applied
sciences
A icle
Model and Algo i hm o Two-S age Dis ibu ion
Loca ion Rou ing wi h Ha d Time Window o Ci y
Cold-Chain Logis ics
Liying Yan 1,2,3,4 , Manel G i oll 5and Pengjun Zheng 1,3,4,*
1Facul y o Ma i ime and T anspo a ion, Ningbo Uni e si y, Collabo a i e Inno a ion Cen e o Ningbo
Po Logis ics Se ice Sys em, Ningbo 315211, Zhejiang, China; [email p o ec ed]
2
Depa men o Basic Cou ses, Ningbo Uni e si y o Finance & Economics, Ningbo 315211, Zhejiang, China
3Ningbo Uni e si y Sub-Cen e , Na ional T a ic Managemen Enginee ing& Technology Resea ch Cen e,
Ningbo 315211, Zhejiang, China
4Collabo a i e Inno a ion Cen e o Mode n U ban T a ic Technologies, Nanjing 211189, Jiangsu, China
5Ba celona Inno a ion in T anspo (BIT), Ba celona School o Nau ical S udies, Uni e si a Poli ècnica de
Ca alunya-Ba celonaTech, 08003 Ba celona, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 15 Ma ch 2020; Accep ed: 3 Ap il 2020; Published: 8 Ap il 2020


Abs ac :
Taking cold-chain logis ics as he esea ch backg ound and combining wi h he o e all
op imisa ion o logis ics dis ibu ion ne wo ks, we de elop wo-s age dis ibu ion loca ion- ou ing
model wi h he minimum o al cos as he objec i e unc ion and a ying ehicle capaci y in di e en
deli e y s ages. A hyb id gene ic algo i hm is designed based on coupling and collabo a ion o he
wo-s age ou ing and ans e s a ions. The alidi y and easibili y o he model and algo i hm
a e e i ied by conduc ing a andomly gene a ed es . The op imal solu ions o di e en objec i e
unc ions o wo-s age dis ibu ion loca ion- ou ing a e compa ed and analysed. Resul s u n ou ha
o di e en dis ibu ion objec i es, di e en dis ibu ion schemes should be employed. Finally, we
compa e he wo-s age dis ibu ion loca ion- ou ing o single-s age ehicle ou ing p oblems. I is
ound ha a wo-s age dis ibu ion loca ion- ou ing sys em is easible and e ec i e o he cold-chain
logis ics ne wo k, and can dec ease dis ibu ion cos s o cold-chain logis ics en e p ises.
Keywo ds: wo-s age dis ibu ion; loca ion- ou ing; hyb id gene ic algo i hm
1. In oduc ion
Many ci ies ind i challenging o se up a high-e iciency ci y logis ics sys em o inc ease eigh
e iciency and dec ease he impac s o ci y dis ibu ion on ci y li ing condi ions [
1
]. Macha is e
al. [
2
] poin ed ou ha u ban goods dis ibu ion (UGD) has an impo an impac on he sus ainable
de elopmen o ci ies. I is necessa y o ind new solu ions o he managemen o eigh dis ibu ion in
o de o each a highe le el o e iciency. So a ci y logis ics model and some u ban eigh a ic policies
ha e been s udied by many esea che s such as Taniguchi e al. [
3
], Allen e al. [
4
], Taniguchi e al. [
5
],
Anand e al. [
6
], Danielis e al. [
7
]. Allen e al. [
8
] showed ha he use o u ban consolida ion cen e s
(UCC) is p esumed o p o ide mo e e icien dis ibu ion in an u ban a ea, and i can dec ease ene gy
use and en i onmen al impac . A UCC can be desc ibed as a logis ics acili y loca ed in ela i ely close
p oximi y o he geog aphic a ea ha i se es, and wi h he ange o e ms used o e e o he UCC
concep main including a public dis ibu ion depo , u ban ans-shipmen cen e , eigh pla o ms,
coope a i e deli e y sys em, u ban dis ibu ion cen e , consolida ion cen e (some imes speci ic, e.g.,
e ail, cons uc ion), pick-up d op-o loca ion, o si e logis ics suppo concep and so on [
9
]. UCCs a e
one o he mos equen ly implemen ed and s udied ci y logis ics ini ia i es, and acco ding o Lago io
Appl. Sci. 2020,10, 2564; doi:10.3390/app10072564 www.mdpi.com/jou nal/applsci
Appl. Sci. 2020,10, 2564 2 o 16
e al. [
10
] many case s udies ha e been p esen ed in li e a u e [
11
,
12
]. One o he mos e icien and
ypical ways o implemen goods consolida ion is o adop mul i-s age dis ibu ion sys ems, especially
wo-s age dis ibu ion sys em, whe e he deli e y om dis ibu ion cen e o cus ome s is managed
by ou ing and consolida ing he eigh h ough in e media e depo s called ans e s a ions [
13
].
The e o e, selec ion o he loca ions o he ans e s a ions and planning o he wo-phase dis ibu ion
ou es a e key p oblems in ci y logis ics sys em op imiza ion.
Wi h he de elopmen o he economy and he con inuous imp o emen o people’s li ing
s anda ds in China, demand o cold-chain p oduc s has also inc eased o a la ge ex en , which
p omo es apid de elopmen o he cold-chain logis ics. The cold chain has become an impo an pa
o he u ban dis ibu ion sys em.
The emainde o his pape is o ganised as ollows. Fi s ly, he loca ion- ou ing p oblem is
desc ibed in a li e a u e e iew. Then a wo-s age loca ion- ou ing model is cons uc ed and a
me aheu is ic algo i hm is implemen ed o sol e he model. The algo i hm is included as a hyb id
gene ic algo i hm [
14
]. Finally, he easibili y and alidi y o he model and algo i hm a e demons a ed
h ough a es example, and esul s o he s udy a e summa ised.
2. Li e a u e Re iew
The concep o he loca ion- ou ing p oblem (LRP) can be aced back o 1961. Von Bo en e
i s discussed he ela ionship be ween loca ion selec ion and anspo a ion cos in anspo a ion
p oblems [
15
]. LRP and i s a ian s ha e been s udied ex ensi ely in he pas . Pe l and Daskin p oposed
a mul i- ehicle and mul i- acili y wa ehouse loca ion- ou ing p oblem model wi h ehicle capaci y
cons ain s [
16
]. Li e al. [
17
] s udied a h ee- ie dis ibu ion sys em wi h supplie s, wa ehouses, and
mul iple geog aphically dispe sed e aile s, whe e e aile s could eplenish goods om wa ehouses
and supplie s. A model was es ablished o minimize he long- e m a e age cos wi hin he sys em
while mee ing he demand o each e aile . P odhon [
18
] pu o wa d a linea p og amming model o
mul i-plan pe iodic loca ion pa hs by easonably de ining a iables and designed a hyb id e olu iona y
algo i hm o sol e i . Xu [
19
] in oduced he damage cos o goods incu ed in ansi and es ablished a
loca ion- ou e op imisa ion model conside ing wo-s age anspo a ion cos , damage cos , and penal y
cos , and se ice le el. A Gene ic Algo i hm-Pa icle Swa m Op imiza ion algo i hm was designed o
sol e he p oblem. Howe e , he ene gy consump ion cos was no conside ed in he model. Song e
al. [
20
] s udied he LRP o a mul i-jou ney ehicle pa h, i.e., conside ing ha he ehicle uns mul iple
ou es wi hin he a el cons ain ime, and a h ee-s age heu is ic algo i hm was designed o sol e
he p oblem. Zhao e al. [
21
] build a he e ogeneous lee wo-echelon capaci a ed loca ion- ou ing
model o join deli e y in ci y logis ics. Wang e al. [
22
] p oposed a bi-objec i e model, and designed
an imp o ed algo i hm. The e ec i eness o he imp o ed algo i hm was demons a ed h ough a
compa ison. Koç e al. [
23
] es ablished a loca ion- ou ing p oblem model conside ing a he e ogeneous
lee and ime windows. A hyb id e olu iona y algo i hm combining mul iple heu is ics was designed
o sol e he p oblem. Leng e al. [
24
] conside ed mul iple condi ions in he egional low-ca bon
loca ion- ou ing p oblem, such as simul aneous pickup and deli e y, ime windows. Yu e al. [
25
] and
Zhao e al. [
26
] s udied loca ion- ou ing p oblem wi h simul aneous pick-up and deli e y. To sol e he
p oblem, a simula ed annealing (SA) heu is ic algo i hm was designed and a hype -heu is ic app oach
based on i e a ed local sea ch was p oposed, espec i ely.
Fo he cold-chain logis ics p oblems, Yang e al. [
27
] s udied a loca ion model o pe ishable
p oduc s in wo- ie dis ibu ion cen e s, in oduced he cos o goods damage in o he objec i e unc ion,
designed an imp o ed gene ic algo i hm o sol e he model, and demons a ed he e ec i eness o he
model and algo i hm h ough an example. Howe e , he dis ance s udied was calcula ed using he
adial shape o he loca ion and demand poin s. Zhao e al. [
28
] designed a sa is ac ion deg ee unc ion
acco ding o se ice ime windows and in oduced a minimum en elope clus e ing analysis me hod
and abu sea ch algo i hm o sol e he p oblem. Zheng e al. [
29
] cons uc ed loca ion in en o y
ou ing p oblem unde a demand en i onmen in a cold-chain logis ics ne wo k, and non-domina ed
Appl. Sci. 2020,10, 2564 3 o 16
so ing in oduced a mul i-objec i e gene ic algo i hm (GA). Wang e al. [
30
] s udied a cold-chain
logis ics dis ibu ion ne wo k conside ing ca bon oo p in , he model o minimum o al cos including
ca bon emission cos was cons uc ed, and a hyb id algo i hm was designed o sol e he model.
Mos o he exis ing LRP esea ch ocused on a gene al logis ics dis ibu ion ne wo k and
single-s age loca ion- ou ing p oblems. When i comes o wo-s age LRP, mos o he models assume
ha he anspo a ion pa h be ween he dis ibu ion cen e and he ans e s a ions is adial, i.e, he
ehicle only p o ides se ices o one ans e s a ion a a ime and hen e u ns o he dis ibu ion
cen e , and mos o he algo i hms a e in ol ed in single-s age op imisa ion o ans e s a ion selec ion
and pa hs wi hou conside ing he coupling and collabo a i e op imisa ion o he wo s ages. In iew o
his, we s udied a wo-s age cold-chain logis ics dis ibu ion ne wo k, including wo-s age op imisa ion
o ans e s a ion loca ions and ou ing. A wo-s age loca ion- ou ing model wi h he minimum o al
cos associa ed wi h he ha d ime window is cons uc ed, and di e en ypes o ehicles a e conside ed
o dis ibu ion asks. An in eg a ed app oach is employed o design he algo i hm o ensu e he
quali y o he solu ion. Finally, h ough a case s udy, we e i y he easibili y and e ec i eness o he
model and algo i hm.
3. Model Fo mula ion
3.1. P oblem Desc ip ion
The cold-chain logis ics p oblem conside ed in his s udy comp ises a cold-chain logis ics
dis ibu ion cen e , mul iple po en ial cold-chain logis ics ans e s a ions and cus ome poin s.
The dis ibu ion cen e need o deli e goods o he cus ome s h ough ans e s a ions wi hin a
speci ied ime window. In he sys em, he dis ibu ion cen e , ans e s a ions, and ou ing be ween
hem cons i u e he i s -s age ci y logis ics ne wo k. The ans e s a ions, cus ome s, and ou ing
be ween hem cons i u e he second-s age ci y logis ics ne wo k. As shown in Figu e 1, we op imise
loca ions o ans e s a ions and he wo-s age dis ibu ion ne wo k ehicle ou ing p oblem unde
he condi ion o minimum o al cos .
Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 o 17
Fo he cold-chain logis ics p oblems, Yang e al. [27] s udied a loca ion model o pe ishable 88
p oduc s in wo- ie dis ibu ion cen e s, in oduced he cos o goods damage in o he objec i e 89
unc ion, designed an imp o ed gene ic algo i hm o sol e he model, and demons a ed he 90
e ec i eness o he model and algo i hm h ough an example. Howe e , he dis ance s udied was 91
calcula ed using he adial shape o he loca ion and demand poin s. Zhao e al. [28] designed a 92
sa is ac ion deg ee unc ion acco ding o se ice ime windows and in oduced a minimum 93
en elope clus e ing analysis me hod and abu sea ch algo i hm o sol e he p oblem. Zheng e al. 94
[29] cons uc ed loca ion in en o y ou ing p oblem unde a demand en i onmen in a cold-chain 95
logis ics ne wo k, and non-domina ed so ing in oduced a mul i-objec i e gene ic algo i hm (GA). 96
Wang e al. [30] s udied a cold-chain logis ics dis ibu ion ne wo k conside ing ca bon oo p in , 97
he model o minimum o al cos including ca bon emission cos was cons uc ed, and a hyb id 98
algo i hm was designed o sol e he model. 99
Mos o he exis ing LRP esea ch ocused on a gene al logis ics dis ibu ion ne wo k and 100
single-s age loca ion- ou ing p oblems. When i comes o wo-s age LRP, mos o he models 101
assume ha he anspo a ion pa h be ween he dis ibu ion cen e and he ans e s a ions is 102
adial, i.e, he ehicle only p o ides se ices o one ans e s a ion a a ime and hen e u ns o he 103
dis ibu ion cen e , and mos o he algo i hms a e in ol ed in single-s age op imisa ion o ans e 104
s a ion selec ion and pa hs wi hou conside ing he coupling and collabo a i e op imisa ion o he 105
wo s ages. In iew o his, we s udied a wo-s age cold-chain logis ics dis ibu ion ne wo k, 106
including wo-s age op imisa ion o ans e s a ion loca ions and ou ing. A wo-s age 107
loca ion- ou ing model wi h he minimum o al cos associa ed wi h he ha d ime window is 108
cons uc ed, and di e en ypes o ehicles a e conside ed o dis ibu ion asks. An in eg a ed 109
app oach is employed o design he algo i hm o ensu e he quali y o he solu ion. Finally, h ough 110
a case s udy, we e i y he easibili y and e ec i eness o he model and algo i hm. 111
3. Model Fo mula ion 112
3.1. P oblem Desc ip ion 113
The cold-chain logis ics p oblem conside ed in his s udy comp ises a cold-chain logis ics 114
dis ibu ion cen e , mul iple po en ial cold-chain logis ics ans e s a ions and cus ome poin s. The 115
dis ibu ion cen e need o deli e goods o he cus ome s h ough ans e s a ions wi hin a 116
speci ied ime window. In he sys em, he dis ibu ion cen e , ans e s a ions, and ou ing be ween 117
hem cons i u e he i s -s age ci y logis ics ne wo k. The ans e s a ions, cus ome s, and ou ing 118
be ween hem cons i u e he second-s age ci y logis ics ne wo k. As shown in Figu e 1, we op imise 119
loca ions o ans e s a ions and he wo-s age dis ibu ion ne wo k ehicle ou ing p oblem unde 120
he condi ion o minimum o al cos . 121
122
Figu e 1. Schema ic map o loca ion- ou ing in wo-s age dis ibu ion. 123
Figu e 1. Schema ic map o loca ion- ou ing in wo-s age dis ibu ion.
3.2. P oblem Assump ions
To acili a e he s udy, he ollowing assump ions a e made:
(1) he geog aphical loca ion o dis ibu ion cen e , po en ial ans e s a ions, ime window, demand
o cus ome s a e known;
(2)
each cus ome can only be se ed by one deli e y ehicle;
(3)
he demand o a single cus ome is less han he ehicle capaci y.
(4)
a ic conges ion is no conside ed;
(5)
he ca go load o a ehicle should no exceed i s a ed load;
Appl. Sci. 2020,10, 2564 4 o 16
(6)
he uni ans e cos o each ans e s a ion is known and is a cons an ; empe a u e changes
and he i s s age-ca go losses a e no conside ed;
(7) ans e s a ions compe e wi h each o he , and en e p ises can choose di e en ans e s a ions o
p o ide se ices acco ding o hei own cos minimiza ion;
(8)
he capaci y o e ige a ed anspo ehicles o dis ibu ion cen e ( i s -s age ou e) is known,
and he demand o a ans e s a ion can be g ea e han he capaci y o a anspo ehicle, i.e.
he demand o he i s -s age dis ibu ion pa h can be spli ;
(9)
he capaci y o e ige a ed anspo ehicles (second-s age ou e) o ans e s a ions is known,
and can no be exceeded. Each ehicle will e u n o he s a ing poin a e comple ing asks;
(10)
ypes o ehicles a e di e en o deli e ies om dis ibu ion cen e and ans e s a ions, and
capaci ies o same ype o ehicles a e he same, and he e a e enough anspo ehicles o
he sys em.
3.3. Pa ame e and Va iables
To build he model, he ollowing pa ame e s and a iables a e de ined:
M: Se o candida e ans e s a ions and dis ibu ion cen e ;
M1: Se o candida e ans e s a ions;
Ng: Se o cus ome s assigned o ans e s a ion g, and ans e s a ion g;
K: Se o e ige a ed ehicles in he dis ibu ion cen e ;
Lg: Se o e ige a ed ehicles in he g ans e s a ion;
dij: Dis ance be ween ipoin and jpoin ;
1: A e age speed o e ige a ed ehicles in he i s -s age ou e;
2: A e age speed o e ige a ed ehicles in he second-s age ou e;
sl
i: Se ice imes o ehicle l o he i- h cus ome (o ans e s a ion);
c1: Use and consump ion cos o ehicles pe kilome e in he i s -s age ou e;
c2: Use and consump ion cos o ehicles pe kilome e in he second-s age ou e;
c0
1: Cos o he d i e in he i s -s age ou e;
c0
2: Cos o he d i e in he second-s age ou e;
uk
j
: Remaining ca go olume o he e ige a ed ehicle kwhen a i ing a he ans e s a ion o he
cus ome poin j;
bk
j
: Remaining ca go olume o he e ige a ed ehicle kwhen lea ing he ans e s a ion o cus ome
poin j;
P1: P ice o uni commodi y;
θ1: Spoilage a e o p oduc in he anspo a ion p ocess;
θ2: Spoilage a e o p oduc in unloading p ocess;
W1: Fuel consumed in ope a ing he e ige a o du ing anspo a ion;
W2: Fuel consumed in e ige a o ope a ion du ing unloading;
θ3: Fuel p ice pe uni weigh ;
l
i: Time when he e ige a ed ehicle l eaches poin i;
l
ij: Time equi ed by a e ige a ed ehicle l o a el be ween wo poin s iand j;
l
0g: Depa u e ime o e ige a ed ehicle l om he ans e s a ion g;
λ1: T ansi cos pe uni o ime;
Wg: Amoun o goods ans e ed a he g ans e s a ion;
3: T ans e poin unloading p ocessing speed;
qi: Demand o ans e s a ion i;
q0
i: Demand o cus ome i;
Q1: Vehicle capaci y in he i s -s age ou e;
Appl. Sci. 2020,10, 2564 5 o 16
Q2: Vehicle capaci y in he second-s age ou e;
Xk
ij
is a 0–1 a iable: when
Xk
ij
=1, he ehicle kpasses he oad be ween ans e s a ion(o dis ibu ion
cen e ) iand ans e s a ion(o dis ibu ion cen e ) j; o he wise, Xk
ij =0;
xk
jis a 0–1 a iable: when xk
j=1, he ehicle kse ices o cus ome i; o he wise, xk
j=0;
xl
ijg
is a 0–1 a iable: when
xl
ijg
=1, he ehicle lo ans e s a ion gpasses he oad be ween cus ome
(o ans e s a ion ) iand cus ome (o ans e s a ion) j; o he wise, Xl
ijg =0;
Zgis a 0–1 a iable: when Zg=1, he ans e s a ion gis used; o he wise, Zg=0;
yl
ig
is a 0–1 a iable: when
yl
ig
=1, he ehicle lo ans e s a ion g p o ides se ice o cus ome i;
o he wise, yl
ig =0.
3.4. Model De elopmen
The wo-s age dis ibu ion loca ion- ou ing wi h ha d ime window o ci y cold-chain logis ics
model cons uc ed in his pape akes he minimum o al cos as he objec i e unc ion. In consequence,
he sub-cos should be analyzed i s ly. Then he o al cos o he wo-s age dis ibu ion loca ion- ou ing
is ob ained by he a ious sub-cos s.
3.4.1. Objec i e Func ion Analysis o Model
(1)
T anspo a ion Cos
The anspo a ion cos mainly includes ehicle use cos , uel used in he p ocess o anspo a ion,
d i e ’s cos , and o he ac o s. Fo he con enience o esea ch, he anspo a ion cos is conside ed
in wo pa s, e e ing o he use and consump ion cos o ehicles pe kilome e and d i e ’s cos . The
anspo a ion cos o a e ige a ed ehicle in he i s -s age and second-s age ou es can be exp essed
as ollows:
C1=c1X
k∈K
X
i,j∈M
Xk
ijdij+c0
1X
k∈K
X
i,j∈M
(Xk
ij
dij
1
+sk
j)(1)
C0
1=c2X
g∈M1
X
l∈Lg
X
i,j∈Ng
Zgdijxl
ijg +c0
2X
g∈M1
X
l∈Lg
X
i,j∈Ng
Zg(dij
2
xl
ijg +sl
j)(2)
(2)
Damage Cos
Commodi ies in he cold-chain dis ibu ion a e easily spoiled and he e o e need o be kep in an
app op ia e low- empe a u e en i onmen . The quali y o pe ishable goods g adually declines o can
lose alue wi h ime. When he quali y o he p oduc declines o a ce ain ex en , spoilage cos will
be incu ed. The cos o damage is di ided in o wo pa s: he damage cos o goods accumula ed
o e ime in he p ocess o anspo a ion and he damage cos incu ed when opening he doo in he
p ocess o unloading. Because he i s -s age deli e y is usually ca ied ou in an enclosed en i onmen ,
spoilage cos will be minimal. We only conside ed he damage cos in he second-s age dis ibu ion.
Hence, he o al damage cos can be exp essed as:
C0
2=P1X
g∈M1
X
l∈Lg
X
j∈Ng
Zgyl
jg(1−e−θ1( l
ij− l
0g))ul
j+P1X
g∈M1
X
l∈Lg
X
j∈Ng
Zgyl
jg(1−e−θ2sl
j)bl
j(3)
The i s pa ep esen s he cos o ca go damage in he anspo a ion p ocess, and he second
pa ep esen s he cos o ca go damage in he unloading p ocess.
(3)
Re ige a ion Cos
The cos o s o age associa ed wi h main aining he empe a u e and humidi y inside he ca iage is
called he e ige a ion cos . The e ige a ion me hods employed in e ige a ed ehicles on he ma ke

Appl. Sci. 2020,10, 2564 6 o 16
oday mainly include liquid ni ogen e ige a ion, mechanical e ige a ion, d y ice e ige a ion, and
cold pla e e ige a ion. The e ige a ion me hod conside ed in his s udy is mechanical e ige a ion,
and he cos gene a ed in he e ige a ion p ocess is ela ed o uel consump ion, ime, and weigh o
goods. Li e a u e [31,32] p oposed a o mula o calcula ing he uel consump ion: W=ωεPe
ξ∗10−3.
The e ige a ion cos includes he cos associa ed wi h he ene gy consump ion o he ehicle
in main aining a low- empe a u e en i onmen du ing deli e y and he cos o addi ional ene gy
supplied o he e ige a ion sys em du ing he unloading p ocess.
The e ige a ion cos in he i s -s age and second-s age ou e can be exp essed as ollows:
C3=θ3W1X
k∈K
X
i,j∈M
dijxk
ijuk
j
1
+θ3W2X
k∈K
X
j∈M
xk
jbk
jsk
j(4)
C0
3=θ3W1X
g∈M1
X
l∈Lg
X
i,j∈Ng
Zg
dijxl
ijgul
j
2
+θ3W2X
g∈M1
X
l∈Lg
X
j∈Ng
Zgsl
jbl
j(5)
whe e,
W
is he uel consump ion (g/h);
ω
is he powe u iliza ion coe icien o he e ige a o ;
ε
is he uel consump ion a e (g/kW
·
h);
Pe
is he e ec i e powe o he e ige a o (kW);
ξ
is he
speci ic g a i y o he uel;
W1
is he uel oil consump ion du ing anspo a ion; and
W2
is he uel
consump ion du ing unloading. The i s pa ep esen s he e ige a ion cos o e ige a ed ehicles
du ing anspo a ion, and he second pa ep esen s he e ige a ion cos o he e ige a ed ehicles
du ing unloading.
(4)
Penal y Cos
In an ac ual dis ibu ion p ocess, he dis ibu ion ehicles may no a i e on ime o a ious
easons, his will incu a penal y cos . The concep o a ime window is in oduced. As he iming o
i s -s age dis ibu ion is lexible, we only conside he ime window equi emen s o he cus ome s in
he second-s age dis ibu ion. The ime window is di ided in o a ha d ime window and a so ime
window. Conside ing he cha ac e is ics o cold-chain dis ibu ion, we calcula ed he penal y cos s
associa ed wi h he ha d ime window. Assuming ha he ea lies se ice ime allowed by cus ome i
is ET
i
, he la es se ice ime allowed by cus ome iis LT
i
, he se ice ime window equi ed by he
cus ome iis [ETi,LTi]. The penal y unc ion equa ion can be exp essed as ollows [33]:
C4=P( ) = 








M <ET
0ET ≤ ≤LT
M >LT
whe e Mis in ini e. is he ime when he ehicle a i es a he cus ome . [ET,LT] is he se ice ime
window equi ed by he cus ome .
(5)
T ans e Cos
The cos a he ans e s a ion mainly consis s o he ime cos incu ed in he ans e o goods
om la ge e ige a ed ehicles o small e ige a ed ehicles. The e o e, he ans e cos can be
exp essed as ollows:
C5=λ1X
g∈M1
Zg
Wg
3
(6)
Appl. Sci. 2020,10, 2564 7 o 16
3.4.2. Model Se ing
Based on he analysis o sub-cos in Sec ion 3.4.1, a wo-s age dis ibu ion loca ion- ou ing model
wi h he ha d ime window cons ains o ci y cold-chain logis ics is es ablished as ollows:
Minz1=C1+C0
1+C0
2+C3+C0
3+C4+C5(7)
Subjec o:
X
i∈M1
Xk
iqi≤Q1k∈K(8)
X
i∈Ng
q0
iyl
ig ≤Q2g∈M1,l∈Lg(9)
X
j∈M
Xk
ij =Xk
ii∈M;k∈K(10)
X
i∈M
Xk
ij =Xk
j,j∈M;k∈K(11)
X
i∈Ng
X
g∈M1
xl
ijg =yl
jg j∈Ng,l∈Lg(12)
X
j∈Ng
X
g∈M1
xl
ijg =yl
ig i∈Ng,l∈Lg(13)
X
g∈M1
X
l∈Lg
yl
jg =1j∈Ng(14)
X
j∈Ng
xl
ijg =X
j∈Ng
xl
jig ≤1i=g∈M1;l∈Lg(15)
X
j∈M1
Xk
ij =X
j∈M1
xk
ji ≤1i=0; k∈K(16)
X
i∈Ng
q0
iyig ≤Zgqgg∈M1(17)
ETi≤ l
i≤LTi(18)
The objec i e unc ion o he model is shown in (7). Cons ain (8) shows ha a ehicle canno
exceed i s maximum load in he i s le el dis ibu ion. Cons ain (9) shows ha ehicle can no exceed
i s maximum load in he second-le el dis ibu ion. The i s s age o dis ibu ion low balance is shown
in (10) and (11). The second-s age o dis ibu ion low balance is shown in (12) and (13). Cons ain
(14) shows ha he e is only one e ige a ed ehicle p o iding a deli e y se ice o one cus ome .
As pe cons ain s (15) and (16), he e ige a ed ehicles s a ing om a ans e s a ion mus e u n o
he same a e se ing he cus ome , and he e ige a ed ehicles s a ing om he dis ibu ion cen e
mus e u n o he dis ibu ion cen e a e se ing ans e s a ions. The o al cus ome equi emen s
assigned o a ans e s a ion mus be lowe han o equal o he s o age capaci y o he ans e s a ion,
which is imposed by (17). Cons ain (18) ep esen s he ime window o cus ome i.
4. Algo i hm Design
Two-s age dis ibu ion loca ion- ou ing belong o he NP-Ha d p oblem, so we use heu is ic
algo i hms o sol e his p oblem. A hyb id gene ic algo i hm is p oposed in he pape combining
heu is ic ules and dis ance clus e ing. The low cha o he algo i hm is shown in Figu e 2.
Appl. Sci. 2020,10, 2564 8 o 16
Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 o 17
Decoded ch omosome
T ans e s a ion capaci y;
ehicle capaci y;
Whe he he e mina ion
c i e ion is me ?
End: S op i e a ion and
ge he bes popula ion
I he gene a ion o he new popula ion
eaches he p e-se e olu iona y
i
d
h
ii
id
Fi ness e alua ion: Fi=1/Zi ;Roule e gambling
Two-poin c osso e ;
l
Gene a ing easible
ini ial popula ion a
andom
S a ing
Ch omosome coding
Each ch omosome
consis s o h ee
Mu a ion
i
C osso e
i
In e change mu a ion;
In e sion mu a ion;
Selec ion ope a ion
4. Algo i hm Design 261
Two-s age dis ibu ion loca ion- ou ing belong o he NP-Ha d p oblem, so we use heu is ic 262
algo i hms o sol e his p oblem. A hyb id gene ic algo i hm is p oposed in he pape combining 263
heu is ic ules and dis ance clus e ing. The low cha o he algo i hm is shown in Figu e 2. 264
Figu e 2. Flow cha o hyb id gene ic algo i hm. 265
4.1. Ch omosome Coding 266
Each ch omosome consis s o h ee sub-s ings: Sub-s ing 1 in ol es encoding leng h J. The 267
in ege o gene 1 − J is a non- epe i i e sequence, ep esen ing he p io i y selec ion o de and 268
deli e y o de o he ans e s a ion. Sub-s ing 2 is encoded as an in ege wi h leng h N and gene 1 269
− N, ep esen ing he dis ibu ion o de o he demand poin s. Sub-s ing 3 is encoded as an in ege 270
code o leng h 1, indica ing he numbe o ans e s a ion selec ed. Fo example, when J = 5 and N = 271
6, namely, he e a e 5 ans e s a ion and 6 cus ome poin , o he ch omosome shown in Figu e 3, 272
he p io i y o he ansi ion o be selec ed is 5, 4, 2, 1, 3. The dis ibu ion pa h o he demand poin s 273
is 2-3-4-1-5-6; 2 indica es ha he i s wo bi s o he i s laye o coding a e selec ed o se up a 274
ans e s a ion. Mo eo e , 5, 4 coded in he i s laye is he o de o dis ibu ion o he ans e 275
s a ion. 276
N
Y
Figu e 2. Flow cha o hyb id gene ic algo i hm.
4.1. Ch omosome Coding
Each ch omosome consis s o h ee sub-s ings: Sub-s ing 1 in ol es encoding leng h J. The in ege
o gene 1
−
Jis a non- epe i i e sequence, ep esen ing he p io i y selec ion o de and deli e y o de
o he ans e s a ion. Sub-s ing 2 is encoded as an in ege wi h leng h Nand gene 1
−
N, ep esen ing
he dis ibu ion o de o he demand poin s. Sub-s ing 3 is encoded as an in ege code o leng h 1,
indica ing he numbe o ans e s a ion selec ed. Fo example, when J=5 and N=6, namely, he e
a e 5 ans e s a ion and 6 cus ome poin , o he ch omosome shown in Figu e 3, he p io i y o
he ansi ion o be selec ed is 5, 4, 2, 1, 3. The dis ibu ion pa h o he demand poin s is 2-3-4-1-5-6;
2 indica es ha he i s wo bi s o he i s laye o coding a e selec ed o se up a ans e s a ion.
Mo eo e , 5, 4 coded in he i s laye is he o de o dis ibu ion o he ans e s a ion.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 o 17

3
21
2|6-5-1-4-3-2|3-1-2-4-5
subs ing
subs ingsubs ing
 277
Figu e 3. Ch omosome coding example. 278
4.2. Gene a e he Ini ial Popula ion a Random 279
Gene ic algo i hm s a s om he popula ion o he p oblem solu ions, so i is necessa y o 280
gene a e an ini ial popula ion as he s a ing poin o e olu ion. Acco ding o he encoding me hod, 281
N ch omosomes a e andomly gene a ed. 282
4.3. C osso e Ope a ion 283
To main ain he di e si y o he popula ion, we conduc ed c osso e ope a ions o each 284
sub-s ing in he ch omosome. Two-poin c osso e ope a ion is ca ied ou in sub-s ing 1. Cycle 285
c osso e ope a ion is chosen o pe o m on sub-s ing 2. Two-poin c osso e ope a ion is used in 286
sub-s ing 3. The de ailed desc ip ion o c osso e ope a ion is as ollows: 287
(1) C osso e ope a ion o sub-s ing 1 288
Two-poin c osso e : i s ly, wo ch omosomes a e andomly selec ed as he pa en , and 289
gene a ing wo andom na u al numbe s 1 and 2. Secondly, he gene agmen s be ween 1 and 2 290
o wo pa en ch omosomes a e exchanged, wo o sp ing ch omosomes a e ob ained. Finally, he 291
ch omosomes o he wo o sp ing a e modi ied so ha no con lic occu s. Fo example, pa en 292
ch omosome 1: 1 3 2 5 4, pa en ch omosome 2: 1 2 4 5 3, andom numbe s 1 = 2, 2 = 4. The 293
o sp ing ch omosomes a e c ossing a e o sp ing ch omosome 1: 1 2 4 5 4 and o sp ing 294
ch omosome 2: 1 3 2 5 3. Finally, wo new o sp ing ch omosome a e ob ained by epai ing, new 295
o sp ing ch omosome 1: 1 2 4 5 3, new o sp ing ch omosome 2: 1 3 2 5 4. 296
(2) C osso e ope a ion o sub-s ing 2 297
Cycle c osso e : i s ly, a cycle will be ound acco ding o he co esponding gene posi ion o 298
he pa en ch omosome. Secondly, he ci cula ing gene is eplica ed o he o sp ing. Thi dly, he 299
emaining genes a e iden i ied o he o sp ing, and he emaining genes ou sides he pa en 300
ch omosome cycle a e used o ill wi h he o iginal o sp ing. Finally, o ming new descendan s. Fo 301
example : Fi s ly, pa en ch omosome 1 : 2 7 5 6 4 8 9 1 3, pa en ch omosome 2 : 4 3 6 8 9 7 1 2 5, 302
ind cycle 1 : 2 4 9 1 2, cycle 2：4 2 1 9 4. Secondly, o sp ing ch omosome 1: 2 * * * 4 * 9 1 * , o sp ing 303
ch omosome 2 : 4 * * * 9 * 1 2 *. Thi dly, pa en 1 emaining ch omosome: * 3 6 8 * 7 * * 5, pa en 2 304
emaining ch omosome: * 7 5 6 * 8 * * 3. Finally, new o sp ing ch omosome 1: 2 3 6 8 4 7 9 1 5; new 305
o sp ing ch omosome 2: 4 7 5 6 9 8 1 2 3. 306
(3) C osso e ope a ion o sub-s ing 3 307
Two-poin c osso e : andom selec ion o wo ch omosomes as pa en s and wo o sp ing 308
ch omosomes a e ob ained by di ec exchange o wo pa en ch omosomes. Fo example, selec wo 309
pa en ch omosomes 1 and 2, he ch omosomes o he c ossed o sp ing a e 2 and 1. 310
4.4. Mu a ion Ope a ion 311
To main ain he di e si y o he popula ion, we conduc ed mu a ion ope a ion o each 312
sub-s ing in he ch omosome. In e change mu a ion ope a ion is ca ied ou in sub-s ing 1. 313
In e sion mu a ion ope a ion is chosen o pe o m on sub-s ing 2. Single-poin mu a ion ope a ion 314
is used in sub-s ing 3. The de ailed desc ip ion o mu a ion ope a ion is as ollows: 315
(1) Mu a ion ope a ion o sub-s ing 1 316
Figu e 3. Ch omosome coding example.
Appl. Sci. 2020,10, 2564 9 o 16
4.2. Gene a e he Ini ial Popula ion a Random
Gene ic algo i hm s a s om he popula ion o he p oblem solu ions, so i is necessa y o
gene a e an ini ial popula ion as he s a ing poin o e olu ion. Acco ding o he encoding me hod, N
ch omosomes a e andomly gene a ed.
4.3. C osso e Ope a ion
To main ain he di e si y o he popula ion, we conduc ed c osso e ope a ions o each sub-s ing
in he ch omosome. Two-poin c osso e ope a ion is ca ied ou in sub-s ing 1. Cycle c osso e
ope a ion is chosen o pe o m on sub-s ing 2. Two-poin c osso e ope a ion is used in sub-s ing 3.
The de ailed desc ip ion o c osso e ope a ion is as ollows:
(1)
C osso e ope a ion o sub-s ing 1
Two-poin c osso e : i s ly, wo ch omosomes a e andomly selec ed as he pa en , and gene a ing
wo andom na u al numbe s 1 and 2. Secondly, he gene agmen s be ween 1 and 2 o wo pa en
ch omosomes a e exchanged, wo o sp ing ch omosomes a e ob ained. Finally, he ch omosomes o
he wo o sp ing a e modi ied so ha no con lic occu s. Fo example, pa en ch omosome 1: 1 3 2 5 4,
pa en ch omosome 2: 1 2 4 5 3, andom numbe s 1 =2, 2 =4. The o sp ing ch omosomes a e
c ossing a e o sp ing ch omosome 1: 1 2 4 5 4 and o sp ing ch omosome 2: 1 3 2 5 3. Finally, wo
new o sp ing ch omosome a e ob ained by epai ing, new o sp ing ch omosome 1: 1 2 4 5 3, new
o sp ing ch omosome 2: 1 3 2 5 4.
(2)
C osso e ope a ion o sub-s ing 2
Cycle c osso e : i s ly, a cycle will be ound acco ding o he co esponding gene posi ion o he
pa en ch omosome. Secondly, he ci cula ing gene is eplica ed o he o sp ing. Thi dly, he emaining
genes a e iden i ied o he o sp ing, and he emaining genes ou sides he pa en ch omosome cycle
a e used o ill wi h he o iginal o sp ing. Finally, o ming new descendan s. Fo example: Fi s ly,
pa en ch omosome 1: 2 7 5 6 4 8 9 1 3, pa en ch omosome 2: 4 3 6 8 9 7 1 2 5, ind cycle 1: 2 4 9 1 2, cycle
2: 4 2 1 9 4. Secondly, o sp ing ch omosome 1: 2 * * * 4 * 9 1 *, o sp ing ch omosome 2: 4 * * * 9 * 1 2 *.
Thi dly, pa en 1 emaining ch omosome: * 3 6 8 * 7 * * 5, pa en 2 emaining ch omosome: * 7 5 6 * 8 * *
3. Finally, new o sp ing ch omosome 1: 2 3 6 8 4 7 9 1 5; new o sp ing ch omosome 2: 4 7 5 6 9 8 1 2 3.
(3)
C osso e ope a ion o sub-s ing 3
Two-poin c osso e : andom selec ion o wo ch omosomes as pa en s and wo o sp ing
ch omosomes a e ob ained by di ec exchange o wo pa en ch omosomes. Fo example, selec wo
pa en ch omosomes 1 and 2, he ch omosomes o he c ossed o sp ing a e 2 and 1.
4.4. Mu a ion Ope a ion
To main ain he di e si y o he popula ion, we conduc ed mu a ion ope a ion o each sub-s ing
in he ch omosome. In e change mu a ion ope a ion is ca ied ou in sub-s ing 1. In e sion mu a ion
ope a ion is chosen o pe o m on sub-s ing 2. Single-poin mu a ion ope a ion is used in sub-s ing 3.
The de ailed desc ip ion o mu a ion ope a ion is as ollows:
(1)
Mu a ion ope a ion o sub-s ing 1
In e change mu a ion: Two andom poin s a e selec ed in he encoding s ing, and hei posi ions
a e swapped. Fo example, selec exchange poin s 3 and 7 om ch omosomes 1 4
3|
926
7|
5 8, he
ch omosome ob ained a e exchange mu a ion is 1 4 7|9263|5 8.
(2)
Mu a ion ope a ion o sub-s ing 2
Appl. Sci. 2020,10, 2564 16 o 16
32. Wang, G. Op imiza ion Resea ch o Common Dis ibu ion Rou ing P oblem o Cold Chain Logis ics Based
on Join Dis ibu ion Mode. Mas e ’s Thesis, Beijing Jiao ong Uni e si y, Beijing, China, 2018.
33.
Shi, C.C.; Wang, X.; Ge, X.L. Resea ch on ehicle scheduling p oblem o mul i-dis ibu ion cen e s wi h ime
window. Compu . Eng. Appl. 2009,45, 21–24.
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2020 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access
a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion
(CC BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).