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C EngineeringFacutly,ShenyangJianzhuUniverstiy,Shenyang,Liaoning,110168,China
: s d r o w y e
K Vehicle rouitng problem wtih itme windows, Strate yg hybrid algortihm, Merge sort, t
e s d e t a n i m o d n o
n , Dfiferenita levoluitonalgortihm.
.t c a r t s b
A Fort hecharacteristicsoft heVehicleRoutingProblemwithTimeWindows(VRPTW)
�a i
t l u
m -objectivehybrid Differentia lEvolution algorithm forVRPTWisproposed .Firstly ,througha d n a y g e t a r t s n o i t a t u m 1 / d n a r / E D f o e c i o h c f o y t i l i b a b o r p e h t s l o r t n o c r e t e m a r a p g n i y r a v y l r a e n i l
s s o r c a , y l d n o c e S . y g e t a r t s n o i t a t u m 1 / t s e b / E
D over operation based on merge sor tis designed . o
t e r a P y o l p m e s n o i t a r e p o n o i t c e l e s , y l l a n i
F -dominatedconceptsandr ingr ulest or anki ndividualsand n
o n t u p t u
o -dominated solutions .The experimenta l results compared with single strategy DE s
m h t i r o g l a C B A d n a m h t i r o g l
a showt hatt heproposedalgorithmi seffectivei nsolvingt heVRPTW.
Introduciton
n o i t a t i m i l a s i W T P R V e h T s c i t s i g o l f o e r o c t n a t r o p m i t s o m e h t s a ) P R V ( m e l b o r P g n i t u o R e l c i h e V
P N n a s i d n a P R V e h t o t d e d d a w o d n i w e m i t e h t f
o -hard problem .DE(Differentia lEvolution) ,
e u q i n h c e t h c r a e s w e n a s a m h t i r o g l
a i t has many advantages such as high efficiency ,fas t s
r e t e m a r a p s s e l , m u m i t p o l a c o l o t n i g n i l l a f f o e c n a h c s s e l , e c n e g r e v n o
c a ndsoon.
E o a
C rbao[1]andothersapplyt hei mproveddiscrete-typedifferentia levolutionalgorithmt osolve
P R
V . WangJun[2]initializedrandomstowagetoconstruc tanew mutationoperationand crossover
n o i t a r e p
o andcombinedwitht abusearchalgorithm .Fort hefirstt ime ,Changandothers[3]usedDE
i t l u m n
o -objectiveoptimizationproblemswith goodresults, heshared thefitnessvaluetoimprove .t
e s n o i t u l o s l a m i t p o e h t f o y t i l a u q e h t
i t l u m e h t n o d e s a
B -objectiveVRPTWfeatures,t hispaperconsiderst het woobjectivesofvehicle e
c n e r e f f i d e h t t a g n i m i A . e g n a r g n i v i r d d n a r e b m u
n fo explorationanddevelopmen tabilitybetween .
d e s o p o r p s i m h t i r o g l a E D d i r b y h a , y g e t a r t s 1 / d n a r / E D d n a y g e t a r t s 1 / t s e b / E
D The experimenta l
d n a d e e p s e c n e g r e v n o c f o s m r e t n i s e g a t n a v d a s u o i v b o s a h m h t i r o g l a E D d i r b y h e h t t a h t w o h s s t l u s e r
o i t u l o s f o y t i l a u
q n .
m e l b o r
P Descrip itonandEstablsihaMathema itca lModel
N : s w o l l o f s a P R V f o s i s a b e h t n o d e b i r c s e d e b n a c W T P R
V ={0,1,2,…,n} in undirected graph (
=
G N ), E representsase tofnodes-Each customerhas itsfixed geographica lcoordinatesand time r
t s e r w o d n i
w ictions ,andeachcustomercanonlybedeliveredonceacar I. nt heserviceprocess,t he e
c i v r e s y r e v i l e d w o d n i w e m i t e h t f o s n o i s i v o r p e h t h t i w e c n a d r o c c a n i e b t s u m e l c i h e
v . With these
f o e g a e l i m l a t o t e h t f o n o i t a z i m i t p o e h t , t i m i l e h t s a s n o i t i d n o c o w
t vehicles and the number of
s e l c i h e
v a retwoobjectives. Incombinationwitht healgorithm,t heoptima ldrivingroutei sselected .
e l b i s s o p e r a e c n a t s i d g n i v i r d t s e t r o h s e h t d n a r a c t s a e l e h t t a h t o s
n o i t c n u f e v i t c e j b o e h
T :
. )(1
. (2)
s t n i a r t s n o
C :
(4)
. )(5
. (6)
. )(7
. )(8
(9)
. (10)
. (11)
h t n
I eaboveformula ,(1)(2)representst woobjectivefunctions ,(1)representst heshortes tvehicle e h t n i r e m o t s u c e h t f o t h g i e w l a t o t e h t s t n e s e r p e r ) 3 ( ; s e l c i h e v f o r e b m u n t s a e l e h t s t n e s e r p e r ) 2 ( , h t a p
x e t o n n a c k e l c i h e v e h t f o e c i v r e s y r e v i l e d d e t e l p m o
c ceedt hemaximuml oadoft hevehicle ;(4)(5) e
m o t s u c e n o o t e c i v r e s r e v i l e d y l n o n a c e l c i h e v h c a e t a h t t n e s e r p e r ) 6 ( d n
a r ,andt hevehicler eturnst o
d e r e v i l e d e r a s r e m o t s u c e h t l l a r e t f a e g a r a g e h
t ;constrain t(7) represents the elimination error ; r
t s n o
c ain t(8)(9)representsthetimewindowlimit :timewindowstar ttimeandend time ;(10)(11) .
s e l c i h e v f o r e b m u n m u m i x a m e h t d n a s e l c i h e v e h t n e e w t e b p i h s n o i t a l e r e h t s t n e s e r p e r
it l u
M -objecitveHybridDifferenceEvolu itonaryAlgortihm d
n a g n i d o
C Decoding
This paper u s se a natura lnumber encoding based on vehicle path order .One chromosome is s
a d e s s e r p x
e (y1,y2,y3,…,yi,…,yn) .Oneoft hegenesyiinachromosomer epresentsoneoft hecustomer
, s t n i o
p and yi is a natura lnumber tha tdoes no toverlap in the interva l[1 ,n]. In fact ,these n
o
n -repetitivenatura lnumbersrepresen tthecustomerpointstha tthevehiclepassesthrough to the r
e m o t s u c e h t o t e c i v r e s y r e v i l e
d .
a t e g o t e l b a e b t o n y a m d o h t e m g n i d o c n e e v o b a e h t g n i s u e s u a c e
B feasiblesolutiont oaproblem ,a e h T . h t a p e l c i h e v e l b a i v a o t n i e m o s o m o r h c e h t n i s e n e g e h t t r e v n o c o t d e s o p o r p s i d o h t e m g n i d o c e d
e n e g h c a e s e c a l p d n a ) e g a r a g e h t r o f 0 ( e g a r a g e h t m o r f s t r a t s e l c i h e v e h t t a h t s i g n i d o c e d f o a e d i n i a m
e d e l c i h e v e h t f o h t a p e h t n i ) t n i o p t n e i l c
( liveryin turn accordingto the orderof each genein the l a t o t r e m o t s u c e h t r o t i m i l e m i t f o w o d n i w e h t s k a e r b d e t r e s n i t n i o p r e m o t s u c y n a f I . e m o s o m o r h c
a y o l p e d e r d n a e g a r a g e h t o t n r u t e r t s u m e l c i h e v e h t , t h g i e w e l c i h e v m u m i x a m e h t s d e e c x e t h g i e w
v w e
n ehiclewitht hesamel oadcapacityt oservet hecustomerunti lal lcustomerspointsarei nserted .
h t a p e h t o t n
i Ifthechromosomehasarank of5 87 94 62 1 3and atota lof4 vehicles ,thepath 0
: s i g n i d o c e
d →5→8→7→0;0→9→4→ ;00 →6→2→0;0→1→3→ .0
a t u
M itonOpera iton
e h t n i E D l a n o i t i d a r t e h t m o r f t n e r e f f i d s i h c i h w , g n i d o c n e r e b m u n l a r u t a n e s u e w , r e p a p s i h t n I
: s w o l l o f s a s i r e p a p s i h t n i n o i t a r e p o n o i t a t u m c i f i c e p s e h t . n o i t a r e p o n o i t a t u m f o d o h t e m
)
1 ThemutationstrategyDE/rand/1i sgiven:
(12)
, a l u m r o f e v o b a e h t n
(13)
r dan ( )representsarandomnumberbetween[0,1] ,Ins()representsacustomi nser toperation .The d
e t a r e n e g s i l a u d i v i d n i t e g r a t d e b r u t r e p a t a h t s w o h s a l u m r o f e v o b
a . Specifically, is
: s w o l l o f s a d e s s e r p x
e randomly selec tagroup of gene segmentsin thetarge tindividual a nd n
i n o i t i s o p y n a o t n i y l m o d n a r t n e m g e s e n e g s i h t t r e s n
i . Finally find thesamecustomerin the n
i n o i t a r e p o e t e l e d e h t e t u c e x e d n a t n e m g e s e n e
g .
f o t r a p t x e n e h
T , the operation method is consisten t with tha t of .
)
2 ThemutationstrategyDE/best/1i sgiven
( 4 1 ) ,
m e h t g n o m
A is the bes tindividua lin this generation .The specific method selected is t
a h
t ischosenby et nh n -o dominatedlayerthat si constructedbythering ruleandthepareto e
u l a v e c n a t s i d d e d w o r c d n a r e i t n o r
f s .Thespecificoperationmethodi sconsisten twitht heoperation f
o d o h t e
m .
Muta itonStrategyContro lParameters
m s i y g e t a r t s 1 / d n a r / E
D u ch prominen tin theability of globa lsearch ,which helps to maintain the t n e m p o l e v e d l a c o l e h t n i t n e n i m o r p e r o m s i 1 / t s e b / E D f o y g e t a r t s e h T . s n o i t a l u p o p f o y t i s r e v i d
t n i s e i l e g a t n a v d a s t I . y t i l i b
a hatt hebesti ndividuali nt hepopulationi susedast heguide. Thesearch t n e r e f f i d n o s t c e f f e t n e r e f f i d e v a h s e i g e t a r t s o w t e h t t a h t t c a f e h t n o d e s a B . h g i h s i s s e n s u o l u c i t e m
e r u t x i m e h t l o r t n o c o t d e s o p o r p s i r e t e m a r a p e g n a h c r a e n i l a , s e z i s n o i t a l u p o
p ofthetwostrategies .
: s w o l l o f s a s i a l u m r o f e h T
( 51 )
e r e h
W are constants , represents the curren t evolution algebra , m
i x a m e h t s t n e s e r p e
r um evolution algebra .thestepsto be applied to the hybrid algorithm are as :
s w o l l o
f first ,generatearandomnumberrand()between(0,1);i ft herandomnumber , ;
y g e t a r t s 1 / d n a r / E D e s o o h
c otherwiseselectt heDE/best/1strategyformutationoperation.
s s o r
C overOpera iton
t r o S e g r e M n o d e s a b n o i t a r e p o r e v o s s o r c a s e s o p o r p r e p a p s i h t n
I [4].Accordingt ot hei deaofmerge
: s w o l l o f s a e r a s p e t s c i f i c e p s e h T , t r o s
)
1 Randomlygeneratearandomnumberrand()between(0,1) ,se tjdtobearandomnumberi nt he e
z i s n o i t a l u p o p e h t t e s d n a , s t n e i l c f o r e b m u n e h t s t n e s e r p e r D , ) D , .. . 2 , 1 ( f o e g n a
r NP=1.
)
2 Judge the size of rand () and crossover probability Cr for each gene of the mutation l
a u d i v i d n
i . Ifrand ()<Cr ro j=j d ,thegene ofthevarian tindividuals ispu tintotheempty s
l a u d i v i d n i l a t n e m i r e p x e e h t f o s n o i t i s o p e n e
g in turn .Otherwise ,the gene of the targe t l
a u d i v i d n
i isputi ntot heemptygeneposition of int urn. )
3 Ift hegene isselectedt obeplacedi nt heemptygeneposition of ,thensett hesamegenei n l
a u d i v i d n i t e g r a t e h
t asgene to0 ;otherwise ,se tthesamegeneinthemutant ni dividuals d
n
a as 0 ;i f or ,then le tj=j+1 .When al lthe genes filled the locus of the s
l a u d i v i d n i l a t n e m i r e p x
e ,a new experimenta lpopulation was generated and the crossover .
)
4 Ift herei sadeletiongenei nt het esti ndividua l ,allt hegeneswhicharenotequalt o0i n or l
a u d i v i d n i t s e t e h t o t d e i p o c e r
a togenerateanewtesti ndividua l ,andcrossoveroperation d
e h s i n i
f , NP=NP+1.
n o it c e l e
S Opera iton
i t l u m e h t t n u o c c a o t n i g n i k a t , r e p a p s i h t n
I -objectivecharacteristicsoft heproblem ,allt hei ndividuals h
t l l a d n a n o i t a r e n e g e h t f o n o i t a l u p o p l a i t i n i e h t n
i e experimenta lindividuals in the crossover e
n o o t n i d e g r e m e r a n o i t a r e p
o doublesizepopulation .Thenthenex tgenerationoft arge tpopulations h
g u o r h t d e t a r e n e g s
i t nhen -o dominated layerthat si constructed by thering rule[5]and thepareto
a r e i t n o r
f ndcrowdeddistancevalues[6].
m h ti r o g l
A Flow
)
1 Firstly ,sett heparametersoft healgorithm ,randomliyinitializet hepopulation ,andsett hecurren t a
r b e g l a n o i t u l o v
e Cycle= ;0 n
o i t a t u M )
2 operation: Randomly generate a random number rand() between (0,1) .The h
t i w d e r a p m o c s i ) 5 1 ( a l u m r o f o t g n i d r o c c a d e t a r e n e
g rand().If therandom number y
g e t a r t s 1 / d n a r / E D e s u n e h
t a end xecute formula (12); otherwise choose DE/best/1 strategy for n
o i t a r e p o n o i t a t u
m andexecuteformula(14). )
3 rC ossoveroperation :According to section crossoveroperation, performacrossoveroperation .
n o i t a l u p o p t s e t e h t e t a r e n e g o t t r o s e g r e m n o d e s a b
)
4 Selectionoperation :Thet arge tpopulationandt heexperimenta lpopulationaremixedi ntoanew n
a , n o i t a l u p o
p dt heselectionoperationi sperformedaccordingt oselectionoperation. )
5 Inordert omaintaint hediversityofi ndividualsi nt hepopulation ,selec tindividualst ha tgenerate u
p o p e h t r e t f a r e h t o h c a e p a l r e v o o h w s l a u d i v i d n i e h t e c a l p e r o t y l m o d n a r s h t a
p lationi smixed.
e g d u J )
6 Cycle� ,if yes ,outpu tthe non-dominated solution se tgenerated by the ,
p e t s d n o c e s e h t o t o g , e s i w r e h t o ; n o i t a r e p o n o i t c e l e
s Cycle=Cycle+ 1.
l a t n e m i r e p x
E DataandAnalyssi
f n o c e h t s e r i u q e r n e t f o m h t i r o g l a d e s o p o r p e h
T irmationofexperimenta ldata .Theexperimentalt es t p
t t h ( t e s a t a d n o m o l o S d e t p e c c a y l l a n o i t a n r e t n i e h t m o r f d e t c e l e s s a w t e s a t a
d ://web.cba.neu.edu /
r i e h t o t g n i d r o c c a s e i r o g e t a c t n e r e f f i d 6 o t n i d e d i v i d e r a s t e s a t a d e h T . ) m t h . s m e l b o r p / n o m o l o s m ~
s n o i t a c o
l :C1 ,R1 ,RC1 ,C2 ,R2 ,RC2.
s r e m o t s u c f o r e b m u n e h t n e h w : s w o l l o f s a s i m h t i r o g l a s i h t f o s r e t e m a r a p l a t n e m i r e p x e f o t e s e h T
5 2 s
i ,thepopulation sizeNP=40 ,themaximumnumberofiterations =400 ;when the 0
5 s i s t n e i l c f o r e b m u
n ,thepopulationsizeNP=100and =1000 ;when thenumberof 0
0 1 = P N e z i s n o i t a l u p o p e h t , 0 0 1 s i s r e m o t s u
c and = 1000 .When using DE/rand/1 ,
y g e t a r t
s F = 0.7 ,andwhenusingDE/best/1strategy ,F= 30 d. a n Cr=0.2.I nt hehybridDEalgortihm , , andF =0.5 ,Cr=0.2.
W T P R V e v l o s o t y t i l i b a e h t d n a E D d i r b y h f o y t i l i b a t s d n a s s e n e v i t c e f f e e h t y f i r e v o t r e d r o n
I ,with
m h t i r o g l a C B A e h
t [7]forfurthercomparison ,Fort hegeneralityoft heexperiment ,sixofC ,six fo R
x i s d n
a fo RC,r espectively ,wereselectedf romt heSolomondatase tbasedont heclien tsizeof25 ,50 s
e m i t 0 2 n u r s a w t n e m i r e p x e e h T . 0 0 1 d n
a ,comparet heaverageandthespecificdatashowni nTable .
2
l a r e n e g n i , 2 e l b a T m o r
F ,t hereare11datain18 datasetstesttha tusingthehybrid DEalgorithm n
a h t s s e
l thealgorithmusingmutationstrategyandDE/best/1mutationstrategyont hemileage, and e
r a e r e h
, s e l c i h e v f o r e b m u n e h t g n i z i m i n i m f
o thesolutionobtainedbyt hehybridDEalgorithmi nt he7 data .
m h t i r o g l a C B A e h t y b d e n i a t b o s e l c i h e v f o r e b m u n l a m i t p o e h t n a h t s s e l s i s t e s
e l b a
T 1. ComparisonofhybridDEalgorithmandABCalgorithmexperimenta lresults.
t e s a t a d
u N
r e b m
f o
o t s u c
r e m
1 / t s e b / E
D DE/rand/1 ABC hybridDEalgorithm
e g a r e v a
e u l a v
e g a r e v a
e u l a v
l a m i t p O
n o i t u l o s
e g a r e v a
e u l a v
l a m i t p O
n o i t u l o s
e g a r e v a
e u l a v f
o r e b m u N
t s i d / s r a
c ance
f o r e b m u N
e c n a t s i d / s r a c
f o r e b m u N
e c n a t s i d / s r a c
f o r e b m u N
e c n a t s i d / s r a c
f o r e b m u N
e c n a t s i d / s r a c
f o r e b m u N
e c n a t s i d / s r a c 4
0 1
C 2 5 3.5/205.8 3.7/222.18 3/187.5 3/189.0 3/188.3 3 4. 1/ 88 . 26 1
0 2
C 2 5 2.1/240.4 2/315.56 2/215.5 2/216.7 2/215.5 2 15 6/2 . 4 3
0 1
R 2 5 4.8/467.8 5.5/532 5/455.7 5.4/468.1 5/459.7 5/469.2 2
. 9 7 4 /
4 5.4/468.1 4/495.6 5/469.2 1
0 2
R 2 5 3.8/478.1 4.5/526.4 4/464.4 3.6/482.7 4/464.4 3 /.8482.65 5
. 1 8 4 /
3 3.6/482.7 2/526.8 3 /.8482.65 1
0 1 C
R 2 5 4.4/493.2 5.7/541.8 4/463.6 4.9/478.7 4/462.2 4 /.3478.79 8
0 2 C
R 2 5 1.8/289 2.1/320.6 2/269.6 2/270.8 2 7 6/2 9 . 1 /.9270.7 6
. 9 0 3 / 8 .
1 1 /.9270.7 3
0 1
C 5 0 5.7/416.89 6.2/428.76 5/392.2 6.1/439.9 5/372.3 6 /.1417.25 1
0 2
C 5 0 3.8/396.5 3.5/439.73 3/373.8 3.8/411.5 3/371.6 3 /.6406.5 /
2428.0 3 /.6406.5 1
0 1
R 5 0 13.5/1072.8 15/1112.2 13/1049.5 13.1/1081.8 12/1056.7 13.5/1069.8 2
. 1 6 0 1 / 2
1 13.1/1081.8 13.5/1069.8 1
0 2
R 5 0 6.9/899.6 7.7/915.8 7/829.8 7.4/872.7 6/844.3 6 /.5865.5 8
. 2 7 8 /
5 7.4/872.7 3/954.0 6 /.5865.5 1
0 1 C
R 5 0 10.95/981. 5 11.3/1108 10/977.1 10/986.6 8/958.3 1 /0986.65 6
. 2 8 9 /
9 10/986.6 4
0 2 C
R 5 0 2.8/496.6 5.6/525.7 3/459.4 3/508.1 3/459.2 2.85/460.6 4
2 5 /
2 3/508.1 2/553.7 2.85/460.6 1
0 1
C 1 00 10.2/887.36 9.8/867.3 10/828.9 10.2/846.2 10/858.9 9 /.9838.3 1
0 2
C 1 00 6.2/768.4 4/621.9 4/618.6 4/631.8 3/621.6 4 /.5622.3 3
0 1
R 1 00 15.8/1567.8 15.6/1280 15/1264.5 15.3/1273.9 15/1235.5 15.6/1288.3 1
0 2
R 1 00 11.7/1405.8 9.3/1246.3 11/1227.2 10.8/1245.9 9/123 76 . 9 /.61230.34 7
. 1 6 2 1 /
9 10.8/1245.9 7/1291.8 9 /.61230.34 1
C
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y t i l i b a b o r p e h t l o r t n o c o t r e t e m a r a p g n i y r a v y l r a e n i l a g n i s u , y g e t a r t s 1 / d n a r / E D f o y t i l i b a p a c h c r a e s
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o f DE/rand/1 mutation strategy and DE/best/1 mutation strategy ,achieve the balance g n i s u y b d e s a e r c n i s i s l a u d i v i d n i n o i t a l u p o p f o y t i s r e v i d e h T . t n e m p o l e v e d d n a n o i t a r o l p x e n e e w t e b
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P Projec tnumber :20170540767.
t c e j o r
P source :Provincia lNatura lScienceFoundation
n e r e f e
R c es
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