A particle swarm optimization for the capacitated vehicle routing problem.
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ISSN1881-5456
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Volume2Numberl Nov..2007
InternationalJournalof
Logisticsand
SCMSystems
OfficialJournal:
TheInternationalFederationof LOGISI/CSand SCM Sysfems
The AsiaPacificFederationof LOGISTICS
and SCM Sysfems
Yutaka Karasawa
Foreword
Jessie Chung, Koichi Osawa, Yutoka
Karasawa, Akihiro Watanabe, and Keizo
Wakabayashi
A Study on Redesignof ReverseLogisticsNetwork in Electric Home
AppliancesRecycling System
Yu Fujila, Keizo Wokabayashi, Yutaka
Karasawa , Akihiro Watanabe and
Koichi Osawa
Decision Making aboutGetting BackhaulLoad and Improvementof Load
Efficiency for TruckloadCarriers
Makoto Yamashita, Katsuki Fui is awa
and Kazuhide Nqkata
Parallel Solver for SemidefiniteProgramming
Yusuke Satou, Jun Usuki, Masatoshi
Kitaoka and Satoshi Tange
Comparisonon the EconomicalEfficiency betweenFinished
Productsand PartsStock in ProductionSystems
Shusaku Hirqki, Thkaya lchimura,
Hiroshi Katayama and Kazuyoshi Ishii
The Effects of Shorteningthe TransportationLead-timein the International
CooperativeGlobal ComplementaryProductionSystems
Mitsuyoshi Horikawa, Takeo Takeno and
Mitsumasa Sugawara
Inventory ManagementSystemUsing RFID Tagsfor Supply Chain
Management
The Jin Ai and Voratas
Kachinichyanukul
A Particle SwarmOptimization for the CapacitatedVehicleRouting
Problem
Navee Chiadamrong and Priyanka
Kohly
Optimal Design of a Man-MachineManufacturingCell Using Statistical
Approachand EconomicConsideration
Jun Toyotani, Kyoichi Mine, Akihiro
Watanabe,Keizo Wakabayashi, Koichi
Osawa, Yu Fujita and Yutaka Karasqwa
XMLApplication Method and ProcessingCharacteristicsfor Physical
Distribution Systemusing HandicapTerminal
Takeo Takeno, Mitsumasa Sugawara and
Yasuyuki Murakami
Developmentof ElectronicData InterchangeSystemfor FreshSeafood
Supply Chain Enhancingthe Functionof CentralWholesaleMarket
S. Komolavanij, K. Mingsakun and W
Kongprawechnon
Fuzzy Bineray Programingfor Outsourcingin SupplyChainManagement
Shingo Ishikawa and Masanobu
Matsumaru
ExchangeRateForecastingin World Wide Trade
-
InternationalJournalof Logisffcs and SCM Sysfems
EDITOR
IN CHIEF
Dr.KatsuhisaOhno
GUESTEDITORS
Dr. KeizoWakabayashi
Dept. of Applied Information Science,
Aichi Institute of Technology
Toyota,Aichi 470-0392, Japan
Nihon University
Dr.Yutaka Karasawa
Kanagawa University
ADVISORY
BOARD
Dr.Gheng-MinFeng
NationalChiaoTungUniversity,Taiwan
Dr.HiroshiKatayama
WasedaUniversity,Japan
Dr.TakahiroFujimoto
TheUniversityof Tokyo,Japan
Dr.YoungHae Lee
HanyangUniversity,Korea
Dr.Hark Hwang
KoreaAdratcedhstofScience&Teclmologr,Korrea
Dr.Mario Tabucanon
Asian Instituteof Technology,Thailand
Dr.Yutaka Karasawa
KanagawaUniversity,Japan
Dr. DingWelWang
NothernUniversity,China
MANAGING
EDITOR
Dr.Mitsuo Gen
WasedaUniversity, Japan
REGIONAL
EDITORS
Dr.BongiuJeong
YonseiUniversity,Korea
Dr.VoratasKachitvichyanukul
Asian Instituteof Technology,Thailand
Dr.ZhihongJin
DalianMaritimeUniversity,China
RajeshPoplani
NanyangTech.University,Singapore
Dr.Lie-GhienLin
NationalKaohsiungUniversityof Scienceand
Technology,Taiwan
Dr.ShamsRahman
RMIT University,Australia
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PATING
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IZATIONS
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scM systetns
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of Logistics
&
AIMS& SCOPE
Logistics and SCM Systemsins an internationaljoumal for reportingoriginal quality works on Logistics and Supply
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SCM systems.
Copyright O The InternationalFederationof Logistics and SCM Systems.All right reserved.
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lnternationalEditorialBoard Members
of The International Federation of Logisfrcs and S.C.M.Sysfems
Australiar
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Schoolof Management
Science,
Univ.
of
Technology,
Australia
Queensland
Dr. ShamsRahman
Schoolof Management
RMIT University,Australia
Ghina:
Dr. Jian Chen
Dept.of Management
ScienceandEngineering
TsinghuaUnivirsity, China
Dr. Peng Tian
Dept.of ManagementScienceandEngineering
Shanghai
JiaotongUniversiryChina
Dr. MasatoshiKitaoka
Dept.of InformationSystemCreation,
KanagawaUniversity,Japan
Dr. MasanobuMatsumaru
Dept.of Industial& SystemsEngineering
TokaiUniversity,Japan
Korea:
Dr. Kap Hwan Kim
Dept.of IndustialEngineering
PusanNationalUniversity,Korea
Dr. Suk-Chul Rim
Dept.of Industial& InformationSystems
Engineering
Ajou University,Korea
Taiwan:
Hong Konq:
Dr. Xiande Zhao
Dept.of DecisionSci.& ManagerialEconomic,
Thechineseuniversityof HongKong' H' K'
Dr. Yon-Chun Chou
Instituteof IndustrialEngineering
NationalTaiwanUniversity,Taiwan
lndia:
Dr. yen-chen Jim wu
NationalKaoshiungFirst Scienceand
Technology,Taiwan
Dr. PramondK. Jain
Dept.of Mechanical& IndustrialEngineering
IndianInstituteof TechnologyRoorkee,India
Thailand:
Japan:
Dr. ShusakuHiraki
Dept.of Economics& InformationScience
HiroshimaShudoUniversity,Japan
Dr. VoratasKachitvichyanukul
Asian Institute of Technology,Thailand
Dr. paisal yenradee
SirithornInstituteof Technology,
Thailand
Organization
of TheInternationalFederationof Logisficsand S.c.M. Sysfems
Ghairman
Karasawa,Yutaka
Chairtnan of The Japon Society of Logistics SystemsAnd Dr & Emeritus Professor of
Kanagawa Univers ity, Japan
Vice Chairman
Feng,Cheng-Min
Chairman of The Taiwan Society of Logistics Systems And Ph.D, & Professor of
National Chiao Tung (Jniversity, Taiwan
Fujimoto, Takahiro
WceChairmanof TheJapan Societyof LogisticsSystemsAnd DBA & Professorof The
Universityof Tolryo,Japan
Katayama,Hitoshi
Wce Chairmanof The Japan Societyof LogisticsSystemsAnd'Dn & Professorof
WasedaUniversity, Japan
LeeoYoung Hae
Chairman of The Korea Societyof SCM And Dr. & Professorof Hanyang (Jniversity,
Korea
Pongchai Athikomrattanakul
Ph.D. & Professorof King Monglafi'sUniversityof TechnologtThonburi,Thailand
lnternationalJournalof Logisfics and SCM Sysfems
Vofume2NumberlNov.,2007
CONTENTS
I
YutakaKorasawa
Foreword
2 JessieChunS,Koichi Osawa,YutakaKarctsowo,
Akihilo Watarnbe,
andKeizoWatrabayashi
A Studyon Redesignof ReverseLogistics Network in Electric Home AppliancesRecycling System
12 YuFujita,Kein Wakabayashi,
YutokaKarasowa, Akihirc Watonnbe
andKoichi Osawa
DecisionMaking aboutGetting BackhaulLoad and Improvementof Load Efficiency for TruckloadCarriers
22 Makoto Yamoshita,Katsuki Fujisawa and Kazuhide Nakata
ParallelSolver for SemidefiniteProgramming
30 YusukeSatou,Jun Usuki, Masatoshi Kitaoka and Satoshi Tange
Comparisonon the EconomicalEfficiency betweenFinishedProductsand PartsStock in ProductionSystems
ICLS 2007 SpecialSession
38 ShusakuHiraki, Takayalchimura, Hiroshi Katayama and Kazuyoshi Ishii
The Effects of Shorteningthe TransportationLead-timein the InternationalCooperativeGlobal Complementary
ProductionSystems
44 Mitsuyoshi Horikawq, TakeoTakenoond Mitsumasa Sugawara
Inventory ManagementSystemUsing RFID Tagsfor Supply chain Management
50 TheJin Ai and VoratasKachinichyanulul
AParticle Swarm Optimization for the CapacitatedVehicleRouting Problem
56 Navee Chiadamrong and Priyanka Kohly
Optimal Design of a Man-MachineManufacturingCell Using StatisticalApproachand EconomicConsideration
66 Jun Tbyotani,Kyoichi Mine, Akihiro Watanqbe,Keizo Wakabayashi,Koichi Osawa, YuFujita and YutaksKarasqwa
XML Application Method and ProcessingCharacteristicsfor PhysicalDistribution Systemusing Handicap
Terminal
74
.i.
Takeo Thkeno, Mitsumqsa Sugawara and Yasuyuki Murakami
Development of Electronic Data Interchange System for Fresh Seafood Supply Chain Enhancing the Function of
Central Whalesale Market
80 S. Komolqvanij, K. Mingsakun and W Kongltrawechnon
Fuzzy Bineray Programing for Outsourcing in Supply Chain Management
88 Shingo Ishikowa and Masanobu Matsumqru
Exchange Rate Forecasting in World Wide Trade
lnternational Journal of Logistics and SCM Systems
A Particle Swarm Optimization for the Capacitated
Vehicle Routing Problem
The Jin Ai.A and Voratas Kachitvichyanukul*B
Asian Institute of Technology, Thailand
*A
e-mail:thejin.ai@ait.ac.th
*B
e-mail :v or atas@,ait.ac.th
Abstract
This paperproposeda random-keybasedsolution
representationand decoding method for solving the
CapacitatedVehicle Routing Problem (CVRP) using
Particle Swarm Optimization (PSO). The solution
representation is (n+2m)-dimensional particle for
CVRP with r customers and m vehicles. The
decodingmethod startswith transformingthe particle
to a priority list of customer to enter route and
priority matrix of vehicle to serveeachcustomer.The
vehicle routes are constructedbasedon the customer
priority list and vehicle priority matrix. The proposed
representationis applied using GLNPSO, a PSO
algorithm with multiple social learning structures.
The proposed algorithm is tested using the
benchmark data set provided by Christofides. The
computational result shows that this representation
and decoding method is promising to be applied for
CVRP.
Keywords: CapacitatedVehicle Routing Problem,
Particle Swarm Optimization, Random Key
DecodingMethod
Representation,
I .Introduction
The Capacitated Vehicle Routing Problem
(CVRP), which was firstly introducedby Danzig and
Ramser[], is a problem to designa set of vehicle
routes in which a fixed fleet of delivery vehicles of
uniform capacity must service known customer
demands for a single commodity from a common
depot at minimum cost. In the literature [2)-141,
CVRP can be formally defined as follows. Let
G:(VA be a graphwhere V:{vs,v1,...,v,}is a vertex
set,andA:{(vi,v)lv;,v1eV,it'j} is an arc set.Vertexvs
represents a depot, while the remaining vertices
correspondto customers.Associatedwith A are a cost
or distancematrix (d;) and a travel time matrix (r,y).
Each customerhas a non-negativedemand q; and a
service time si. A fleet of m identical vehicles of
capacity Q is based at the depot. The number of
vehicles is either known in advanceor treated as a
decisionvariable.The CVRP consistsof designinga
set of at most tn delivery or collection routessuchthat
Received Date:
Accepted Date:
JuJy 12,2007
September.20,2007
(l) each route starts and ends at the depot, (2) each
customer is visited exactly once by exactly one
vehicle, (3) the total demandof each route does not
exceed Q, g) the total duration of each route
(including travel and servicetimes) doesnot exceeda
preset limit D, and (5) the total routing cost is
minimized.
CVRP is an NP-hard problem, which meansthat
any optimization algorithm for their solution has
worst case running time which is likely to grow
exponentiallywith problem size l5]. In other words,
there is a high possibility that an optimal solution of a
large VRP problem may only be found with
excessively long solution time. To overcome this
situation,evolutionary computingmethodshave been
used to find a near optimal solution in a reasonable
amount of time, for example: Genetic Algorithm
16l-[7), Ant Colony Optimization [8]-[9], and
ParticleSwarmOptimization[0].
Particle Swarm Optimization (PSO) is a
population based search method proposed by
Kennedyand Eberhart[11], which were motivatedby
the group organismbehavior such as bee swarm, fish
school, and bird flock. PSO imitated the physical
movements of the individuals in the swann as a
searchingmethod, altogether with its cognitive and
social behavior as local and global exploration
abilities.In the PSO, a solutionof a specificproblem
is being representedby n-dimensionalposition of a
particle. The ability of particlesto searchfor solution
is represented
by meansof velocity vector.The PSO
algorithm starts with population of particles with
random initial position and velocity. The population
of particlesis usually called a swarrn.In one iteration
step, every particle is moved from previous position
to the new position based on its velocity; and its
velocity is updatedbasedon its personalbestposition
and the global best position obtained so far. Once a
particle reach a position which has a better objective
function than the previousbest objective function for
this particle, the personalbest position is updated.
Also, if it found better objective function than the
previousbest objective function for whole swarm,the
global best position is updated.A brief and complete
survey on PSO mechanism, technique, and
application is provided by Kennedy & Eberhart [2]
and alsoClerc [3].
It is obviously seen that the PSO works on
lnternational Journal of Logistics and SCM Systems
finding the best position and the position is
representedby real number.To make PSO applicable
to specific problem; the relationship between the
position of particlesand the solutionsof that problem
must be clearly defined.In CVRP case,the particle's
position represents the vehicle route. This paper
proposed a specific solution representationand its
decoding method to relate position in PSO with
CVRP solution. The proposed representation is
different from representationof Chen et al. [l0l in
two aspects. First, the proposed representationis
designedfor the classicvariant of PSO, i.e. directly
using real value of position, insteadof the discrete
version. Second, the proposed representationis
implementedin a pure PSO without any local search
or other hybrid method.
The remainder of this paper is organized as
follow: Section 2 explains the proposed solution
representation and decoding method. Section 3
reviews PSO framework for solving CVRP. Section4
discussesthe computationalexperiment of PSO that
applied the proposed solution representation on
benchmarkdata set. Finally, Section 5 discussesthe
result of this study and suggestsfurther improvement
of proposedalgorithm.
2. Proposed Solution Representation and
Decoding Method
In PSO, a problem specific solution is
represented by
position
particle
of
in
2
multi-dimensional space. The proposed solution
representationof CVRP with n customers and m
vehicles consists of (n+2m) dimensional particle.
Eachparticledimensionis encodedas a real number.
The first n dimension is related to customers,each
customerwill be represented
by one dimension.The
last 2m dimensionis relatedto vehicles,eachvehicle
will be represented by two dimensions as the
referencepoint in Cartesiandiagrarnlmap.
In order to decode this solution representation
into the CVRP solution, three steps are taken. First,
extract the information from the first n dimension to
make a priority list of customers.Second,extract the
information from the last 2m dimension to set the
referencepoint of vehiclesand use this information to
createpriority matrix of vehicles.Third, constructthe
vehicle routesbasedon the customerpriority list and
vehicle priority matrix.
For illustration, consider CVRP problem with 6
customersand 2 vehicles. A particle representation
for the solutionof this problemwill consistof (6+2.2)
: l0 dimensions.The first six dimensionsare related
to the customersand the four remaining are relatedto
the vehicles.Supposea particlehas certainpositional
valuesas describebelow in part (a) of Figure 1. The
decoding process of this particle begins by
constructinga priority list of customersby sorting in
ascending order the position value and taking the
dimensionindex asthe list.
The next step is extracting the referencepoint
5
6
0.4 0.2 0.9 0.5 0 . 6 0 . 1
I
Dimension ( d )
Value( x,, )
1
J
7
0.8
8
9
1 . 5 0.4
t0
1.0
I
Customers
(a) Particle position
d
Vehicles
A
6
2
a
J
5
0 . 1 0.2 0.4 0 . 5 0 . 6 0 . 9
CustomerPriority List
z
6
1
4
5
(b) First Step: Customer Prioritv List
J
Vehicle Reference Points
(0.8.0.4)
Vehicle I
Vehicle2
Customer
1.5,1.0)
Relative Distance
Vehicle I
Vehicle 2
0.66
0.70
J
0.13
4
0.21
5
0.35
6
0.35
(c) Second Step:
2
Vehicle Prioritv
Priority I
Prioritv 2
0.26
2
0.40
2
1.05
2
0.84
2
0.59
z
1.08
2
Vehicle Prioritv Matrix
Vehicle Routes
Vehicle I
Vehicle 2
0
0
6
2
J
4
I
0
5t0
@) Third Sten: VehicleRoute
Figure 1: conversion Processof SolutionRepresentationto vehicle Routes
Systems
InternationalJournalof Logisticsand SCM
dimensionof
for vehicles;In this case,the 7thand9th
of Vehicle I
ii, iini"l" will be the refetencepoint
and 10'hdimensionof the particlewill be
;;;ils.h
matrix
tt.,.f.t"n.. point of Vehicle 2'The priority
relative
of vehicles is constructedbased on the.
betweenthesepoints and customerslocatron'
distance
case oI
The distancecould be calculated in every
usually
anRP, since the location of customer is
prioritized
is
customer
A
map'
olacedin a Cartesian
For
io f. ."*"4 by vehicle which has closerdistance'
vehicle
is
1.
customer
of
priority
.^u-pf", the vehicle
closerto
I thenvehicle 2, sincecustomer1 location is
1' This
the vehicle 2 referencepoint than to vehicle
(b) of
part
the
in
illustrated
is
conversionresult
Figurel.
The last decoding step is to construct a route
vehicle
basedon the customer priority list and the
in the
cu-stoler
each
one
by
OA""W matrix. One
on
based
vehicle
a
to
is
assigned
i"tio.". priority list
time
route
and
constraint,
its priority, vehille capacity
is inserted
.o*ouini.-This newly assignedcustomer
route
vehicle
existing
the
in
to the best sequence
This
distance'
or
cost
basedon the least additional
insertion
cheapest
the
heuristic is usually called
6 is
heuristic.In the illustration in Figure 1: customer
2'
vehicle
to
2
customer
,rtGt.A to vehicle 1,
1'
vehicle
to
4
customer
2,
rot,"o-at 1 to vehicle
1'
vehicle
to
3
customer
and
.urtotn.t 5 to vehicle 1,
the
in
displayed
is
route
,ft" final vehicle
ii.*",
'0' represents
bottompart of Figure 1. In this figure'
heuristic
tn. a"pot. Note ihat the cheapestinsertion
route
the
in
sequence
may make the customer
for
assignment'
different from the order of customer
assigned
last
it
example: customer 3, which
thg
because
,urto-"t, appearedin the middle of the route
of this
result
frnal
and
of this treuiiitic. The process
of this
details
Th9
2
exampleare illustrateditt Fig"t"
1'
Algorithm
in
decodingprocedureare described
Atgorithm 1: DecodingMethod
n;- - ,rh
oI tne I
De*codefrom ParticlePosition (ttd posltlon
VehicleRoute.(Ra;
;;i"1" a,t thedh dimension)into
to the i'
corresponding
vehicle
ioute of the i'h
particle)
(tf
i. Constructthe priority list of customers
t : { 1 , 2 , " ' , n }a n d U : A
a . B u i l d s eS
set 'Swherexi": min(xia)'deS
from
c
Select
b.
c. Add c to the last position in set U'
d. Removec from set S'
S: A
e. RePeatsteP1.b.:ur;.rt[
matrix (V)
priority
vehicle
the
2. Construct
position' For
reference
vehicle
a. Set the
andyrefi:xi,n+6+i
j:l ...m, seIxrefi:xi,n+1
b. For eachcustomeri, i:| " 'n
distance between
Calculate the
i.
referencepoints
vehicle
and
i
customer
formula
following
using
-ffi
a, = f,(xpot, - xref,)- +\tnost Yre.f
i)
ii. Build setS : {1,2,"',m\ and'Vi: O
iii. Selectc from set S where 6': min(6)'
deS
iv. Add c to the last position in set tr/i
v. Removec from set S
vi. RePeatsteP2'b'iii until S: O
3. Constructvehicle route'
a. Setft:l
b. Add customerone by one to the route
Setl:UrandP:l
i.
Setj:i1'o' Evaluate the vehicle load if
ii'
customei/ is addedto the route R;7'
iii. Make a candidate of new route by
insertingcustomer/ to the best sequence
in the route Ri;, which has the .smallest
additional cost (cheapest rnsenron
heuristic).
iv. Check ihe capacity and route time
F
m
q
'*iT:"*'
\
I
\
**tbr
70
rlo
llt
112
1
Map
Figure2:GraphicallllustrationofVehicleRoutesConstructiononCartesian
lnternational Journalof Logistics ond SCM Systems
constraintofthe candidateroute.
If a feasible solution is reached,update
the route R,;with the candidateroute; go
to step3.c.
vi. lf p:m, go to step 3.c. Otherwise,set
p--p+l andrepeat3.b.ii
c. If k:n, stop.Otherwise,setk:k+l andrepeat
3.b.
v.
3. PSO Framework for Solving CVRP
The proposed representationis applied using
GLNPSO, a PSO Algorithm with multiple social
structures 114], with slight modification on the
velocity and position limitation. In this application,
no velocity and position limitation is incorporated.It
can be done due to the flexibility of this
representation in which all real value can be
converted into vehicle route. By applying no
restriction on velocity and position value, the
additional effort for checking and adjusting these
valuescan be eliminated.The details of the GLNPSO
for solving CVRP are explainedbelow
Notation
t
I
d
u
w(t)
vi{t)
xi{t)
Pia
Psd
L
n.,
rta
Pia
9n
Ct
Wn
Iterationindex,t:I...7
Particleindex,i:l ...I
Dimensionindex,d:|...D
Uniform random number in the interval
[0,1]
Inertia weieht inthe t'n iteration
Velocity o1 the ith particle at the dh
dimensioninrhe t'n iteration
Position of the ith particle at the dh
dimensioninthe t'n iteration
Personalbest position (pbest) of the ith
particleat thed' dimension
Global best position (gbest) at the d'
dimension
Local best position (lbest) of lhe i'n
particleat thedn dimension
Near neighborbest position (nbest)of the
i'h parlicle atthe dh dimension
Personal best position acceleration
constant
Global bestposition accelerationconstant
Local bestposition accelerationconstant
Near neighborbest position acceleration
constant
Algorithm 2: GLNPSO Framework for CVRP
l. Initialize 1 particles as a population,generatethe
i'n particle with random position X; in the range
[,f'n,trflu*], velocity V;:0 andpersonalbestP;X1
for i:l...L Setiterationl:1.
2. For i: I . . .1, decodeXi(t) to a set of vehicle route
R;. (Algorithm I in Section2)
3. For i:\ ..J,
compute the performance
measurementof R'. and set this as the fitness
value ofX;, representedby q(Xt).
update P.Xi, if
4. Update pbest: For i:|..J,
q(xr)
ISSN1881-5456
I
'J.
I
rr L--
I
l___. l
I
l
1
I
,-*
---\--
+-
1
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\
tt*,.
-/
\\
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ll
t/'
V
Volume2Numberl Nov..2007
InternationalJournalof
Logisticsand
SCMSystems
OfficialJournal:
TheInternationalFederationof LOGISI/CSand SCM Sysfems
The AsiaPacificFederationof LOGISTICS
and SCM Sysfems
Yutaka Karasawa
Foreword
Jessie Chung, Koichi Osawa, Yutoka
Karasawa, Akihiro Watanabe, and Keizo
Wakabayashi
A Study on Redesignof ReverseLogisticsNetwork in Electric Home
AppliancesRecycling System
Yu Fujila, Keizo Wokabayashi, Yutaka
Karasawa , Akihiro Watanabe and
Koichi Osawa
Decision Making aboutGetting BackhaulLoad and Improvementof Load
Efficiency for TruckloadCarriers
Makoto Yamashita, Katsuki Fui is awa
and Kazuhide Nqkata
Parallel Solver for SemidefiniteProgramming
Yusuke Satou, Jun Usuki, Masatoshi
Kitaoka and Satoshi Tange
Comparisonon the EconomicalEfficiency betweenFinished
Productsand PartsStock in ProductionSystems
Shusaku Hirqki, Thkaya lchimura,
Hiroshi Katayama and Kazuyoshi Ishii
The Effects of Shorteningthe TransportationLead-timein the International
CooperativeGlobal ComplementaryProductionSystems
Mitsuyoshi Horikawa, Takeo Takeno and
Mitsumasa Sugawara
Inventory ManagementSystemUsing RFID Tagsfor Supply Chain
Management
The Jin Ai and Voratas
Kachinichyanukul
A Particle SwarmOptimization for the CapacitatedVehicleRouting
Problem
Navee Chiadamrong and Priyanka
Kohly
Optimal Design of a Man-MachineManufacturingCell Using Statistical
Approachand EconomicConsideration
Jun Toyotani, Kyoichi Mine, Akihiro
Watanabe,Keizo Wakabayashi, Koichi
Osawa, Yu Fujita and Yutaka Karasqwa
XMLApplication Method and ProcessingCharacteristicsfor Physical
Distribution Systemusing HandicapTerminal
Takeo Takeno, Mitsumasa Sugawara and
Yasuyuki Murakami
Developmentof ElectronicData InterchangeSystemfor FreshSeafood
Supply Chain Enhancingthe Functionof CentralWholesaleMarket
S. Komolavanij, K. Mingsakun and W
Kongprawechnon
Fuzzy Bineray Programingfor Outsourcingin SupplyChainManagement
Shingo Ishikawa and Masanobu
Matsumaru
ExchangeRateForecastingin World Wide Trade
-
InternationalJournalof Logisffcs and SCM Sysfems
EDITOR
IN CHIEF
Dr.KatsuhisaOhno
GUESTEDITORS
Dr. KeizoWakabayashi
Dept. of Applied Information Science,
Aichi Institute of Technology
Toyota,Aichi 470-0392, Japan
Nihon University
Dr.Yutaka Karasawa
Kanagawa University
ADVISORY
BOARD
Dr.Gheng-MinFeng
NationalChiaoTungUniversity,Taiwan
Dr.HiroshiKatayama
WasedaUniversity,Japan
Dr.TakahiroFujimoto
TheUniversityof Tokyo,Japan
Dr.YoungHae Lee
HanyangUniversity,Korea
Dr.Hark Hwang
KoreaAdratcedhstofScience&Teclmologr,Korrea
Dr.Mario Tabucanon
Asian Instituteof Technology,Thailand
Dr.Yutaka Karasawa
KanagawaUniversity,Japan
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NothernUniversity,China
MANAGING
EDITOR
Dr.Mitsuo Gen
WasedaUniversity, Japan
REGIONAL
EDITORS
Dr.BongiuJeong
YonseiUniversity,Korea
Dr.VoratasKachitvichyanukul
Asian Instituteof Technology,Thailand
Dr.ZhihongJin
DalianMaritimeUniversity,China
RajeshPoplani
NanyangTech.University,Singapore
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NationalKaohsiungUniversityof Scienceand
Technology,Taiwan
Dr.ShamsRahman
RMIT University,Australia
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IZATIONS
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scM systetns
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Federation
of Logistics
&
AIMS& SCOPE
Logistics and SCM Systemsins an internationaljoumal for reportingoriginal quality works on Logistics and Supply
Chain Management(SCM)systems.It is publishedby the InternationalFederationsof Logistics and SCM Systems.The
aims of thejournal are to
I publish original, high-quality researchthat will have a significant impact on Logistics and SCM systems.
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Copyright O The InternationalFederationof Logistics and SCM Systems.All right reserved.
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lnternationalEditorialBoard Members
of The International Federation of Logisfrcs and S.C.M.Sysfems
Australiar
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Schoolof Management
Science,
Univ.
of
Technology,
Australia
Queensland
Dr. ShamsRahman
Schoolof Management
RMIT University,Australia
Ghina:
Dr. Jian Chen
Dept.of Management
ScienceandEngineering
TsinghuaUnivirsity, China
Dr. Peng Tian
Dept.of ManagementScienceandEngineering
Shanghai
JiaotongUniversiryChina
Dr. MasatoshiKitaoka
Dept.of InformationSystemCreation,
KanagawaUniversity,Japan
Dr. MasanobuMatsumaru
Dept.of Industial& SystemsEngineering
TokaiUniversity,Japan
Korea:
Dr. Kap Hwan Kim
Dept.of IndustialEngineering
PusanNationalUniversity,Korea
Dr. Suk-Chul Rim
Dept.of Industial& InformationSystems
Engineering
Ajou University,Korea
Taiwan:
Hong Konq:
Dr. Xiande Zhao
Dept.of DecisionSci.& ManagerialEconomic,
Thechineseuniversityof HongKong' H' K'
Dr. Yon-Chun Chou
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NationalTaiwanUniversity,Taiwan
lndia:
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NationalKaoshiungFirst Scienceand
Technology,Taiwan
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Dept.of Mechanical& IndustrialEngineering
IndianInstituteof TechnologyRoorkee,India
Thailand:
Japan:
Dr. ShusakuHiraki
Dept.of Economics& InformationScience
HiroshimaShudoUniversity,Japan
Dr. VoratasKachitvichyanukul
Asian Institute of Technology,Thailand
Dr. paisal yenradee
SirithornInstituteof Technology,
Thailand
Organization
of TheInternationalFederationof Logisficsand S.c.M. Sysfems
Ghairman
Karasawa,Yutaka
Chairtnan of The Japon Society of Logistics SystemsAnd Dr & Emeritus Professor of
Kanagawa Univers ity, Japan
Vice Chairman
Feng,Cheng-Min
Chairman of The Taiwan Society of Logistics Systems And Ph.D, & Professor of
National Chiao Tung (Jniversity, Taiwan
Fujimoto, Takahiro
WceChairmanof TheJapan Societyof LogisticsSystemsAnd DBA & Professorof The
Universityof Tolryo,Japan
Katayama,Hitoshi
Wce Chairmanof The Japan Societyof LogisticsSystemsAnd'Dn & Professorof
WasedaUniversity, Japan
LeeoYoung Hae
Chairman of The Korea Societyof SCM And Dr. & Professorof Hanyang (Jniversity,
Korea
Pongchai Athikomrattanakul
Ph.D. & Professorof King Monglafi'sUniversityof TechnologtThonburi,Thailand
lnternationalJournalof Logisfics and SCM Sysfems
Vofume2NumberlNov.,2007
CONTENTS
I
YutakaKorasawa
Foreword
2 JessieChunS,Koichi Osawa,YutakaKarctsowo,
Akihilo Watarnbe,
andKeizoWatrabayashi
A Studyon Redesignof ReverseLogistics Network in Electric Home AppliancesRecycling System
12 YuFujita,Kein Wakabayashi,
YutokaKarasowa, Akihirc Watonnbe
andKoichi Osawa
DecisionMaking aboutGetting BackhaulLoad and Improvementof Load Efficiency for TruckloadCarriers
22 Makoto Yamoshita,Katsuki Fujisawa and Kazuhide Nakata
ParallelSolver for SemidefiniteProgramming
30 YusukeSatou,Jun Usuki, Masatoshi Kitaoka and Satoshi Tange
Comparisonon the EconomicalEfficiency betweenFinishedProductsand PartsStock in ProductionSystems
ICLS 2007 SpecialSession
38 ShusakuHiraki, Takayalchimura, Hiroshi Katayama and Kazuyoshi Ishii
The Effects of Shorteningthe TransportationLead-timein the InternationalCooperativeGlobal Complementary
ProductionSystems
44 Mitsuyoshi Horikawq, TakeoTakenoond Mitsumasa Sugawara
Inventory ManagementSystemUsing RFID Tagsfor Supply chain Management
50 TheJin Ai and VoratasKachinichyanulul
AParticle Swarm Optimization for the CapacitatedVehicleRouting Problem
56 Navee Chiadamrong and Priyanka Kohly
Optimal Design of a Man-MachineManufacturingCell Using StatisticalApproachand EconomicConsideration
66 Jun Tbyotani,Kyoichi Mine, Akihiro Watanqbe,Keizo Wakabayashi,Koichi Osawa, YuFujita and YutaksKarasqwa
XML Application Method and ProcessingCharacteristicsfor PhysicalDistribution Systemusing Handicap
Terminal
74
.i.
Takeo Thkeno, Mitsumqsa Sugawara and Yasuyuki Murakami
Development of Electronic Data Interchange System for Fresh Seafood Supply Chain Enhancing the Function of
Central Whalesale Market
80 S. Komolqvanij, K. Mingsakun and W Kongltrawechnon
Fuzzy Bineray Programing for Outsourcing in Supply Chain Management
88 Shingo Ishikowa and Masanobu Matsumqru
Exchange Rate Forecasting in World Wide Trade
lnternational Journal of Logistics and SCM Systems
A Particle Swarm Optimization for the Capacitated
Vehicle Routing Problem
The Jin Ai.A and Voratas Kachitvichyanukul*B
Asian Institute of Technology, Thailand
*A
e-mail:thejin.ai@ait.ac.th
*B
e-mail :v or atas@,ait.ac.th
Abstract
This paperproposeda random-keybasedsolution
representationand decoding method for solving the
CapacitatedVehicle Routing Problem (CVRP) using
Particle Swarm Optimization (PSO). The solution
representation is (n+2m)-dimensional particle for
CVRP with r customers and m vehicles. The
decodingmethod startswith transformingthe particle
to a priority list of customer to enter route and
priority matrix of vehicle to serveeachcustomer.The
vehicle routes are constructedbasedon the customer
priority list and vehicle priority matrix. The proposed
representationis applied using GLNPSO, a PSO
algorithm with multiple social learning structures.
The proposed algorithm is tested using the
benchmark data set provided by Christofides. The
computational result shows that this representation
and decoding method is promising to be applied for
CVRP.
Keywords: CapacitatedVehicle Routing Problem,
Particle Swarm Optimization, Random Key
DecodingMethod
Representation,
I .Introduction
The Capacitated Vehicle Routing Problem
(CVRP), which was firstly introducedby Danzig and
Ramser[], is a problem to designa set of vehicle
routes in which a fixed fleet of delivery vehicles of
uniform capacity must service known customer
demands for a single commodity from a common
depot at minimum cost. In the literature [2)-141,
CVRP can be formally defined as follows. Let
G:(VA be a graphwhere V:{vs,v1,...,v,}is a vertex
set,andA:{(vi,v)lv;,v1eV,it'j} is an arc set.Vertexvs
represents a depot, while the remaining vertices
correspondto customers.Associatedwith A are a cost
or distancematrix (d;) and a travel time matrix (r,y).
Each customerhas a non-negativedemand q; and a
service time si. A fleet of m identical vehicles of
capacity Q is based at the depot. The number of
vehicles is either known in advanceor treated as a
decisionvariable.The CVRP consistsof designinga
set of at most tn delivery or collection routessuchthat
Received Date:
Accepted Date:
JuJy 12,2007
September.20,2007
(l) each route starts and ends at the depot, (2) each
customer is visited exactly once by exactly one
vehicle, (3) the total demandof each route does not
exceed Q, g) the total duration of each route
(including travel and servicetimes) doesnot exceeda
preset limit D, and (5) the total routing cost is
minimized.
CVRP is an NP-hard problem, which meansthat
any optimization algorithm for their solution has
worst case running time which is likely to grow
exponentiallywith problem size l5]. In other words,
there is a high possibility that an optimal solution of a
large VRP problem may only be found with
excessively long solution time. To overcome this
situation,evolutionary computingmethodshave been
used to find a near optimal solution in a reasonable
amount of time, for example: Genetic Algorithm
16l-[7), Ant Colony Optimization [8]-[9], and
ParticleSwarmOptimization[0].
Particle Swarm Optimization (PSO) is a
population based search method proposed by
Kennedyand Eberhart[11], which were motivatedby
the group organismbehavior such as bee swarm, fish
school, and bird flock. PSO imitated the physical
movements of the individuals in the swann as a
searchingmethod, altogether with its cognitive and
social behavior as local and global exploration
abilities.In the PSO, a solutionof a specificproblem
is being representedby n-dimensionalposition of a
particle. The ability of particlesto searchfor solution
is represented
by meansof velocity vector.The PSO
algorithm starts with population of particles with
random initial position and velocity. The population
of particlesis usually called a swarrn.In one iteration
step, every particle is moved from previous position
to the new position based on its velocity; and its
velocity is updatedbasedon its personalbestposition
and the global best position obtained so far. Once a
particle reach a position which has a better objective
function than the previousbest objective function for
this particle, the personalbest position is updated.
Also, if it found better objective function than the
previousbest objective function for whole swarm,the
global best position is updated.A brief and complete
survey on PSO mechanism, technique, and
application is provided by Kennedy & Eberhart [2]
and alsoClerc [3].
It is obviously seen that the PSO works on
lnternational Journal of Logistics and SCM Systems
finding the best position and the position is
representedby real number.To make PSO applicable
to specific problem; the relationship between the
position of particlesand the solutionsof that problem
must be clearly defined.In CVRP case,the particle's
position represents the vehicle route. This paper
proposed a specific solution representationand its
decoding method to relate position in PSO with
CVRP solution. The proposed representation is
different from representationof Chen et al. [l0l in
two aspects. First, the proposed representationis
designedfor the classicvariant of PSO, i.e. directly
using real value of position, insteadof the discrete
version. Second, the proposed representationis
implementedin a pure PSO without any local search
or other hybrid method.
The remainder of this paper is organized as
follow: Section 2 explains the proposed solution
representation and decoding method. Section 3
reviews PSO framework for solving CVRP. Section4
discussesthe computationalexperiment of PSO that
applied the proposed solution representation on
benchmarkdata set. Finally, Section 5 discussesthe
result of this study and suggestsfurther improvement
of proposedalgorithm.
2. Proposed Solution Representation and
Decoding Method
In PSO, a problem specific solution is
represented by
position
particle
of
in
2
multi-dimensional space. The proposed solution
representationof CVRP with n customers and m
vehicles consists of (n+2m) dimensional particle.
Eachparticledimensionis encodedas a real number.
The first n dimension is related to customers,each
customerwill be represented
by one dimension.The
last 2m dimensionis relatedto vehicles,eachvehicle
will be represented by two dimensions as the
referencepoint in Cartesiandiagrarnlmap.
In order to decode this solution representation
into the CVRP solution, three steps are taken. First,
extract the information from the first n dimension to
make a priority list of customers.Second,extract the
information from the last 2m dimension to set the
referencepoint of vehiclesand use this information to
createpriority matrix of vehicles.Third, constructthe
vehicle routesbasedon the customerpriority list and
vehicle priority matrix.
For illustration, consider CVRP problem with 6
customersand 2 vehicles. A particle representation
for the solutionof this problemwill consistof (6+2.2)
: l0 dimensions.The first six dimensionsare related
to the customersand the four remaining are relatedto
the vehicles.Supposea particlehas certainpositional
valuesas describebelow in part (a) of Figure 1. The
decoding process of this particle begins by
constructinga priority list of customersby sorting in
ascending order the position value and taking the
dimensionindex asthe list.
The next step is extracting the referencepoint
5
6
0.4 0.2 0.9 0.5 0 . 6 0 . 1
I
Dimension ( d )
Value( x,, )
1
J
7
0.8
8
9
1 . 5 0.4
t0
1.0
I
Customers
(a) Particle position
d
Vehicles
A
6
2
a
J
5
0 . 1 0.2 0.4 0 . 5 0 . 6 0 . 9
CustomerPriority List
z
6
1
4
5
(b) First Step: Customer Prioritv List
J
Vehicle Reference Points
(0.8.0.4)
Vehicle I
Vehicle2
Customer
1.5,1.0)
Relative Distance
Vehicle I
Vehicle 2
0.66
0.70
J
0.13
4
0.21
5
0.35
6
0.35
(c) Second Step:
2
Vehicle Prioritv
Priority I
Prioritv 2
0.26
2
0.40
2
1.05
2
0.84
2
0.59
z
1.08
2
Vehicle Prioritv Matrix
Vehicle Routes
Vehicle I
Vehicle 2
0
0
6
2
J
4
I
0
5t0
@) Third Sten: VehicleRoute
Figure 1: conversion Processof SolutionRepresentationto vehicle Routes
Systems
InternationalJournalof Logisticsand SCM
dimensionof
for vehicles;In this case,the 7thand9th
of Vehicle I
ii, iini"l" will be the refetencepoint
and 10'hdimensionof the particlewill be
;;;ils.h
matrix
tt.,.f.t"n.. point of Vehicle 2'The priority
relative
of vehicles is constructedbased on the.
betweenthesepoints and customerslocatron'
distance
case oI
The distancecould be calculated in every
usually
anRP, since the location of customer is
prioritized
is
customer
A
map'
olacedin a Cartesian
For
io f. ."*"4 by vehicle which has closerdistance'
vehicle
is
1.
customer
of
priority
.^u-pf", the vehicle
closerto
I thenvehicle 2, sincecustomer1 location is
1' This
the vehicle 2 referencepoint than to vehicle
(b) of
part
the
in
illustrated
is
conversionresult
Figurel.
The last decoding step is to construct a route
vehicle
basedon the customer priority list and the
in the
cu-stoler
each
one
by
OA""W matrix. One
on
based
vehicle
a
to
is
assigned
i"tio.". priority list
time
route
and
constraint,
its priority, vehille capacity
is inserted
.o*ouini.-This newly assignedcustomer
route
vehicle
existing
the
in
to the best sequence
This
distance'
or
cost
basedon the least additional
insertion
cheapest
the
heuristic is usually called
6 is
heuristic.In the illustration in Figure 1: customer
2'
vehicle
to
2
customer
,rtGt.A to vehicle 1,
1'
vehicle
to
4
customer
2,
rot,"o-at 1 to vehicle
1'
vehicle
to
3
customer
and
.urtotn.t 5 to vehicle 1,
the
in
displayed
is
route
,ft" final vehicle
ii.*",
'0' represents
bottompart of Figure 1. In this figure'
heuristic
tn. a"pot. Note ihat the cheapestinsertion
route
the
in
sequence
may make the customer
for
assignment'
different from the order of customer
assigned
last
it
example: customer 3, which
thg
because
,urto-"t, appearedin the middle of the route
of this
result
frnal
and
of this treuiiitic. The process
of this
details
Th9
2
exampleare illustrateditt Fig"t"
1'
Algorithm
in
decodingprocedureare described
Atgorithm 1: DecodingMethod
n;- - ,rh
oI tne I
De*codefrom ParticlePosition (ttd posltlon
VehicleRoute.(Ra;
;;i"1" a,t thedh dimension)into
to the i'
corresponding
vehicle
ioute of the i'h
particle)
(tf
i. Constructthe priority list of customers
t : { 1 , 2 , " ' , n }a n d U : A
a . B u i l d s eS
set 'Swherexi": min(xia)'deS
from
c
Select
b.
c. Add c to the last position in set U'
d. Removec from set S'
S: A
e. RePeatsteP1.b.:ur;.rt[
matrix (V)
priority
vehicle
the
2. Construct
position' For
reference
vehicle
a. Set the
andyrefi:xi,n+6+i
j:l ...m, seIxrefi:xi,n+1
b. For eachcustomeri, i:| " 'n
distance between
Calculate the
i.
referencepoints
vehicle
and
i
customer
formula
following
using
-ffi
a, = f,(xpot, - xref,)- +\tnost Yre.f
i)
ii. Build setS : {1,2,"',m\ and'Vi: O
iii. Selectc from set S where 6': min(6)'
deS
iv. Add c to the last position in set tr/i
v. Removec from set S
vi. RePeatsteP2'b'iii until S: O
3. Constructvehicle route'
a. Setft:l
b. Add customerone by one to the route
Setl:UrandP:l
i.
Setj:i1'o' Evaluate the vehicle load if
ii'
customei/ is addedto the route R;7'
iii. Make a candidate of new route by
insertingcustomer/ to the best sequence
in the route Ri;, which has the .smallest
additional cost (cheapest rnsenron
heuristic).
iv. Check ihe capacity and route time
F
m
q
'*iT:"*'
\
I
\
**tbr
70
rlo
llt
112
1
Map
Figure2:GraphicallllustrationofVehicleRoutesConstructiononCartesian
lnternational Journalof Logistics ond SCM Systems
constraintofthe candidateroute.
If a feasible solution is reached,update
the route R,;with the candidateroute; go
to step3.c.
vi. lf p:m, go to step 3.c. Otherwise,set
p--p+l andrepeat3.b.ii
c. If k:n, stop.Otherwise,setk:k+l andrepeat
3.b.
v.
3. PSO Framework for Solving CVRP
The proposed representationis applied using
GLNPSO, a PSO Algorithm with multiple social
structures 114], with slight modification on the
velocity and position limitation. In this application,
no velocity and position limitation is incorporated.It
can be done due to the flexibility of this
representation in which all real value can be
converted into vehicle route. By applying no
restriction on velocity and position value, the
additional effort for checking and adjusting these
valuescan be eliminated.The details of the GLNPSO
for solving CVRP are explainedbelow
Notation
t
I
d
u
w(t)
vi{t)
xi{t)
Pia
Psd
L
n.,
rta
Pia
9n
Ct
Wn
Iterationindex,t:I...7
Particleindex,i:l ...I
Dimensionindex,d:|...D
Uniform random number in the interval
[0,1]
Inertia weieht inthe t'n iteration
Velocity o1 the ith particle at the dh
dimensioninrhe t'n iteration
Position of the ith particle at the dh
dimensioninthe t'n iteration
Personalbest position (pbest) of the ith
particleat thed' dimension
Global best position (gbest) at the d'
dimension
Local best position (lbest) of lhe i'n
particleat thedn dimension
Near neighborbest position (nbest)of the
i'h parlicle atthe dh dimension
Personal best position acceleration
constant
Global bestposition accelerationconstant
Local bestposition accelerationconstant
Near neighborbest position acceleration
constant
Algorithm 2: GLNPSO Framework for CVRP
l. Initialize 1 particles as a population,generatethe
i'n particle with random position X; in the range
[,f'n,trflu*], velocity V;:0 andpersonalbestP;X1
for i:l...L Setiterationl:1.
2. For i: I . . .1, decodeXi(t) to a set of vehicle route
R;. (Algorithm I in Section2)
3. For i:\ ..J,
compute the performance
measurementof R'. and set this as the fitness
value ofX;, representedby q(Xt).
update P.Xi, if
4. Update pbest: For i:|..J,
q(xr)