Soh abi, Somayeh; Zia a i, Koo ush; Kesh ka an, Mo eza
A icle
ACS-OPHS: An Colony Sys em o he O ien ee ing
P oblem wi h ho el selec ion
EURO Jou nal on T anspo a ion and Logis ics (EJTL)
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Sugges ed Ci a ion: Soh abi, Somayeh; Zia a i, Koo ush; Kesh ka an, Mo eza (2021) : ACS-OPHS: An
Colony Sys em o he O ien ee ing P oblem wi h ho el selec ion, EURO Jou nal on T anspo a ion
and Logis ics (EJTL), ISSN 2192-4384, Else ie , Ams e dam, Vol. 10, Iss. 1, pp. 1-10,
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ACS-OPHS: An Colony Sys em o he O ien ee ing P oblem wi h
ho el selec ion
Somayeh Soh abi, Koo ush Zia a i
*
, Mo eza Kesh ka an
School o Elec ical and Compu e Enginee ing, Shi az Uni e si y, Shi az, I an
ARTICLE INFO
Keywo ds:
Rou ing p oblems
O ien ee ing p oblem
O ien ee ing p oblem wi h ho el selec ion
In e media e acili ies
An colony sys em
ABSTRACT
In his pape , an algo i hm, called ACS-OPHS, is p oposed o ackle he O ien ee ing P oblem wi h Ho el Selec ion
(OPHS). This algo i hm is s ongly based on he An Colony Sys em (ACS); howe e , i di e s om he ACS in he
way he pa hs a e cons uc ed, in uning a pa ame e o he ansi ion ule and in he phe omone ails upda ing
ules. The ACS-OPHS uses a bi-di ec ional sea ch s a egy and employs a no el and as app oach o iden i y all
easible in e media e ho els in an o fline manne . Mo eo e , in he ACS-OPHS, he ela i e impo ance o
exploi a ion e sus explo a ion is de e mined acco ding o he p og ess o he algo i hm in app oaching o he
global op ima. The ACS-OPHS is a simple and well-pe o ming app oach o sol e he OPHS. Conce ning he
s anda d benchma k ins ances, i ou pe o ms he s a e-o - he-a algo i hms in se e al ins ances and p oduces
compe i i e solu ions in easonable ime. This algo i hm also imp o es he bes known esul s o ou ins ances
wi h unknown op imal solu ions.
1. In oduc ion
The O ien ee ing P oblem (OP), which has a ac ed he a en ion o
many esea che s, is a combina o ial op imiza ion p oblem (Tsiligi ides,
1984). The O ien ee ing P oblem wi h Ho el Selec ion (OPHS) is a
a ian o he OP (Di sala e al., 2013). The OPHS is defined on a
comple e g aph G¼ðV;EÞwhe e Vis he se o e ices and Edeno es
he se o edges. Vhas wo subse s including HOTELS ¼ H0;H1;…;
Hh1gand NODES ¼ 1;…;ng. The sco e o each e ex in HOTELS is
ze o, and he e is no es ic ion on he numbe o imes ha each o hese
e ices can be isi ed. On he o he hand, each e ex in NODES has a
specific sco e, siði2NODESÞ, and mus be isi ed a mos once.
The goal o he OPHS is o find a ou in g aph Gwi h a maximum
sco e. A ou is a con inuous pa h ha mus be s a ed om H0and
finished in H1. I mus also con ain a pa icula numbe o ips, D. Each
ip has i s own ime es ic ion, Tjðj2 1;…;DgÞ, ha limi s he subse
o NODES ha can be isi ed along he ip. The ime limi o each ip
mus be sa isfied ega ding he Euclidean dis ance be ween he e ices.
The o igin and des ina ion o each ip mus be chosen om HOTELS and
he des ina ion o ip i(1 i<D1) is he o igin o ip iþ1. Mo e-
o e , acco ding o he defini ion o a easible ou , he s a -ho el o he
fi s ip and he end-ho el o he las ip mus be H0and H1, espec i ely.
Hence o h he “in e media e ho els” e m e e s o he end-poin o ip
1, he s a -poin o ip D, and he o igins and des ina ions o he o he
ips. The wo d “node” e e s only o a membe o he NODES se while
“ e ex”is used o e e o a ho el o a “node”.
The e a e se e al p ac ical applica ions o he OPHS (Di sala e al.,
2013)(Di sala e al., 2014). Planning a mul i-day ou o a ou is who
wan s o isi a egion du ing a specific numbe o days is one o hem.
Assume a a eling salesman who has a maximum numbe o wo king
hou s pe day and wan s o se e some eques s sen om a specific a ea.
Since he spends a limi ed numbe o days in ha a ea, eques s mus be
selec ed acco ding o hei associa ed p ofi s. Planning a ou o his
salesman is ano he applica ion o he OPHS.
In his pape , a new algo i hm, called ACS-OPHS, is p oposed o
sol ing he OPHS. This algo i hm is based on he s anda d An Colony
Sys em (ACS) (Do igo and Gamba della, 1997); howe e , i di e s om
he ACS in some aspec s including:
The pa h cons uc ion p ocedu e: simila o he s anda d ACS p o-
posed o he T a eling Salesman P oblem (TSP) (Do igo and Gam-
ba della, 1997), easible ou s a e buil sequen ially in he
cons uc ion phase o he ACS-OPHS. Howe e , in con as o he
s anda d ACS, wo an colonies, which a e assumed o be loca ed a
he o igin o he fi s ip and he des ina ion o he las ip, a e used.
In o he wo ds, he ACS-OPHS uses a bi-di ec ional sea ch s a egy.
The me hod used o une pa ame e q0o he ansi ion ule: in
con as o he s anda d ACS, he alue o q0is modified o e he
* Co esponding au ho .
E-mail add esses: s.soh abi@shi azu.ac.i (S. Soh abi), [email p o ec ed] (K. Zia a i), [email p o ec ed] (M. Kesh ka an).
Con en s lis s a ailable a ScienceDi ec
EURO Jou nal on T anspo a ion and Logis ics
jou nal homepage: www.jou nals.else ie .com/eu o-jou nal-on- anspo a ion-and-logis ics
h ps://doi.o g/10.1016/j.ej l.2021.100036
Recei ed 5 Sep embe 2020; Recei ed in e ised o m 2 Ma ch 2021; Accep ed 8 Ap il 2021
2192-4376/©2021 The Au ho (s). Published by Else ie B.V. on behal o Associa ion o Eu opean Ope a ional Resea ch Socie ies (EURO). This is an open access
a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
execu ion ime o he ACS-OPHS o ha e a good comp omise be ween
explo a ion and exploi a ion.
Phe omone ail upda ing ules: since ou s a e cons uc ed in wo
di e en di ec ions, a bi-di ec ional s a egy is employed o upda e
phe omone ails.
Mo eo e , a simple and as o fline app oach is in oduced o iden-
i ying he easible in e media e ho els. Using his app oach, a new poin
o iew on he OPHS ou cons uc ion is p o ided. The p e ious me hods
sugges ed o he OPHS, cons uc a easible ou as ollows. Ini ially, a
easible sequence o ho els is de e mined and hen, nodes a e inse ed in
his sequence wi h espec o he ime limi a ions o he ips. Howe e ,
in he ACS-OPHS, a easible ou is easily cons uc ed using a sequen ial
me hod. I means ha ips can be cons uc ed one a e he o he and he
ho el selec ion occu s when he ime limi o a ip p e en s he o he
nodes om being isi ed.
The emainde o his pape is o ganized as ollows: he ela ed wo ks
a e e iewed in Sec ion 2, and he ACS-OPHS is in oduced in Sec ion 3.
Sec ion 4is de o ed o he expe imen al esul s. The conclusion and
u u e wo k a e p esen ed in Sec ion 5.
2. Rela ed wo ks
So a , ou algo i hms ha e been sugges ed o he OPHS. Di sala
e al. (2013) p oposed a me hod based on he skewed a iable neigh-
bo hood sea ch, called SVNS. The SVNS includes h ee phases: ini iali-
za ion, imp o emen , and e-cen e ing. In he ini ializa ion phase, h ee
s eps a e aken o cons uc an ini ial solu ion. Fi s , conside ing each
easible pai o ho els as he s a - and end-poin o each ip, an inde-
penden OP ins ance is sol ed wi h espec o he ime limi o he ip. In
his way, he po en ial sco e ha can be ob ained wi hin each pai o
ho els is de e mined. A e wa ds, all easible sequences o ho els a e
p oduced and a numbe o hem a e selec ed o cons uc he ini ial so-
lu ion and also o use hem in he imp o emen phase. Fo his pu pose,
he sequences wi h he highes po en ial sco es a e p e e ed. In he
imp o emen phase, he e ices and ho els a e shaken and he solu ion is
imp o ed using a local sea ch p ocedu e. The ocus o he local sea ch
p ocedu e, ha con ains nine ope a o s, is o imp o e he o de o he
isi ed nodes along a ou . Finally, in he e-cen e ing phase, he solu ion
ha is used o s a he nex i e a ion o he algo i hm is selec ed.
In addi ion o he p oposed SVNS, Di sala e al. (2013) ha e in o-
duced 229 p oblem ins ances o he OPHS. These ins ances ha e been
designed using he s anda d OP ins ances wi h a ying sizes. Conside ing
hese ins ances, he SVNS p oduces high quali y solu ions and ha e an
accep able compu a ion ime.
In 2014, Di sala e al. (2014) p oposed a meme ic algo i hm o he
OPHS, called MA. Since he gene ic algo i hm ope a o s a e only used o
imp o e he sequence o ho els, each ch omosome p esen s a sequence o
ho els in a ou . The fi ness o he ch omosome is equal o he sco e
ob ained along he ou . The MA has wo pa s: ini ializa ion and main
loop. The same as he SVNS, in he ini ializa ion pa , he po en ial sco e
be ween each pai o ho els is de e mined and hen he ini ial popula ion
is c ea ed. Two c osso e and one mu a ion ope a o s a e applied in he
main loop o c ea e new solu ions om he cu en popula ion. Each o
hese new solu ions, which is called o sp ing, is imp o ed using a local
sea ch p ocedu e. Then, conce ning he cu en popula ion and he
o sp ing, he nex popula ion is selec ed. I should be men ioned ha , in
he MA, he sequence o ho els is imp o ed using he c osso e and
mu a ion ope a o s while he local sea ch p ocedu e imp o es he o de
o he nodes placed among he ho els.
Since he ins ances in oduced in (Di sala e al., 2013) we e small in
e ms o he To al Numbe o Feasible Sequences o ho els (TNFS), 176
new and la ge ins ances ha e been designed by Di sala e al. (2014).
The expe imen s indica e ha he TNFS is an impo an p ope y o he
OPHS ins ances. The di ficul y o an ins ance has a di ec ela ionship
wi h i s TNFS alue. Using he new ins ances, he main d awback o he
SVNS has been disclosed. Al hough his algo i hm p o ides e en be e
solu ions han he MA o small ins ances, conside ing he new la ge
ins ances, he compu a ion ime o he SVNS is no accep able a all. The
bo leneck o he SVNS is a he ini ializa ion phase when i finds all
easible sequences o ho els. In con as o he SVNS, he MA has a
easonable compu a ion ime e en o ins ances wi h la ge TNFS alues.
In 2019, Toledo e al. (2020) p oposed a hype -heu is ic o ackle he
OPHS. Sol ing he OPs, he same as he SVNS and MA, he po en ial
sco es be ween each pai o ho els a e de e mined. Then, he
hype -heu is ic c ea es he ini ial popula ion and selec s an indi idual
om he popula ion using ou namen selec ion. The selec ed solu ion is
shaken employing ou easible heu is ics. A he nex s ep, a local sea ch
p ocedu e con aining ele en low-le el heu is ics is used o imp o e he
ou . Finally, a membe o he popula ion is eplaced by he newly
gene a ed solu ion. The algo i hm is e mina ed when a s opping c i e ia
has been me . In ac , he p oposed hype -heu is ic is based on he la ge
neighbo hood sea ch. This algo i hm p oduces compe i i e solu ions o
he s anda d OPHS ins ances in easonable ime. I should be men ioned
ha en ins ances ha e been igno ed by Toledo e al. (2020) du ing hei
expe imen s.
Soh abi e al. (2019) p oposed an algo i hm based on he g eedy
andomized adap i e sea ch p ocedu e, called GRASP. As opposed o he
p e ious algo i hms, GRASP does no de e mine he po en ial sco e be-
ween each pai o ho els. Ins ead, a new app oach based on dynamic
p og amming ha e been in oduced o he ho el selec ion. GRASP has
wo phases: cons uc ion and local sea ch. In he cons uc ion phase, a
easible solu ion is c ea ed. Fo his pu pose, using he dynamic p o-
g amming idea, a easible sequence o ho els is c ea ed. Then, wo p o-
cedu es a e used o inse nodes among he ho els. When he e is no node
le ha can be added, he local sea ch phase is s a ed o imp o e he
solu ion. GRASP also uses he p oxima e op imali y p inciple (Fleu en
and Glo e , 1999). Thus, each ime a node is added o he solu ion, a
p ocedu e including ou ope a o s is used o imp o e he ou : wo op-
e a o s o imp o e he o de o he nodes, one ope a o based on he
dynamic p og amming idea o make be e he sequence o ho els, and
ano he ope a o o imp o e he sequence o ho els as well as he selec-
ion o he nodes.
Using he s anda d ins ances o he T a eling Salesman P oblem wi h
Ho el Selec ion (TSPHS), Soh abi e al. (2019) ha e also c ea ed new 76
benchma k ins ances. These ins ances di e om he p e ious ones in he
way hey ha e been c ea ed and in e ms o he numbe o nodes. The
esul s o he expe imen s indica e ha GRASP ou pe o ms he MA
especially when he numbe o ips is less han six.
The An Colony Op imiza ion (ACO), which akes inspi a ion om he
o aging beha io o an s, is a swa m in elligence algo i hm. So a ,
se e al ACO me hods ha e been applied success ully o he OP (Liang
e al., 2002)(Ke and Feng, 2008) and i s a ian s, including he Team
O ien ee ing P oblem (TOP) (Ke e al., 2008), mul i-objec i e e sion o
he OP (Schilde e al., 2009), Team O ien ee ing P oblem wi h Time
Windows (TOPTW) (Gamba della e al., 2012), Time-Dependen O ien-
ee ing P oblem (TD-OP) (Ve beeck e al., 2014), Time-Dependen
O ien ee ing P oblem wi h Time windows (TD-OPTW) (Ve beeck e al.,
2017), and Mul i-Objec i e Time-Dependen O ien ee ing P oblem
(MOTDOP) (Mei e al., 2016). In mos o hese algo i hms, solu ions a e
c ea ed sequen ially, and each an successi ely chooses nodes o be
added o he pa h ollowing a unidi ec ional sea ch s a egy. Fo mo e
in o ma ion abou he a ian s o he OP and he o he ypes o solu ion
me hods applied o sol ing hem, we e e he in e es ed eade o
(Gunawan e al., 2016) and (Vans eenwegen and Gunawan, 2019).
F om ano he poin o iew, he OPHS is a a ian o ou ing p oblems
wi h in e media e acili ies. We e e he in e es ed eade o (Schi e
e al., 2019) in which his ype o ou ing p oblems has been desc ibed
and ca ego ized.
Since, acco ding o he li e a u e, he ACO has a good backg ound in
sol ing he OPs wi h espec o hei g aph based na u e, in his pape ,
he ACO has been success ully applied o ackle he OPHS as an example
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
2
o he ou ing p oblem wi h in e media e acili ies.
3. P oposed algo i hm
The i e a i e p ocedu e o he ACS-OPHS is s a ed by ini ializing he
pa ame e s o he algo i hm. In o de o cons uc easible ou s, he ACS-
OPHS uses wo colonies o an s, whose nes s a e loca ed in H0and H1
(Sec ion 3.1). Each an changes phe omone le els o edges wi hin i s ou
locally (Sec ion 3.3). To imp o e he solu ion c ea ed by each an , a local
sea ch p ocedu e is employed (Sec ion 3.4). A he end o each i e a ion,
he global bes solu ion is upda ed i he bes cons uc ed solu ion is
be e han ha . A e wa ds, phe omone ails a e upda ed globally o
ampli y exploi a ion (Sec ion 3.3). Then, he nex i e a ion o he algo-
i hm is s a ed. The ACS-OPHS is i e a ed un il a specific s opping
c i e ia has been me . The pseudocode o he ACS-OPHS is shown in
Fig. 1. In he ollowing sub-sec ions, each s ep o he ACS-OPHS is
desc ibed in mo e de ail. We ha e used Tx;y o show he Euclidean dis-
ance be ween e ex xand e ex y.Tdand sja e he ime limi o ip d
and he sco e o node j, espec i ely. Mo eo e , as men ioned be o e,
HOTELS is he se o all a ailable ho els.
3.1. Cons uc ion phase
In he ACS-OPHS, wo an colonies, called C0and C1, wi h a simila
indi iduals numbe , m, a e used. Nes s o C0and C1a e loca ed in H0and
H1, espec i ely. Each an o colony C0s a s building a easible ou om
he fi s ip while he ou cons uc ion p ocedu e is begun om he las
ip by he an s o colony C1.
Assume ha an an om colony C0, a e walking along a pa ial ou ,
is now in ip dand he las isi ed ho el and e ex a e Hsand i,
espec i ely. The se FVðiÞo easible e ices ha can be isi ed a e iis
defined as ollows:
FVðiÞ¼j2UNVISITED9Hk2FEH½d½s:LþTi;jþTj;HkTd(1)
In his defini ion, UNVISITED is he se o no ye isi ed nodes wi hin
he cu en pa ial ou and FEH½d½sis he se o easible end-ho els o
ip dwi h Hsas he o igin o he ip. Addi ionally, Ldeno es he cu en
leng h o ip d. I he e is no easible node ha can be isi ed a e e ex
i,FVðiÞis defined wi h espec o he easible end-ho els ha can be
isi ed a e i:
FVðiÞ¼Hk2FEH½d½sLþTi;HkTd(2)
Ou p oposed me hod o cons uc FEH is discussed in Sec ion 3.2.
This me hod ollows an o fline ashion, which means ha FEH½d½sis
de e mined wi hou conce ning e ex iand he o he e ices isi ed
be o e i.
The same defini ions a e alid o an an om colony C1 ha a e
a e sing a e e sed pa ial ou om H1is now in ip dand he las
isi ed ho el and e ex a e Heand i, espec i ely.
FVðiÞ¼j2UNVISITED9Hk2FSH½d½e:LþTj;iþTHk;jTd(3)
o
FVðiÞ¼Hk2FSH½d½eLþTHk;iTd(4)
FSH½d½e ep esen s he se o easible s a -ho els o ip dwhen he
des ina ion o his ip is He. Ou p oposed me hod o cons uc he FSH is
also desc ibed in Sec ion 3.2.
Recall ha i is he las isi ed e ex. Each an selec s he nex e ex
o i s pa ial ou by applying he ollowing ansi ion ule:
j¼8
<
:
a gmax
k2FVðiÞ
τ
α
ik*
η
β
ikqq0ðexploi a ionÞ
J o he wise ðexplo a ionÞ
(5)
He e, q02½0;1is a pa ame e ha de e mines he impo ance o
exploi a ion e sus explo a ion, q is a uni o m andom numbe o e ½0;1,
and Jis a e ex ha is chosen andomly using he ollowing p obabilis ic
dis ibu ion:
pik ¼8
<
:
τ
α
ik*
η
β
ik
X
k02FVðiÞ
τ
α
ik0*
η
β
ik0
k2FVðiÞ
0o he wise
(6)
In o he wo ds, wi h p obabili y q0, he nex e ex is he membe o
FVðiÞwi h he highes alue o
τ
α
ik*
η
β
ik; o he wise, a oule e wheel p o-
cedu e is used o selec a e ex om FVðiÞ. In his case, he p obabili y o
choosing each e ex k2FVðiÞis de e mined using equa ion (6).
As i can be seen in equa ions (5) and (6), each an uses wo kinds o
in o ma ion in o de o make a decision: he phe omone in o ma ion
(
τ
ik), and he heu is ic in o ma ion (
η
ik). In he ACS-OPHS,
η
ik is defined
as ollows:
η
ik ¼sjTi;k e ex j is a node
1Ti;k e ex j is a ho el;(7)
α
and β, in equa ions (5) and (6), a e he o he pa ame e s o he ACS.
They de e mine he ela i e impo ance o he phe omone ails and he
heu is ic in o ma ion.
3.2. Iden i ying easible in e media e ho els
As desc ibed in he p e ious sec ion, wo ma ices, called FSH
(Feasible S a -Ho els) and FEH (Feasible End-Ho els), a e used o
cons uc easible ou s. The FSH and FEH a e wo-dimensional ma ices
wi h hcolumns and D ows. FSH½i½eis he se o easible s a -ho els o
ip iwhen he des ina ion o his ip is Heand FEH½i½sincludes easible
end-ho els o ip iwhen his ip is s a ed om Hs.
These ma ices a e cons uc ed be o e he s a o he main p ocedu e
o he ACS-OPHS. In o he wo ds, ini ially, conce ning he ime limi s,
he easible s a - and end-ho els o each ip a e de e mined. Then,
ega ding his in o ma ion, easible ou s a e cons uc ed in he con-
s uc ion phase a each i e a ion o he algo i hm.
The FSH and FEH a e defined using an auxilia y g aph, Ga¼ðV0;E0Þ.
V0is he se o e ices and con ains hðD1Þþ2 nodes ha a e loca ed
in Dþ1 le els. Ve ex i
j2V0is in le el iand ep esen s Hjas he s a
ho el o ip i. Since he o igin o ip 1 mus be H0, he fi s le el o Gahas
only one node, called 1
0. Simila ly, since H1is he only easible des i-
na ion o ip D, only one node, called Dþ1
1, is in he las le el o Ga. The
numbe o nodes in each o he o he D1 le els is equal o h.Fig. 2
demons a es he g aphical ep esen a ion o he e ices in Ga.
Conce ning he ime limi s o he ips, he se o edges, E0, is con-
s uc ed as shown in Fig. 3.
D awing an edge be ween i
jand iþ1
kindica es ha he e migh be a
while
o
Fig. 1. The gene al s uc u e o he ACS-OPHS.
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
3
easible sequence o ho els in which ip iis s a ed om Hjand ends in
Hk. The e is a doub in he exis ence o his sequence since conce ning he
ime limi s o ips iþ1 oD, i is possible ha he e is no any easible
sub-sequence o ho els s a ing om Hk. Howe e , he e is a leas one
easible sub-sequence o ho els o ips 1 o iin which ip iends in Hk.
An example wi h 8 ho els and 4 ips is shown in Fig. 4. As i can be seen,
conce ning only he fi s wo ips, he e a e h ee easible sub-sequences
o ho els in which ip 2 ends in H2(do ed lines). Howe e , he e is no
any easible sequence o ho els in which H2is he end-ho el o ip 2.
Now he FSH and FEH can be cons uc ed. Ini ially, o each iand j,
FSH½i½jand FEH½i½ja e emp y. S a ing om Dþ1
1, he g aph is explo ed,
and i edge ð i
j; iþ1
kÞis isi ed hen:
ho el Hkis a easible end-ho el o ip iwhen he o igin o his ip is
Hj. Thus, FEH½i½j¼FEH½i½j[ Hk}.
ho el Hjis a easible s a -ho el o ip iwhen his ip mus be ended
in Hk. Thus, FSH½i½k¼FSH½i½k[ Hj}.
In o he wo ds, his edge indica es ha he e is a leas one easible
sequence o ho els in which ip is a s om Hjand ends in Hk. The
cons uc ion p ocedu e o he FSH and FEH is as shown in Fig. 5.
The ime complexi y o he p ocedu e u ilized o cons uc ing he
FSH and FEH is OðDH2Þ. Recall ha Dand Ha e he numbe o ips and
ho els, espec i ely. Since hese wo pa ame e s usually do no ha e a
e y la ge alue in he applica ions o he OPHS, he p ocedu e in o-
duced in his sec ion is no ime consuming.
3.3. Phe omone ails upda ing ules
A e c ea ing a easible ou , he phe omone le els o he edges in he
ou a e upda ed locally using he ollowing equa ion:
τ
new
ij ¼ð1
ρ
Þ
τ
old
ij þ
τ
0(8)
ρ
2½0;1is he e apo a ion a e. The ini ial alue o he phe omone le el
(
τ
0) is se o 1=P
i2NODES
si. The local upda e occu s in he di ec ion ha he
ou is cons uc ed (i.e., om H0 o H1o ice e sa). In o he wo ds, i
a c ði;jÞis a e sed by an an o colony C0,
τ
ij mus be upda ed; howe e ,
i an an o colony C1goes h ough his edge,
τ
ji mus be changed.
As discussed in Sec ion 3.1, each an o bo h colonies cons uc a
easible ou . A e ha , he global bes solu ion is upda ed i a be e
solu ion is ound. Finally, acco ding o he ollowing equa ion, he
phe omone le els a e globally upda ed:
τ
new
ij ¼ð1
ρ
Þ
τ
old
ij þ
ρ
Δ
τ
ij
Δ
τ
ij ¼S0
ibS1
ibSgb (9)
S0
ib and S1
ib a e he sco es o he bes solu ions cons uc ed by he an s o C0
and C1, espec i ely. Mo eo e , Sgb is he sco e o he global bes solu ion.
Wi h espec o he global bes solu ion, he global upda e is pe o med in
bo h di ec ions. I means ha i a c ði;jÞis a e sed, bo h
τ
ij and
τ
ji a e
upda ed. Howe e , simila o he local upda ing, an unidi ec ional
scheme is employed o change he phe omone alues ega ding S0
ib and
S1
ib (i.e., upda ing only occu s in he di ec ion ha he solu ion is c ea ed).
3.4. Local sea ch ope a o s
Fig. 6 shows he pseudocode o he p ocedu e ha is employed o
imp o e a cons uc ed ou . This p ocedu e consis s o ou ope a o s. In
he ollowing, hese ope a o s a e desc ibed in o de o usage:
a) Imp o eHo els-DP: ollowing a dynamic p og amming s a egy, his
p ocedu e imp o es he sequence o ho els wi h espec o he nodes
isi ed along he ou . This local sea ch ope a o has been in oduced
by Soh abi e al. (2019) o he OPHS.
b) Exchange: ollowing he bes imp o emen s a egy, i swaps wo
nodes om wo di e en ips, whose exchange makes he mos
dec ease in he a eled ime. This ope a o is i e a ed un il no mo e
imp o emen s a e possible (Soh abi e al., 2019).
c) 2-op : his ope a o educes he leng h o each ip as a as possible
ollowing he bes imp o emen s a egy. The leng h o he ip is
educed i e a i ely. A each i e a ion, he pai o nodes, o which
e e sing he o de o isi ed nodes among hem makes he mos
educ ion in leng h, is de e mined. Then, acco dingly, he ou is
modified (Soh abi e al., 2019).
d) Inse : his ope a o de e mines he bes inse ion place o each un-
isi ed node conce ning ime es ic ions o he ips. Then, ollowing
he bes imp o emen s a egy, he node ha makes he mos inc ease
in he sco e and hen leads o he leas inc ease in he leng h, is
included in he ou . This ope a o is i e a ed un il no mo e inse ions
a e possible.
4. Expe imen al esul s
Acco ding o (Do igo and Gamba della, 1997),
α
,β,
ρ
, and ma e se o
1, 5, 0.9, and 10, espec i ely. I a be e solu ion is no ound o e 1000
consecu i e i e a ions, he algo i hm is s opped. The ini ial alue o q0is
se o 0.9. Du ing he execu ion o he ACS-OPHS, he alue o q0is
changed o es ablish an equilib ium be ween exploi a ion and explo a-
ion. I he global bes solu ion is no imp o ed o e 400 consecu i e
i e a ions, he alue o q0is al e ed o 0.2 o inc ease explo a ion. This
pa ame e is se o 0.9 when a new global bes ou is ound. This
modifica ion encou ages exploi a ion (i.e., sea ching he space a ound
he new global bes solu ion).
The ACS-OPHS has been applied o 405 benchma k ins ances c ea ed
by Di sala e al. (2013) (Di sala e al., 2014) and 76 ins ances designed
by Soh abi e al. (2019). The ins ances c ea ed by Di sala e al. (2013)
(Di sala e al., 2014) a e di ided in o 17 se s ega ding he numbe o
ho els and he numbe o ips. 16 p oblem se s a e e e ed o as SET
X–Y, whe e X is he numbe o ho els (excluding H0and H1) and Y is he
numbe o ips. 395 o he OPHS ins ances a e included in hese se s.
One o he se , which is SET 4, con ains en ins ances wi h h ee ho els
(excluding H0and H1) and wo o h ee ips. The op imal solu ions a e
unknown o fi e ou o en ins ances in SET 4. All he ins ances can be
accessed om h p://www.mech.kuleu en.be/en/cib/op. Using he
TSPHS ins ances wi h a mos 500 cus ome s, Soh abi e al. (2019) ha e
c ea ed 76 ins ances. These ins ances a e di ided in o fi e se s. The se s
a e e e ed o as “SET X-Y-Z”whe e X is he numbe o ho els excluding
H0and H1, Y is he numbe o ips, and Z is he name o he p oblem
(VRP o TSP) whose ins ances ha e been used by Vans eenwegen e al.
Fig. 2. The g aphical ep esen a ion o he e ices inGa.
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
4

(2012) o cons uc he TSPHS ins ances.
As desc ibed in Sec ion 2, ou algo i hms ha e been sugges ed o he
OPHS un il now: SVNS, MA, a hype -heu is ic ha is called HH in he
ollowing subsec ions, and GRASP. The ACS-OPHS is compa ed wi h all
o hese algo i hms excep he SVNS since he SVNS canno cons uc
easible solu ions o all OPHS ins ances in a easonable ime (Di sala
e al., 2014). In he ollowing subsec ions, he esul s associa ed wi h he
MA and GRASP a e he alues epo ed in (Soh abi e al., 2019). The
esul s epo ed in (Toledo e al., 2020) ha e also been used o compa e
he ACS-OPHS wi h HH. Table 1 p esen s he configu a ions o he PCs on
which he code o each algo i hm has been un. I should be men ioned
ha all he algo i hms ha e been implemen ed in Cþþ and ha e been
un h ee imes o e each ins ance. Mo eo e , o ha e some addi ional
compa isons, he ACS-OPHS is also un o 15 and 30 imes on each
o
o
i
Fig. 3. The cons uc ion p ocedu e o he edges inGa.
Fig. 4. Feasible sequences o ho els andGa.
o
i
o
o
i
o
i
Fig. 5. Cons uc ion p ocedu e o he FSH and FEH.
while
i
else
Fig. 6. The local sea ch p ocedu e.
Table 1
The configu a ions o he PCs on which each algo i hm has been un.
Algo i hm CPU RAM
GRASP,MA In el Co e i7 wi h 3.5 GHz 4 GB
ACS In el Co e i7 wi h 2.5 GHz 8 GB
HH In el Co e i7 wi h 3.40 GHz 16 GB
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
5
ins ance. In he ollowing sec ions, hese a ian s o ou p oposed
me hod a e called ACS15 and ACS30, espec i ely.
The bes solu ion o an algo i hm o an ins ance is he one ha is
ound du ing all specified numbe o uns o he algo i hm on he
ins ance. In all he expe imen s, he bes solu ions p o ided by he
algo i hms a e compa ed. To compa e he algo i hms in e ms o he
execu ion ime, a e age CPU imes o he uns a e conside ed in he
ollowing sec ions. To ob ain he o al execu ion ime o a specific
algo i hm, he a e age CPU ime is mul iplied by he numbe o imes
he algo i hm has been un. In he appendix, Tables A1 o A.22 show
he bes , a e age, and wo s solu ions ha he ACS-OPHS p oduces
o e h ee uns on each ins ance. The a e age execu ion ime o each
ins ance is also a ailable in hese ables.
4.1. The solu ions o he 405 benchma k ins ances in oduced by
(Di sala e al. (2013) (Di sala e al., 2014)
In Table 2, ega ding he execu ion ime and quali y o solu ions,
he ACS-OPHS, MA, GRASP and HH ha e been compa ed. A measu e
called gap is also used ha is de e mined o each ins ance as ollows:
gap ¼op imal alue bes Alg
op imal alue 100 (10)
bes Alg is he alue o he bes solu ion cons uc ed by algo i hm Alg.
The ACS-OPHS ou pe o ms he MA in e ms o :
a. The numbe o ins ances o which he op imal solu ions a e ound,
b. The numbe o ins ances o which he solu ions p o ided by he
ACS-OPHS is be e han ha o he MA,
c. The a e age gap in each se .
The sligh di e ences be ween he compu a ion imes o hese wo
algo i hms a e negligible.
Acco ding o he esul s, he solu ions p o ided by GRASP ha e
highe quali y han hose o he ACS-OPHS when he numbe o ips is
less han six. Howe e , he ACS-OPHS ou pe o ms his algo i hm in
SET 15-6, 15-8 and 15-10. Rega ding se e al compa isons, Soh abi
e al. (2019) claimed ha GRASP is a be e app oach o sol e he
OPHS han he MA i he numbe o ips is less han six. In mos o he
se s including ins ances wi h less han six ips, he e is a li le di -
e ence be ween he a e age gap o he ACS-OPHS and GRASP.
Mo eo e , hese wo algo i hms p o ide he op imal solu ions o
almos he same numbe o ins ances. Thus, we elici ha he
ACS-OPHS is a leas as good as GRASP.
In con as o GRASP, ACS-OPHS can p o ide high quali y solu-
ions o ins ances wi h e en mo e han six ips. Mo eo e , he
execu ion ime o ACS-OPHS is significan ly be e han ha o GRASP.
Conce ning he execu ion ime o he ACS-OPHS and GRASP, o
ha e a ai compa ison, he ACS-OPHS has also been un 15 imes on
each ins ance. This e sion o he ACS-OPHS is called ACS15. As
shown in Table 2, he ACS15 ou pe o ms GRASP in 38 ins ances while
he e a e 17 ins ances o which GRASP finds be e solu ions han he
ACS15. Mo eo e , he ACS15 can find he op imal ou s o mo e in-
s ances han GRASP. The supe io i y o he ACS15 wi h espec o he
a e age gaps especially in he se s wi h mo e han six ips is also
ema kable. Acco ding o hese esul s, we can conclude ha he
ACS15 is as powe ul as GRASP and, in some cases, i pe o ms e en
significan ly be e han GRASP.
Un o una ely, he de ailed esul s o he HH a e no a ailable. This
issue makes i impossible o ha e some comp ehensi e compa isons
wi h his algo i hm. The ACS-OPHS is compa ed wi h he HH only in
e ms o he a e age gap and he a e age CPU ime in each se . The
a e age gap o he HH is be e han ha o he ACS-OPHS only in six
se s. I should be no ed ha he esul s o he HH o en ins ances in
SET 4 a e una ailable. Acco ding o (Toledo e al., 2020), he HH
Table 2
The compa ison o he ACS-OPHS and he o he h ee algo i hms.
Ins ances Op imal Solu ions MA s. ACS GRASP s. ACS GRASP s. ACS15 gap A g. CPU Time (s)
SET #Ins. ACS ACS15 MA GRASP MA ACS GRASP ACS GRASP ACS ACS ACS15 MA GRASP HH ACS ACS15 MA GRASP HH
1–235 27 28 14 28 0 21 1 0 0 0 0.25 0.24 2.82 0.24 1.81 1.70 1.86 0.82 2.41 1.9
2–3 35 24 26 23 26 1 6 2 0 0 0 0.34 0.29 0.43 0.29 0.34 1.77 1.84 0.67 3.20 1.19
4 (OPT) 5 4 4 4 4 0 0 0 0 0 0 0.76 0.76 0.76 0.76 –1.70 1.70 0.56 2.46 –
5–335 26 26 24 26 0 4 0 0 0 0 0.29 0.29 0.41 0.29 0.34 1.91 1.94 0.68 4.17 1.19
3–4 35 25 27 23 26 0 5 3 0 0 1 0.35 0.30 0.46 0.31 0.31 1.75 1.81 0.52 3.94 0.87
6–4 35 26 26 23 26 0 4 0 0 0 0 0.33 0.33 0.39 0.33 0.31 1.78 1.88 0.56 5.06 0.9
10–4 22 10 12 7 11 0 9 3 0 2 1 0.30 0.25 0.96 0.25 0.39 7.92 8.54 3.08 31.60 7.11
12–4 22 11 12 6 10 1 11 4 2 1 3 0.37 0.24 1.24 0.31 0.72 7.85 8.99 2.97 34.95 7.1
15–422 7 11 5 12 3 12 7 0 3 0 0.65 0.29 1.32 0.23 0.61 8.46 8.91 3.05 39.60 7.43
10–5 22 9 11 6 10 3 7 5 0 1 2 0.91 0.51 1.30 0.54 0.67 7.33 8.23 2.61 33.12 5.18
12–5 22 8 9 6 9 1 8 5 0 3 2 0.94 0.88 1.43 0.60 0.65 8.28 8.56 2.44 35.85 5.37
15–522 11 11 7 10 1 10 3 3 1 3 0.68 0.51 1.40 0.67 0.88 7.63 8.43 2.55 40.56 5.87
10–6 22 10 10 9 10 2 8 5 2 2 2 0.51 0.39 1.24 0.38 0.56 9.21 9.04 2.17 34.28 4.22
12–6 22 10 11 6 11 1 10 5 1 2 2 0.47 0.35 1.63 0.47 0.71 7.80 8.31 2.15 37.51 4.5
15–6 22 10 10 8 10 0 10 4 6 2 6 0.68 0.60 1.39 0.81 0.92 8.45 8.69 2.35 43.62 5.08
15–8 13 4 4 1 2 0 8 0 7 0 7 0.87 0.75 2.95 2.72 1.87 10.13 11.09 2.52 69.63 5.43
15–10 9 2 3 2 0 1 5 0 9 0 9 2.10 0.91 3.78 6.15 1.82 12.97 13.92 2.29 93.73 5.28
All Ins. 400 224 241 174 231 14 138 47 30 17 38 0.52 0.40 1.24 0.60 –5.42 5.78 1.71 24.24 –
Column “Ins ances”p esen s he name and he numbe o ins ances o each se . Fo his compa ison, only fi e ou o en ins ances wi h known op imal alues in SET 4 ha e been conside ed. Column “Op imal Solu ions”
shows he numbe o ins ances o which each algo i hm p oduces he op imal solu ions. In he nex columns, wi h espec o he quali y o he bes solu ions, each pai o algo i hms a e compa ed. Fo example, in column
“MA s. ACS”, i is p esen ed ha o how many o he ins ances in each se he ACS-OPHS p o ides be e solu ions han he MA and ice e sa. Column “gap”shows he a e age gap o each algo i hm in each se . The las
column also con ains he a e age CPU imes in each se . To compu e he a e age CPU ime o each algo i hm in each se , we pe o m as ollows. Fo each ins ance, he a e age ime aken o each un o he algo i hm is
compu ed. Then, he mean o hese alues in each se is epo ed as he a e age CPU ime in ha se . A simila app oach is used o ob ain he a e age gap in each se . Besides epo ing hese measu es o each se , in he las
ow o columns “gap”and “A g. CPU Time (s)”, he a e age gaps and a e age CPU imes o e all 400 ins ances a e shown. Please no e ha he a e age o al execu ion ime o he algo i hms can be ob ained by mul iplying
hese alues by h ee excep o he ACS15 o which he co esponding alue mus be mul iplied by 15.
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
6
p oduces he op imal solu ions o 217 ins ances while ou p oposed
me hod finds he op imal ou s o 224 ins ances.
The de ailed esul s o he ACS-OPHS o he fi e ins ances wi h un-
known op imal solu ions in SET 4 a e shown in Table 3. The solu ions
p oduced by he ACS-OPHS o hese ins ances a e he same as hose o
GRASP. The e o e, ega ding hese ins ances, no imp o emen has been
achie ed using he ACS-OPHS.
To alida e he abo e compa isons, some s a is ical es s, which a e
explained as ollows, ha e been pe o med using he gap measu e. The
esul s o hese es s a e shown in Table 4.
Tes 1- ACS-OPHS s. MA:
The desc ip ion o he es is he same as he one conside ed by
Soh abi e al. in (Soh abi e al., 2019). This es indica es ha he pe -
o mance o he ACS-OPHS is significan ly be e han ha o he MA.
Ini ially, he ollowing hypo heses ha e been conside ed and hen he
Z- es has been used:
H0. : The p opo ion o he ins ances o which he ACS-OPHS p oduces
he op imal solu ion is less han o equal o ha o he MA
H1. : The p opo ion o he ins ances o which he ACS-OPHS p oduces
he op imal solu ion is la ge han ha o he MA
The es esul s show ha H
0
is ejec ed since P-Value is less han
0.05. The p opo ion o he ins ances o which he ACS-OPHS can find
he op imal esul is a leas 6.7% mo e han ha o he MA.
Conside ing 172 ins ances o which bo h algo i hms canno find he
op imal ou , he ollowing hypo heses ha e also been conside ed:
H1. The quali y o he solu ions p o ided by he ACS-OPHS is highe
han ha o he MA
In his es , H
0
is also ejec ed using he Mann–Whi ney as a non-
pa ame ic es . A nonpa ame ic es has been used since, acco ding o
Ande son-Da ling es , he dis ibu ions o he gaps a e no no mal. Using
he ACS-OPHS ins ead o he MA, he a e age gap o hese 172 ins ances
is educed om 2.04% o 1.17%.
Tes 2- ACS15 s. GRASP:
Rega ding he solu ions p oduced by ACS15 and GRASP o he in-
s ances o SET 15-6, SET 15-8, and SET 15-10, he ollowing hypo heses
ha e been defined:
H0. When bo h algo i hms canno find he op imal solu ion, he quali y
o he solu ion p o ided by ACS15 is he same as ha o he GRASP
H1. When bo h algo i hms canno find he op imal solu ion, he quali y
o he solu ion p o ided by he ACS15 is highe han ha o he GRASP.
This es has been pe o med conce ning 25 ins ances in he h ee se s
o which bo h algo i hms canno find he op imal esul . Since he
Mann–Whi ney es is significan a 0.0019, i can be concluded ha ,
conside ing he la ges ins ances in e ms o he TNFS alue, ACS-OPHS
ou pe o ms GRASP. Howe e , in he o he se s, he pe o mance o hese
wo me hods a e e y simila .
4.2. The solu ions o he 76 benchma k ins ances in oduced by (Soh abi
e al. (2019)
The esul s o he compa ison o he MA, GRASP, and ACS-OPHS a e
shown in Table 5. Rega ding he execu ion imes o he algo i hms, he
esul s o GRASP wi h 20 i e a ions (Soh abi e al., 2019) ha e been used.
The p ope ies o he ins ances in each se a e in he fi s h ee columns.
Simila o Table 2, in he nex columns, each pai o he algo i hms a e
compa ed wi h espec o he quali y o he solu ions. The las column
shows he a e age CPU ime o each algo i hm in seconds.
Simila o he MA and GRASP, he ACS-OPHS finds he op imal so-
lu ions o 12 ins ances in SET 4-4-VRP. The numbe o ins ances o
which he ACS-OPHS p o ides a be e solu ion han he MA is mo e han
he numbe o ins ances o which he MA finds a be e solu ion han he
ACS-OPHS. The pe o mance o he ACS-OPHS is be e han GRASP wi h
Table 3
The fi e ins ances wi hou known op imal solu ions in SET 4.
Ins ances Resul s A g. Execu ion Time o Each Run (s)
Name BKS ACS ACS15 MA GRASP ACS ACS15 MA GRASP
100-20-3-3.ophs 368 368 368 368 368 3.31 2.54 0.66 3.02
100-25-3-3.ophs 528 526 528 524 528 3.35 3.31 1.26 6.45
102-35-3-3.ophs 324 324 324 324 324 0.95 1.05 0.52 1.90
102-40-3-3.ophs 389 389 389 386 389 1.78 1.98 0.58 3.01
102-45-3-3.ophs 447 447 447 444 447 2.75 2.66 0.58 3.93
Table 4
S a is ical es s.
Z-Tes
Tes ID
α
Z P-Value Lowe bound o
di e ence
1 0.05 P
ACS
–P
MA
¼
0.125
3.56 0.000 0.067
No mali y Tes
Tes ID Ande son-
Da ling
P-Value
1 ACS; MA 12.354; 9.084 <0.005; <0.005
2 ACS15; GRASP 0.476; 1.439 0.218; <0.005
Mann-Whi ney Tes
Tes ID
α
Wilcoxon P-Value
1 0.05 24148 0.0000
2 0.05 488 0.0019
Table 5
The solu ions o he 76 benchma k ins ances.
SET #Ho els #T ips MA s. ACS GRASP-20 i e s. ACS A g CPU Time (s)
MA ACS GRASP ACS ACS MA GRASP-20 i e
4-4-VRP 6 4 22 4 0 52.51 37.76 58.42
2-3-TSP 4 3 2 12 9 3 48.95 62.50 45.75
4-4-TSP 6 4 3 93860.79 50.25 36.98
9-7-TSP 10,11 7 4 8114 37.18 30.12 30.24
8-4-TSP 10 4 66 6 5 59.46 54.21 55.71
To al Ins. ––17 37 23 30 51.79 46.85 45.59
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
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20 i e a ions especially when he numbe o ips is mo e han six (i.e.
SET 9-7-TSP). Be o e, he same conclusion has also been d awn ega ding
he ins ances in oduced by Di sala e al. (2013) (Di sala e al., 2014).
The e a e 20 ins ances o which he ACS-OPHS p oduces be e so-
lu ions han bo h he MA and GRASP wi h 20 i e a ions. As shown in
Table 6, he ACS-OPHS also imp o es he bes -known esul s o ou in-
s ances wi h unknown op imal solu ions. P e iously, he bes known
esul s o hese ins ances ha e been p oduced by GRASP wi h 500 i e -
a ions. Al hough he execu ion ime o he ACS-OPHS is ema kably less
han his e sion o GRASP, he ACS-OPHS can find be e esul s o
hese ins ances. The solu ions ound by he ACS-OPHS o hese ins ances
a e a ailable in he appendix. This is ano he e idence ha indica es he
powe o he ACS-OPHS in compa ison wi h he s a e-o - he-a
algo i hms.
4.3. E ec s o he wo main cha ac e is ics o he ACS-OPHS
Ou p oposed ACS me hod has wo main cha ac e is ics: i uses wo
colonies and modifies he alue o q0du ing he execu ion ime. To
e alua e he influences o hese p ope ies on he quali y o he solu ions,
wo expe imen s a e pe o med using 400 ins ances wi h known op imal
solu ions in oduced by Di sala e al. (2013) (Di sala e al., 2014):
a) The e ec o using wo colonies:
Th ee e sions o he ACS-OPHS a e compa ed. In he fi s one, only
colony C0is used (i.e., a each i e a ion o he algo i hm, 20 an s s a
hei ou s om H0). This e sion is called ACS_C0. ACS_C1 is he second
e sion ha only uses colony C1. The hi d e sion is he o iginal algo-
i hm. Wi h espec o gap, hese algo i hms ha e been compa ed. The
esul s o his es a e shown in Fig. 7. While he a e age gap o e 400
ins ances wi h known op imal solu ions is 0.52% o he ACS-OPHS, his
measu e equals o 1.35% and 1% o ACS_C0 and ACS_C1, espec i ely.
b) The e ec o changing he alue o q0:
As men ioned be o e, he ACS-OPHS changes he alue o q0du ing
i s execu ion o es ablish an equilib ium be ween exploi a ion and
Table 6
The ACS-OPHS imp o es he bes -known solu ions o ou ins ances.
Ins ances BKS ACS-OPHS esul s
Value Time (s) Value Time(s)
sp225-H4-T3-S3 10590 1037.30 10620 81.59
d100-H6-T4-S3 4490 37.46 4500 8.91
eil76-H11-T7-S3 3800 28.38 3810 5.66
eil76-H10-T4-S4 4290 32.62 4300 8.48
Fig. 7. E alua ing he e ec o using wo colonies.
Fig. 8. The influence o changing he alue o q0.
S. Soh abi e al. EURO Jou nal on T anspo a ion and Logis ics 10 (2021) 100036
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