he app oxima ion se
becomes e y la ge. Empi ical es s p o ed ha i all he so-
lu ions in he app oxima ion se we e used, he phe omone
based selec ion becomes e y noisy, which delays he con-
e gence owa d he Pa e o se . Fo he single objec i e
case, a se o he kbes solu ions can be kep , using a subse
o ha se o do phe omones upda e.
The e o e, he phe omone ec o upda e is made acco d-
ing o o mula
τ(e)=ρτ(e)+Δ(e),e∈E,
whe e
•τ(e)is a phe omone ec o associa ed o edge e;
•ρ∈[0,1] is called he pe sis ence ac o (1−ρis he
e apo a ion ac o ). The smalle he alues o ρa e,
he smalle quan i y o in o ma ion, used in one cycle,
is ansmi ed o ollowing cycle;
•Δ(e)=(Δ
1(e),Δ2(e),...,Δm(e)) is he ein o ce-
men phe omone ec o associa ed o edge eand is
compu ed using he elemen s o he app oxima ion se ,
P, and o mula
Δk(e)= 
T∈Pe
Q
wk(T),
whe e
–Qis a alue wi h he same magni ude o he solu-
ions. Fo example, i he weigh s a e balanced i
can be used he a e age o he minimum weigh s,
1
m
m

k=1
min
T∈P wk(T);
–Pea e he elemen s o he app oxima ion se ha
con ain edge eand lie in a subangle o he angle
de ined by he o igin and he ex eme solu ions
(o weigh s kand k+1), ha is,
Pe={T∈P:e∈T∧γ1≤φk(T)≤γ2}
whe e
φk(T) = a c an wk+1(T)
wk(T),
39
Pa ame e Values
D1
d3|V|
4
M2
0
αi,βi{0,0.03,0.06,...,3.0}
ρ0.1
q0{0.5,0.6,0.7,0.8,0.9}
k3
Numbe o
an s pe cycle
|V|
Numbe o
cycles
2
Maximum un
ime
min{60|V| log |V|,36000}
Seconds
Table 1. Used pa ame e s.
γ1=φmin
k+Ikφh
k
γ2=φmin
k+(Ik+1)φh
k
,
φmin
k=min
T∈P φk(T)
φmax
k=max
T∈P φk(T),
o k=1,2,...,m−1,φh
kis he s ep in e al,
and Ikis a alue ela ed o he egion o be ex-
plo ed which is con olled by he main p ocess.
3.5 Resul s
The algo i hm -DANTE was implemen ed in C++
and es swe e unonaPCwi ha3Ghz In el Pen ium
IVTMp ocesso , 512Mbo RAM and Windows XP OS. Fo
each p oblem he me hod was un 15 imes wi h he pa am-
e e s epo ed in Table 1.
The p oblems ins ances ha we e used o es he imple-
men a ion we e de ined by [9] and he -DANTE solu ions
a e compa ed wi h he solu ions ob ained wi h he Gene ic
Algo i hms p esen ed by he same au ho s. Mo e speci i-
cally, i was used as e e ence se he one composed by he
non-domina ed elemen s o he union o he 30 app oxima-
ion se s p esen ed in [9].
Table 2 p esen s a esume o he esul s o some o he
es ed ins ances o e he 15 uns. This able con ains in o -
ma ion abou he a e age alues o he me ics R1and R3,
he a e age ca dinali y o he app oxima ion se buil wi h
-DANTE (in pa en heses he ca dinal o he e e ence se ),
and he a e age ime o he las upda e o he app oxima-
ion se . F om he same able i is possible o obse e ha
in mos o he cases, -DANTE imp o es he e e ence se
since R1<0.5and R3≈0. The excep ions we e he 10
nodes ne wo ks whe e in one case i has achie ed he exac
Pa e o se (ob ained wi h he B u e Fo ce me hod) and in
P oblems μR1μR3μ|P| (μ|P e |)μTime
10 ac 0.57 0.00 183 (191†)7.6
10 -m-c 0.5 0.00 129 (129†)628
10 conc 0.54 0.00 128 (134†) 8
25 ac 0.47 0.00 517 (439) 3754
25 c 0.39 0.00 2496.6 (1480) 4351.
25 -m-c 0.13 0.01 2168.7 (820) 4743
50 ac 0.40 -0.05 6328.1 (1436) 11960
50 conc1 0.48 0.00 1748.6 (894) 11803
Table 2. Resume o he esul s o he k-
Deg ee Minimum Spanning T ee p oblem.
he o he wo cases i has ob ained 128 and 183 o he 134
and 191 solu ions, espec i ely.
The ime e olu ion o he R1and R3me ics a e de-
pic ed in Figu es 3 and 4, espec i ely, o some o he es ed
ins ances (25 ac, 25 c and 50 ac). I is possible o obse e
ha he con e gence o he solu ions owa d he e e ence
se is made qui e as since R3quickly akes alues nea o
ze o. Ne e heless, he local e inemen akes some ex a
ime, as e lec ed in he las upda e ime o he app oxima-
ion se (μTime), which as al eady e e ed is a cha ac e is-
ic common o mos o he An Colony based algo i hms.
4 Conclusions
This pape was de o ed o he s udy o he -DANTE
me hod. This me hod is based in he An Colony Op imiza-
ion pa adigm and appea s as an e o o p o ide a mo e e -
ec i e way o u he explo ing he bes i ness pe o ming
solu ions. Mo e p ecisely, whene e a solu ion is inse ed
in o he app oxima ion se , o sa is ies an dis ance o he
app oxima ion se c i e ion, i is pe o med a limi ed Dep h
Sea ch using he phe omone alues o guide he sea ch.
Resul s o a e sion o -DANTE applied o he Mul i-
ple Objec i e k-Deg ee Minimum Spanning T ees a e p e-
sen ed. I was e i ied ha he me hod apidly con e ges
owa d he on s achie ed by o he au ho s and ends up by
imp o ing, in gene al, hei esul s.
As in many An Colony algo i hms, he -DANTE uses
a g eedy s a egy, whe e only he bes pe o ming solu ions
a e used o upda e he phe omone alue.
Re e ences
[1] C. Blum. Beam–ACO – hyb idizing an colony op imiza-
ion wi h beam sea ch: an applica ion o open shop schedul-
ing. Compu e s and Ope a ions Resea ch, 32(6):1565–
1591, 2005.
40

0 1000 2000 3000 4000
seconds
0.5
0.6
0.7
0.8
0.9
1
R1
25 ac
0 1000 2000 3000 4000 5000
seconds
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R1
25 c
0 2000 4000 6000 8000 1000012000
seconds
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R1
50 ac
Figu e 3. Time e olu ion o R1me ic o e
he15 uns.
[2] P. Ca doso. An Colony Algo i hms o Mul iple Objec i e
Combina o ial Op imiza ion: Applica ions o he Minimum
Spanning T ees P oblems. PhD hesis, Uni e si y o Se ille,
Spain, Ma 2007.
[3] K. Deb. Mul i-objec i e Op imiza ion using E olu iona y
Algo i hms. John Wiley & sons, 2001.
[4] M. Do igo, E. Bonabeau, and G. The aulaz. Swa m In elli-
gence: F om Na u al o a i icial Sys ems. Ox o d Uni e -
si y P ess, 1999.
[5] M. Do igo and T. S u zle. An Colony Op imiza ion.MIT
P ess, 2004.
[6] L. Gamba della and M. Do igo. An an colony sys em hy-
b idized wi h a new local sea ch o he sequen ial o de ing
p oblem. INFORMS, Jou nal on Compu ing, 12(3), 2000.
[7] A. Jaszkiewicz. Mul iple Objec i e Me aheu is ic Algo-
i hms o Combina o ial Op imiza ion. PhD hesis, Poznan
Uni e si y o Technology, 2001.
[8] D. Jungnickel. G aphs, Ne wo ks and Algo i hms, olume 5
o Algo i hms and Compu a ion in Ma hema ics. Sp inge -
Ve lag, Be lin, 1999.
0 1000 2000 3000 4000
seconds
R3
25 ac
-0.74
-0.59
-0.44
-0.29
-0.15
0
0 1000 2000 3000 4000 5000
seconds
R3
25 c
-2.35
-1.88
-1.41
-0.94
-0.47
0
0 2000 4000 6000 8000 1000012000
seconds
R3
50 ac
-2.79
-2.24
-1.68
-1.12
-0.56
0
Figu e 4. Time e olu ion o R3me ic o e
he15 uns.
[9] J. Knowles and D. Co ne. Benchma k p oblem gene a o s
and esul s o he mul iobjec i e deg ee-cons ained min-
imum spanning ee p oblem. In P oceedings o he Ge-
ne ic and E olu iona y Compu a ion Con e ence (GECCO-
2001), pages 424–431. Mo gan Kau mann Publishe s, San
F ancisco, Cali o nia, 2001.
[10] J. Le ine and F. Duca elle. An colony op imisa ion and lo-
cal sea ch o bin packing and cu ing s ock p oblems. Jou -
nal o he Ope a ional Resea ch Socie y, Special Issue on
Local Sea ch, 55(7), 2004.
[11] I. Pa mee. E olu iona y and Adap i e Compu ing in Engi-
nee ing Design. Sp inge -Ve lag, London, 2001. ISNB:1-
85233-029-5.
[12] J. Puchinge and G. R. Raidl. Combining me aheu is ics and
exac algo i hms in combina o ial op imiza ion: A su ey
and classi ica ion. In P oceedings o he Fi s In e na ional
Wo k-Con e ence on he In e play Be ween Na u al and A -
i icial Compu a ion, olume 3562 o LNCS. Sp inge , 2005.
[13] G. R. Raidl. A uni ied iew on hyb id me aheu is ics. In
P oceedings o he Hyb id Me aheu is ics Wo kshop, num-
be 4030 in LNCS. Sp inge , 2006.
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