Ren, Liang; Zhou, Ze ong; Fu, Yaping; Liu, Ao; Ma, Yun eng
A icle
In eg a ed op imiza ion o logis ics ou ing p oblem
conside ing chance p e e ence
Mode n Supply Chain Resea ch and Applica ions
P o ided in Coope a ion wi h:
Eme ald Publishing Limi ed
Sugges ed Ci a ion: Ren, Liang; Zhou, Ze ong; Fu, Yaping; Liu, Ao; Ma, Yun eng (2024) : In eg a ed
op imiza ion o logis ics ou ing p oblem conside ing chance p e e ence, Mode n Supply Chain
Resea ch and Applica ions, ISSN 2631-3871, Eme ald, Bingley, Vol. 6, Iss. 4, pp. 376-392,
h ps://doi.o g/10.1108/MSCRA-05-2023-0016
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In eg a ed op imiza ion o logis ics
ou ing p oblem conside ing
chance p e e ence
Liang Ren
School o Managemen , Wuhan Uni e si y o Science and Technology,
Wuhan, China and
Cen e o Se ice Science and Enginee ing,
Wuhan Uni e si y o Science and Technology, Wuhan, China
Ze ong Zhou
School o Managemen , Wuhan Uni e si y o Science and Technology,
Wuhan, China
Yaping Fu
College Business, Qingdao Uni e si y, Qingdao, China, and
Ao Liu and Yun eng Ma
School o Managemen , Wuhan Uni e si y o Science and Technology,
Wuhan, China and
Cen e o Se ice Science and Enginee ing,
Wuhan Uni e si y o Science and Technology, Wuhan, China
Abs ac
Pu pose –This s udy aims o examine he impac o he decision make s’ isk p e e ence on logis ics ou ing
p oblem, con ibu ing o logis ics beha io analysis and ou e in eg a ion op imiza ion unde unce ain
en i onmen . Due o he unexpec ed e en s and complex en i onmen in mode n logis ics ope a ions, he
logis ics p ocess is ull o unce ain y. Based on he chance unc ion o sa is ying he anspo a ion ime and
cos equi emen s, his pape ocuses on he ou h pa y logis ics ou ing in eg a ed op imiza ion p oblem
conside ing he chance p e e ence o decision make s om he pe spec i e o sa is ac ion.
Design/me hodology/app oach –This s udy used he quan i a i e me hod o in es iga e he ela ionship
be ween ou e decision making and human beha io . The cumula i e p ospec heo y is used o desc ibe he loss, gain
and u ili y unc ion based on con idence le els. A ma hema ical model and an imp o ed an colony algo i hm a e
employed o sol e he p oblems. Nume ical examples show he e ec i eness o he p oposed model and algo i hm.
Findings –The s udy’s indings e eal ha he dual-popula ion imp o emen s a egy enhances he
algo i hm’s global sea ch capabili y and he imp o ed algo i hm can sol e he isk model quickly, e i ying
he e ec i eness o he imp o emen me hod. Mo eo e , he decision-make is mo e sensi i e o losses, and he
u ili y ob ained when conside ing decision-make s’ isk a i udes is g ea e han ha ob ained when he
decision-make exhibi s isk neu ali y.
P ac ical implica ions –In an unce ain en i onmen , he logis ics decision make ’s isk p e e ence di ec ly
a ec s decision making. Di e en pa ame e combina ions in he p oposed model could be se o decision-
MSCRA
6,4
376
© Liang Ren, Ze ong Zhou, Yaping Fu, Ao Liu and Yun eng Ma. Published in Mode n Supply Chain
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This wo k is suppo ed by he Na u al Science Founda ion o Hubei P o ince o China unde G an
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Resea ch P ojec unde G an No. 20Q21; Founda ion o WUST Resea ch on De elopmen o Sma
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Recei ed 1 May 2023
Re ised 27 Janua y 2024
16 Ap il 2024
2 Augus 2024
Accep ed 20 Augus 2024
Mode n Supply Chain Resea ch
and Applica ions
Vol. 6 No. 4, 2024
pp. 376-392
Eme ald Publishing Limi ed
2631-3871
DOI 10.1108/MSCRA-05-2023-0016
make s wi h di e en isk a i udes o i hei needs mo e accu a ely. This could help manage s design
e ec i e anspo a ion plans and imp o e se ice le els. In addi ion, he imp o ed algo i hm can sol e he
p oposed p oblem quickly, s ably and e ec i ely, so as o help he decision make o make he logis ics pa h
decision quickly acco ding o he equi ed con idence le el.
O iginali y/ alue –Conside ing he unce ain y in logis ics and he isk beha io o decision make s, his
pape s udies in eg a ed ou ing p oblem om he pe spec i e o oppo uni y p e e ence. Based on he chance
unc ion o sa is ying he anspo a ion ime and cos equi emen s, a ou h pa y logis ics ou ing
in eg a ed op imiza ion p oblem model conside ing he chance p e e ence o decision make s is es ablished.
Acco ding o he cha ac e is ics o he p oblem, an imp o ed dual-popula ion an colony algo i hm is designed
o sol e he p oposed model. Nume ical examples show he e ec i eness he p oposed me hods.
Keywo ds Fou h pa y logis ics, Rou ing op imiza ion, Dependen -chance p og amming,
An colony algo i hm, Risk a i ude
Pape ype Resea ch pape
1. In oduc ion
Wi h he apid de elopmen o mode n economy and e-comme ce, he compe i ion among
mode n en e p ises is becoming mo e and mo e ie ce (Tian e al., 2022;Fu e al., 2021a,b).
As an impo an means o p omo ing en e p ise indus ial upg ading and enhancing co e
compe i i eness, logis ics has a ac ed widesp ead a en ion om a ious indus ies
(S odola, 2020;Goli e al., 2022). A he same ime, people’s demands o logis ics se ice
le els ha e con inued o inc ease (Qian e al., 2021;Mehmann and Teu ebe g, 2016),
pa icula ly in unce ain logis ics se ices whe e s able and e icien deli e y is one o he
mos e ec i e ways o mode n logis ics companies o win cus ome ecogni ion.
In o de o imp o e logis ics e iciency and co e compe i i eness o en e p ises, mos
en e p ises ou sou ce logis ics business o p o essional hi d-pa y logis ics (3PL) p o ide s.
Howe e , wi h he apid de elopmen o mode n logis ics, cus ome s’ equi emen s o
logis ics se ice le el a e cons an ly inc easing. Logis ics decision is no only abou
accomplishing anspo a ion asks, bu also esou ce sha ing and abili y in eg a ion among
di e en subjec s (Wang e al., 2020,2024). T adi ional 3PL p o ide s lack supply chain
managemen capabili ies, and he coope a ion be ween 3PL p o ide s is no deep enough,
and complemen a y esou ces a e no ully u ilized, making i di icul o mee he cu en
ma ke demand o ie ce compe i ion (Zhang e al., 2021). The e o e, he indus y and
academia a e bo h ocusing on ou h-pa y logis ics (4PL) om a esou ce in eg a ion
pe spec i e (Ga o na, 1998;Yao, 2010). In ecen yea s, 4PL companies and logis ics
en e p ises o med and p o iding se ices based on he 4PL concep ha e g adually
demons a ed hei s ong compe i i eness and in luence (Huang e al., 2009).
The essen ial na u e and co e ad an age o 4PL ope a ion lie in i s abili y o in eg a e
supply chain esou ces (Tao e al., 2017). By coope a ing wi h pa icipan s a a ious s ages
wi hin he supply chain, 4PL can acili a e mu ual p omo ion, alle ia e he phenomenon o
3PL en e p ises ac ing alone, icious compe i ion in de eloped a eas and inadequa e supply
in unde de eloped a eas, and e ec i ely in eg a e and ully u ilize social esou ces (Yin e al.,
2022). Many schola s a home and ab oad ha e conduc ed esea ch on issues ela ed o 4PL,
such as supplie e alua ion p oblems (K ako ics e al., 2008), 3PL supplie selec ion
(Aguezzoul, 2014), con ac design (Wang e al., 2021;Huang e al., 2019), scheduling (Liu
e al., 2014), ne wo k design (Wang e al., 2021), ou ing p oblem (Zhang e al., 2005) and so on.
Rou e planning is one o he co e ac o s a ec ing he o e all e iciency o logis ics (Goli
e al., 2022;Wang e al., 2023). The ou h pa y logis ics ou ing op imiza ion p oblem
(4PLROP) is a c i ical issue in mode n logis ics op imiza ion (Huang e al., 2016). 4PL in ol es
selec ing app op ia e anspo a ion ou es o shipping asks while also selec ing he 3PL
supplie s who p o ide anspo a ion se ices along ha pa h, p esen ing a challenge o
adi ional ou ing p oblems. Some schola s ha e p oposed simpli ying 4PLROP by using a
Mode n Supply
Chain Resea ch
and Applica ions
377
desc ip ion me hod o a di ec ed mul ig aph. Chen e al. (2003) desc ibed each edge in a di ec ed
mul ig aph as a 3PL supplie ha p o ides anspo a ion se ices along ha pa h, p o iding a
clea desc ip ion o 4PLROP and quickly sol ing small-scale p oblems. Cui e al. (2013)
in eg a ed pa h selec ion and 3PL supplie selec ion in o an undi ec ed mul ig aph, also
desc ibing each edge in he g aph as a 3PL supplie , and s udied he mo e complex 4PLROP
while conside ing issues such as ans e uck ime and mul i asking.
Mos exis ing ele an esea ch has ocused on de e minis ic p oblems. Howe e , due o
ac o s such as wea he , a ic, human e o and a ious unexpec ed si ua ions, logis ics
anspo a ion p ocesses ha e s ong unce ain y (Huang e al., 2015), which caused
subs an ial losses in p o i s (Zhou e al., 2023;Goli e al., 2023a,b). Especially in long-dis ance
deli e y such as c oss-bo de anspo a ion, he e a e signi ican dis u bances in logis ics
anspo a ion ime and cos due o geog aphy, ele an sys ems and language easons.
Huang e al. (2013) assumed ha he 4PL sys em had no his o ical da a, so he dis ibu ion o
3PL supplie anspo a ion ime could be desc ibed as a uzzy a iable by ele an expe s
based on his o ical expe ience, and s udied he 4PLROP wi h uzzy p ocessing ime wi h he
objec i e o minimizing o al cos . Howe e , in ano he scena io in logis ics ope a ions,
logis ics companies ha e some his o ical da a, which can be used o es ima e he p obabili y
o p obabili y dis ibu ion o unce ain e en s using andom a iables. Mo eo e , wi h he
apid de elopmen o e-comme ce and ie ce compe i ion in he logis ics indus y in ecen
yea s, adi ional p ice compe i ion be ween logis ics companies has g adually shi ed o
compe i ion based on cus ome se ice le els (Fu e al., 2020,2021a,b).
In an unce ain en i onmen , decision-make s o en hope o maximize he p obabili y
unc ion o e en ealiza ion (Liu, 1997), a he han absolu e e u ns, and hope o achie e
highe cus ome sa is ac ion while ensu ing a ce ain e u n. In o de o accu a ely desc ibe
and measu e he cha ac e is ics o people’s cogni ion, judgmen and choice in unce ain
si ua ions, Simon (1955) p oposed he heo y o bounded a ionali y. On his basis, T e sky
and Kahneman (1992) p oposed he P ospec heo y. P ospec heo y applies psychological
esea ch o economics and p o ides an e ec i e ool o human judgmen and decision
making unde unce ain en i onmen .
This s udy desc ibes he anspo a ion ime and cos o 3PL supplie s as andom a iables
and in es iga es he 4PLROP in an unce ain en i onmen . As he scheme designe , 4PL hopes
o minimize he p obabili y o delay om he cus ome ’s pe spec i e, while also minimizing he
p obabili y o exceeding cos expec a ions om he pe spec i e o he pa icipa ing 3PL
supplie s. The e o e, depending on he isk a i ude o he decision-make , his s udy
es ablishes a ma hema ical model o he Dependen -chance based 4PL Rou ing Op imiza ion
P oblem (DP-4PLROP) by maximizing he o al u ili y o he anspo a ion ime and cos
oppo uni y unc ions, aking in o accoun a i ude owa ds oppo uni ies. The p oposed DP-
4PLROP is an NP-ha d p oblem, which is di icul o be sol ed by adi ional algo i hms.
In elligen op imiza ion algo i hm is one o he e ec i e ways o sol e la ge-scale complex
op imiza ion p oblems (Goli e al., 2023b;Wang e al., 2022). Based on he cha ac e is ics o he
p oblem, an colony algo i hms and dual-popula ion imp o ed an colony algo i hms a e
designed o sol e he model. The nume ical examples demons a e he a ionali y o he
es ablished model and he e ec i eness o he p oposed algo i hm.
2. P oblem desc ip ion
Assuming a ce ain company (4PL) has unde aken a supply chain logis ics pa h in eg a ion
business, i is equi ed o design a se o anspo a ion plans o he clien which will
anspo anspo a ion asks om he s a ing poin o he supply chain o he des ina ion
node. Since 4PL is a logis ics solu ion in eg a o , i is assumed ha i has ce ain pa h
in o ma ion and some coope a ing 3PL supplie s be o e he ask s a s. The anspo a ion
MSCRA
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378
ne wo k in o ma ion and al e na i e 3PL supplie s’ in o ma ion in he supply chain a e
known, and he e may be mul iple 3PL supplie s ha can p o ide anspo a ion se ices
along each pa h. In o de o desc ibe he p oblem mo e clea ly, supplie in o ma ion and pa h
in o ma ion a e in eg a ed. The p oposed DP-4PLROP can be desc ibed by a mul i-g aph
wi h mul iple a ibu es.
An undi ec ed mul ig aph GðV;EÞshown in Figu e 1 is used o desc ibe he p oposed DP-
4PLROP. VðjVj ¼ nÞis he se o nodes, ep esen ing ci ies, wa ehouses, p ocessing plan s
and o he acili ies in he supply chain; E is he se o edges, each edge ep esen s a candida e
3PL supplie who can unde ake anspo a ion asks on ha pa h, and he e may be mul iple
edges be ween adjacen nodes. Bo h nodes and edges ha e cos and ime a ibu es. Due o
a ious unce ain ac o s in logis ics anspo a ion, he anspo a ion ime and cos o he
3PL supplie ha e a ce ain deg ee o dis u bance, which is desc ibed as a andom a iable.
The pa ame e s and decision a iables o he DP-4PLROP based on he desc ip ion o he
undi ec ed mul ig aph a e p esen ed in Table 1.
s e
4
3
2
1
5
2
2
2
1
2
2
2
2
2
2
2
2
1
3
1
3
1
4
1
1
3
4
1
3
11
3
1
1
3
2
1
3
Sou ce(s): Au ho s’ own wo k
Pa ame e s
eijk Rep esen s he k h 3PL supplie (k h edge) be ween nodes iand j, i;j∈ð1;2;���;nÞ
ij Rep esen s he numbe o 3PL supplie s (edges) be ween nodes iand j
Tijk þ ijk Rep esen s he ime be ween nodes iand j o 3PL supplie eijk o comple e he anspo a ion
ask o his sec ion. Whe e, Tijk is cons an and ep esen s basic ime; ijk is a andom a iable,
ep esen ing he dis u bance o ime
Cijk þcijk Rep esen s he cos equi ed by 3PL supplie eijk o comple e his sec ion o anspo a ion ask.
Whe e Cijk is cons an , ep esen ing basic cos ; cijk is a andom a iable ep esen ing he
pe u ba ion o cos
T0
i;C0
iRep esen s he ime and cos equi ed by he anspo a ion ask when i passes h ough node i,
espec i ely
T0;C0Respec i ely ep esen s he decision-make ’s equi emen s on he o al ime and o al cos o he
anspo a ion ask, ha is, he o al anspo a ion ime should no exceed T0and he o al
anspo a ion cos should no exceed C0
α
;βThe con idence le el indica ing he o al ime and o al cos ha mee he equi emen s sepa a ely,
ha is, in an unce ain en i onmen , he p obabili y ha he o al anspo a ion ime does no
exceed T0and he p obabili y ha he o al anspo a ion cos is no g ea e han C0
R I ep esen s a pa h om he s a ing node s o he des ina ion node eo a ask
Decision a iable
eijkðRÞ1 i eijk ∈R, 0 o he wise
yiðRÞ1 i i∈R, 0 o he wise
Sou ce(s): Au ho s’ own wo k
Figu e 1.
Mul i-g aph o DP-
4PLROP
Table 1.
Ma hema ical no a ion
Mode n Supply
Chain Resea ch
and Applica ions
379
He e, ijk and cijk espec i ely ep esen he dis u bance o anspo a ion ime and cos o 3PL
supplie eijk. Fo he pu pose o exposi ion and necessa y ma hema ical simpli ica ion, i is assumed
ha hey ollow no mal dis ibu ions Nð0;
σ
2
ijkÞand Nð0;
σ
2
ijkÞ, and a e independen o each o he [1].
I can be seen ha R is he solu ion o he p oposed p oblem. In he mul i-g aph, each R
uniquely de e mines he pa h aken by he anspo a ion ask and he 3PL supplie who
pe o ms he anspo a ion ask on ha pa h. The ime and cos o Ra e ep esen ed by TðRÞ
and CðRÞ espec i ely, hen:
TðRÞ ¼ X
n
i¼1X
n
j¼1X
ij
k¼1ðTijk þ ijkÞxijkðRÞþX
n
i¼1
T0
iyiðRÞ;(1)
CðRÞ ¼ X
n
i¼1X
n
j¼1X
ij
k¼1ðCijk þcijkÞxijkðRÞþX
n
i¼1
C0
iyiðRÞ:(2)
The e o e, TðRÞand CðRÞalso ollow no mal dis ibu ions, ha is
TðRÞ∼N X
n
i¼1X
n
j¼1X
ij
k¼1
TijkxijkðRÞþX
n
i¼1
T0
iyiðRÞ;X
n
i¼1X
n
j¼1X
ij
k¼1
σ
2
ijkxijkðRÞ!;
CðRÞ∼N X
n
i¼1X
n
j¼1X
ij
k¼1
CijkxijkðRÞþX
n
i¼1
C0
iyiðRÞ;X
n
i¼1X
n
j¼1X
ij
k¼1
σ
2
ijkxijkðRÞ!:
The p oblem o be sol ed in his pape is o p o ide cus ome s wi h a anspo a ion plan ha
anspo s anspo a ion asks om he s a ing node o he des ina ion node unde ce ain
ime and cos equi emen s. Due o he dis u bance o anspo a ion ime and cos o 3PL
supplie s, he plan mee s he o al ime and o al cos equi emen s wi h a ce ain con idence
le el, and maximizes he o al u ili y o he con idence le el while conside ing he decision
make ’s p e e ence.
3. P oblem o mula ion
Fo a gi en pa h R, i s o al ime TðRÞand o al cos CðRÞsa is y he con idence le els
α
and β
(oppo uni y unc ion) equi ed by he cus ome , as shown in Eq. (3) and Eq. (4).
α
¼P (X
n
i¼1X
n
j¼1X
ij
k¼1ðTijk þ ijkÞxijkðRÞþX
n
i¼1
T0
iyiðRÞ≤T0)(3)
β¼P (X
n
i¼1X
n
j¼1X
ij
k¼1ðCijk þcijkÞxijkðRÞþX
n
i¼1
C0
iyiðRÞ≤C0)(4)
People o en pay mo e a en ion o di e ences a he han absolu e e u n alues when
making decisions (T e sky and Kahneman, 1992). The e o e, we i s assume ha he
decision make has expec a ions o he con idence le els o o al ime and o al cos , which
a e he e e ence poin s
α
0and β0. Fo each al e na i e plan R, i ep esen s a loss when he
con idence le el is below he e e ence poin , and a gain when he con idence le el is abo e
he e e ence poin . Thus, d awing on he alue unc ion desc ip ion in cumula i e p ospec
heo y (CPT) (T e sky and Kahneman, 1992), he u ili y unc ions o o al ime and o al cos
can be de ined as ollows:
MSCRA
6,4
380
ð
α
Þ ¼ �ð
α
�
α
0Þγ1;i
α
≥
α
0
�λ1ð
α
0�
α
Þγ1;else (5)
ðβÞ ¼ �ðβ�β0Þγ2;i β≥β0
�λ2ðβ0�βÞγ2;else (6)
He e, he pa ame e s γ1and γ2 ep esen he sensi i i y o he decision make o he
con idence le els o ime and cos , wi h la ge alues indica ing g ea e sensi i i y.
Mo eo e , 0 <γ1<1;0<γ2<1, which e lec s he gene al cha ac e is ic o dec easing
sensi i i y o he decision make . The pa ame e s λ1and λ2a e he ela i e sensi i i y
coe icien s o gain and loss, wi h highe alues indica ing a s onge a e sion o losses. In
addi ion, he ela i e alues o γ1,λ1and γ2,λ2can e lec he ela i e sensi i i y o he
decision make o he con idence le els o ime and cos .
In an unce ain en i onmen , he ma hema ical model o DP-4PLROP can be es ablished
as ollows:
max V¼ ð
α
Þþ ðβÞg (7)
s. .
xijkðRÞ ¼ �1;i eijk ∈R
0;else (8)
yiðRÞ ¼ �1;i i∈R
0;else (9)
R¼ ð s;���; i;k; j;���; eÞ∈G (10)
Among hem, o mula (7) is he objec i e unc ion, which ep esen s maximizing he o al
u ili y o he con idence le els o ime and cos . The con idence le els
α
and βa e exp essed
in o mulas (3) and (4), espec i ely, and he u ili y unc ions ð
α
Þand ðβÞa e exp essed in
o mulas (5) and (6), espec i ely. Fo mulas (8) and (9) a e he 0–1 decision a iables o he
model, which de e mine he selec ed 3PL p o ide o execu ing he ask and he nodes
passed h ough. Fo mula (10) indica es ha he selec ed pa h is a ou e om he s a ing node
o he des ina ion node.
4. Algo i hm design
4.1 Design idea
I can be seen ha DP-4PLROP is an ex ension o he Cons ained Sho es Pa h P oblem
(CSPP). CSPP is an NP-ha d p oblem (Liu e al., 2012), and he e o e DP-4PLROP is also NP-
ha d, making i di icul o sol e using adi ional exac algo i hms (Fu e al., 2022;Tian e al.,
2023). The An Colony Algo i hm (ACA) is an in elligen algo i hm ha mimics he beha io
o an colonies. I has high obus ness, dis ibu ed compu ing and is easily combinable wi h
o he op imiza ion me hods, especially when sol ing sho es pa h p oblems such as VRP
and TSP, showing unique ad an ages (Do igo e al., 1996). Conside ing he p oblem o
p ema u e con e gence and s agna ion in ACA du ing e olu ion, an Imp o ed An Colony
Algo i hm (IACA) is designed by inco po a ing he idea o dual-popula ion independen
sea ching wi h pe iodic in o ma ion exchange based on he cha ac e is ics o undi ec ed
mul ig aphs.
Mode n Supply
Chain Resea ch
and Applica ions
381
4.2 ACA
(1) Coding mechanism
Ris he solu ion o he p oblem, ep esen ing a pa h om he s a ing node s o he des ina ion
node ein a mul i-g aph. I includes a se o edges and a se o nodes. F om he mul ig aph
desc ip ion, i can be seen ha each edge eijk uniquely de e mines a pai o adjacen nodes i
and jin R, ha is, he se o edges in Rcan uniquely de e mine he se o nodes. The e o e, only
he se o edges is encoded. Conside ing ha he numbe o edges in each solu ion may a y, a
a iable-leng h encoding mechanism is designed. NP ep esen s he popula ion size, and he
coding o he m h ðm¼1;2;���;NPÞan can be ep esen ed as ollows:
Rm¼ ðesik;���;ejelÞ∈G:(11)
He e, esik ep esen s he s a ing node so Rm, ejel ep esen s he inal des ina ion node
e.In o de o ensu e he connec i i y o Rm, adjacen elemen s (edges) mus pass h ough he
same in e media e node, ha is, he a i al node o he p e ious elemen and he en y node o
he nex elemen a e he same.
(2) T ans e p obabili y
Du ing he ans e p ocess o an m, i s di ec ion is de e mined based on he in o ma ion on
all easible edges and he pa h heu is ic in o ma ion. Le NG ep esen he maximum numbe
o i e a ions, and allowedm ep esen he se o all easible edges o an m a he cu en
momen . Then, he calcula ion me hod o he ans e p obabili y pm
ijkðNgÞo a ce ain edge
eijk in he cu en easible se a he NgðNg ¼1;2;���;NGÞi e a ion is as ollows:
pm
ijkðNgÞ ¼
½
τ
ijkðNgÞ�
ω
�
η
ijk�φ
Xa c⊂allowedm½
τ
a cðNgÞ�
ω
½
η
a c�φ
0;else
;a c ∈allowedm:
8
>
>
>
<
>
>
>
:
(12)
He e,
τ
ijkðNgÞ ep esen s he concen a ion o in o ma ion phe omones on edge eijk in he Ng- h
i e a ion,
η
ijk ep esen s he pa h heu is ic in o ma ion, whe e
η
ijk ¼1=ðTijk þCijkÞ.
ω
and φ
espec i ely ep esen he phe omone heu is ic ac o and pa h heu is ic ac o , e lec ing he
ela i e impo ance o in o ma ion phe omone concen a ion and pa h heu is ic in o ma ion.
(3) Phe omone upda ing s a egy
A he beginning o algo i hm execu ion, he same ini ial in o ma ion phe omone
concen a ion P0is assigned o each edge in he mul i-g aph. A e e e y gene a ion o
an s comple es he sea ch, he phe omone concen a ion is upda ed using he ollowing
o mula:
τ
ijkðNg þ1Þ ¼
ρτ
ijkðNgÞþ∆
τ
ijk (13)
∆
τ
ijk ¼8
>
<
>
:
α
þβþθ
Q;eijk ∈R
0;else
(14)
He e,
τ
ijkðNg þ1Þ ep esen s he concen a ion o in o ma ion phe omones on edge eijk in he
i e a ion o ðNg þ1Þ,∆
τ
ijk ep esen s he inc emen o phe omone concen a ion, R
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ep esen s he cu en op imal solu ion, ep esen s he cu en op imal alue and θis a
cons an when ≥0, bu θ¼0 when <0.
(4) Maximum and minimum an
As he algo i hm i e a es, i is possible o he concen a ion o in o ma ion phe omones on
ce ain pa hs in he mul i-g aph o con inuously inc ease, while phe omones on o he pa hs
con inuously e apo a e. To a oid highly concen a ed phe omone le els ha cause all an s
in he popula ion o sea ch he same pa h, leading o p ema u e con e gence o a local
op imum, he concen a ion o phe omones on each edge is limi ed o a ce ain ange. When
τ
ijkðNgÞ<
τ
min,
τ
ijkðNgÞ ¼
τ
min; when
τ
ijkðNgÞ>
τ
max,
τ
ijkðNgÞ ¼
τ
max.
(5) Repai s a egy o illegal pa hs
Du ing he an sea ch p ocess, he e may be a si ua ion whe e he cu en easible se
allowedmis emp y, which means ha he e is no alid pa h o each he des ina ion node. In
his case, i is necessa y o epai he in alid pa h. The adi ional me hod is o he an o
back ack o he p e ious node and add he cu en pa h o he aboo lis . Howe e , his
me hod consumes a lo o compu a ional ime due o he back acking p ocess. The e o e,
acco ding o he cha ac e is ics o he p oblem, he ollowing me hod is designed o
epai ing in alid pa hs: o he cu en in alid pa h Rm, s a ing om i s ini ial node, check
whe he he cu en node is di ec ly connec ed o he des ina ion node in he mul i-g aph. I i
is, andomly selec an edge be ween he node and he des ina ion node and add i o he
encoding; o he wise, he an es a s he sea ch.
4.3 IACA
ACA o en shows unique ad an ages in sol ing ou ing p oblems (Do igo e al., 1996).
Howe e , p ema u e con e gence and s agna ion o en cause he algo i hm o ail o ob ain
op imal solu ions. To add ess his p oblem and sol e he p oposed model quickly and
e icien ly, a dual-popula ion independen e olu ion app oach is designed. In IACA, wo
popula ions e ol e independen ly and egula ly in e ac wi h each o he , o p e en local
con e gen beha io o single-popula ion sea ch and imp o e he global sea ch capabili y.
(1) The upda e me hod o phe omone o popula ion A
In he mul i-g aph, each edge eijk has wo ypes o phe omones,
τ
A
ijkðNgÞand
τ
B
ijkðNgÞ, which
ep esen he phe omone concen a ion o popula ions A and B, espec i ely, in he Ng- h
gene a ion.
Popula ion A uses he eli e s a egy o upda e phe omones, as shown in equa ions (13)
and (14).
(2) The upda e me hod o phe omone o popula ion B
In popula ion B, phe omone is upda ed based on he ixed amoun o phe omone le by each
an on he pa h i has passed h ough, and he upda e o mula is as ollows:
τ
B
ijkðNg þ1Þ ¼
ρτ
B
ijkðNgÞþ
μ
∆
τ
;(15)
In which
μ
ep esen s he numbe o an s passing h ough he edge eijk in he Ng- h
gene a ion, and ∆
τ
is he phe omone le by he an s as hey pass by.
(3) Phe omone in e ac ion
When he i e a ion numbe Ng is a mul iple o M, he phe omone in e ac ion be ween
popula ion A and popula ion B is pe o med. The in e ac ion me hod is as ollows:
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Co esponding au ho
Ze ong Zhou can be con ac ed a : [email p o ec ed]
Fo ins uc ions on how o o de ep in s o his a icle, please isi ou websi e:
www.eme aldg ouppublishing.com/licensing/ ep in s.h m
O con ac us o u he de ails: [email p o ec ed]
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