A HYBRID PSO ALGORITHM FOR THE CVRP PROBLEM
Yucheng Kao and Mei Chen
Department of Information Management, Tatung University, 104, Taipei, Taiwan
Keywords: Vehicle routing problems, Particle swarm optimization, Simulated annealing.
Abstract: The Capacitated Vehicle Routing Problem (CVRP) has been studied over five decades. The goal of CVRP
is to minimize the total distance travelled by vehicles under the constraints of vehicles’ capacity. Because
CVRP is a kind of NP-hard problem, a number of meta-heuristics have been proposed to solve the problem.
The objective of this paper is to propose a hybrid algorithm combining Combinatorial Particle Swarm
Optimization (CPSO) with Simulated Annealing (SA) for solving CVRP. The experimental results show
that the proposed algorithm can be viewed as an effective approach for solving the CVRP.
1 INTRODUCTION
There are many combinatorial Optimization problems in
the real world, including portfolio optimization, job
dispatching, vehicle routing problems, etc. The vehicle
routing problem (VRP) is an interesting research subject.
Many researchers have developed a variety of solution
approaches to solve the VRP over past 50 years. The
objective of capacitated vehicle routing problem (CVRP)
is to minimize the total distance traveled by vehicles.
There are two stages used in common CVRP approaches:
customer clustering and sequencing. In clustering stage we
assign each customer into a vehicle, while in sequencing
stage we arrange the visiting order for a vehicle which has
been assigned customers. Apparently, customer clustering
results affect sequencing results, and the sequencing
results determine the objective function values.
CVRP belongs to the category of NP-hard
problems. In recent decades meta-heuristic
approaches attract many researchers’ attention and
have been applied to the VRPs. Meta-heuristic
approaches include genetic algorithms, simulated
annealing, ant colony optimization, particle swarm
optimization, etc. One of important advantages of
using these approaches is that we can obtain optimal
or near optimal solutions in a reasonable
computation time. Applying a meta-heuristic
approach to solve the CVRP, we need to concern
following aspects: current routing solutions may
affect clustering results of the next iteration; initial
solutions may affect the evolution result; checking
solution feasibility may be necessary from time to
time, solution quality and the converging speed of
evolutionary processes are important performance
criteria but both often conflict with each other.
Particle Swarm Optimization (PSO) was
proposed by Kennedy and Eberhart in 1995. PSO
has been applied to solve many problems, including
the VRP. Chen et al. (2006) first proposed a discrete
PSO (DPSO) approach to solve the CVRP. Their
approach follows the two-stage approach, clustering
first and routing second. They use DPSO to perform
customer clustering and use SA to determine the
visiting order of each vehicle. Due to long solution
strings (the product of customer number and vehicle
number) their algorithm often needs a larger amount
of CPU time to find optimal solutions. Ai et al.
(2009) also proposed an algorithm based on PSO for
CVRP. The solution string of a particle represents
coordinate points and coverage radius of vehicles.
Vehicle routes are constructed based on these points
and radiuses. The order of visiting customers of each
route is found by using an insertion heuristic. Thus
these two papers used the cluster-first-route-second
approach to solve CVRP.
This paper proposes a new hybrid PSO with SA
to solve the CVRP. Similarly, it follows the cluster-
first-route-second approach: use a new discrete PSO
to find customer clustering results and then use
simulated annealing to arrange customer visiting
orders. Experimental results show that the proposed
algorithm can solve the CVRPs efficiently.
539
Kao Y. and Chen M..
A HYBRID PSO ALGORITHM FOR THE CVRP PROBLEM.
DOI: 10.5220/0003724005390543
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FEC-2011), pages 539-543
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

2 LITERATURE
2.1 VRP Problems
VRP is an interesting subject which belongs to a
category of combinatorial optimization problems.
Dantzig (1959) first proposed the truck dispatching
problems. The VRP has many variants. Jozefowiez
et al. (2008) presented detailed classifications and
comparisons for VRP problems with respect to
problem definitions, objectives and algorithms. The
article mentioned that VRP has widely applied to
transport delivery routing, urban school bus route
planning, rural school bus routing planning, urban
trash collection, etc. The authors introduced
objectives of VRP from single objective to multiple
objectives. One of common objective functions is to
minimize the total distance traveled by all vehicles.
Other objectives include workload balance, total
number of vehicle, total traveling time and total
waiting time of customers.
Classical VRP problems are classified into two
categories: capacitated vehicle routing problems
(CVRP) and vehicle routing problem with time
windows (VRPTW). CVRP’s objective is to find a
set of routes with minimum total distance traveled to
deliver goods for all customers with different
demands. Its Constraints include: every customer is
to be served exactly once by one vehicle; each
vehicle starts and ends at the depot, every vehicle
has a limited capacity.
VRP is a kind of NP-hard problems. Researchers
developed many solution approaches to solve VRPs,
such as exact algorithms and heuristic algorithms
(Cordeau et al., 2007). Meta-heuristics are a kind of
heuristics, which have widely been applied to VRPs.
The performance of meta-heuristics is often better
than classical heuristics. Popular meta-heuristics
include Genetic Algorithm (GA), Tabu Search (TS),
Simulated Annealing (SA), Ant colony systems
(ACS), Particle Swarm Optimization (PSO), Scatter
Search (SS), etc.
Baker et al. (2003) proposed a simple GA for
solving CVRP. The length of solution string is equal
to the total number of customers. Each gene has an
integer number which ranges from 1 to the total
number of vehicles. The algorithm selects parent
solutions by using a binary tournament method,
produces offspring solutions by using two-point
crossover operation, mutates solutions by swapping
two randomly selected genes, selects better solutions
for next generation by using a ranking replacement
method.
Bell et al. (2004) proposed an ACO algorithm for
solving the CVRP. They proposed two methods to
improve the performance of their algorithm,
including local exchange and candidate list. The
authors also proposed multiple ant colonies method
for solving large size problems (more than 100
customers). Zhang et al. (2009) proposed an
algorithm which integrates scatter search with ACO
for solving CVRP. That paper uses scatter search as
the framework and applies ACO to construct route
solutions.
2.2 Particle Swarm Optimization
(PSO)
PSO is a population-based evolution algorithm and
is a member of swarm intelligence techniques. PSO
was proposed by Kennedy and Eberhart (1995) and
inspired by the birds’ foraging behavior. The
searching process is affected by current positions,
personal best positions (pbest) and the best position
of the flock (gbest). Original PSO algorithm has the
ability of fast convergence whereas it does not
guarantee to find the optimal solution eventually.
After some modifications, one of popular PSO
version proposed by Shi and Eberhart (1999) is
defined as follows:
)(
2
)(
1
id
X
best
GRandC
id
X
best
PRandC
id
VW
id
V


(1)
ididid
VXX


(2)
This paper adopts a discrete PSO model,
combinatorial particle swarm optimization (CPSO)
to solve the CVRP problem. CPSO was proposed by
Jarboui et al. (2007) for solving combinatorial
optimization problems. It permits solutions transited
from a discrete variable space to a continuous
variable space. After a standard PSO evolution
process, new continuous solutions are transited back
to the discrete space in order to obtain new discrete
solutions. It has been proved that CPSO is a useful
solution approach for solving most of discrete
combinatorial optimization problems.
3 CPSO-SA FOR CVRP
This paper proposes a new algorithm combining
CPSO and SA, which is called CPSO-SA. It follows
the two-stage approach for solving CVRP. CPSO
deals with customer clustering while SA arranges
customer visiting sequence. At the end of iterations,
the algorithm conducts local searches for top three
FEC 2011 - Special Session on Future of Evolutionary Computation
540

best particles. Particle’s best solutions and the global
best solution obtained at current iteration will be
used to generate new particles in the next iteration.
This paper adopts similar basic approach used by
Chen et al. (2006), but it tries to improve the latter
by using a short solution representation and a more
efficient discrete PSO algorithm. For customer
clustering, it doesn’t need to check solution
feasibility to prevent that a customer is assigned to
more than one vehicle, and, as a result, it does save
some computational time.
3.1 Mathematical Model
The CVRP considered in this paper has a symmetric
network. The objective is to minimize the total
distance traveled by all vehicles. This problem has
been formulated as follows:
Notation
0：depot；
n：total number of customers；
N：customers set，N=
n}{1,2,...,
；
v：total number of vehicles；
V：vehicles set，
v}{1,2,...,V  ；
ij
d
：distance between customer I and j,
jiij
dd

，
}0{,  Nji ；
i
q ：demand for customer I, 0
0
q ；
Q ：maximum capacity for each vehicle；
k
ij
X ：vehicle k travels edge i-j or not, 0: no,
1: yes；
R
k
：customer set served by vehicle k，
},...,,{R
,
2,1,k
k
Rk
kk
jjj ;
|R
k
|: cardinality of R
k
;
j
k,m
：mth customer served by vehicle k.
Objective function
Minimize

 
v
k
n
i
n
j
k
ijij
d
10 0
X
(3)
Subject to



v
k
n
i
k
ij
njX
10
..., ,2 ,1 ,1
(4)



v
k
n
j
k
ij
niX
10
..., ,2 ,1 ,1
(5)



n
i
n
j
k
uj
k
iu
nuvkXX
00
..., ,2 ,1 ; ..., ,2 ,1 ,0
(6)



n
j
n
i
k
ijj
vkQXq
10
..., ,2 ,1 , ) (
(7)



n
j
k
j
vkX
1
0
..., ,2 ,1 ,1
(8)



n
i
k
i
vkX
1
0
..., ,2 ,1 ,1
(9)
Eqs. (4) and (5) ensure that each customer must be
served by a vehicle exactly once. Eq. (6) considers
that the continuity for every vehicle can be
maintained. Eq. (7) ensures that the total customer
demand of a vehicle does not exceed its maximum
capacity. Eqs. (8) and (9) mean that every vehicle
can be used once or not be used.
3.2 Solution Representation
This paper adopts a cluster-first-route-second
approach for solving CVRP. Thus a solution has two
sections. The first one is for customer clustering and
the second one is for customer sequencing. The
length of each of these two sections equals to
n. XV
p
= (xv
p1
, xv
p2
,…, xv
pn
) stands for the result of
customer clustering of particle
p; xv
pd
indicates the
vehicle which serves customer d. Tour
p
= (0, j
1,1
, …,
j
1,|R1|
, 0, j
2,1
, …, 0, j
v,1
, …, j
v,|Rv|
, 0) is the customer
sequencing result of all vehicles found by particle
p.
In the first stage, some other vectors have to be
defined for executing PSO algorithms.
V
p
= (v
p1
, v
p2
,
…, v
pn
) indicates the velocity of particle p; P
p
= (P
p1
,
P
p2
, …, P
pn
) indicates the personal best solution ever
found of particle
p; G = (G
1
, G
2
, …, G
n
) indicates the
global best solution ever found of the swarm.
3.3 CPSO-SA Algorithm
The proposed algorithm starts conducting evolution
after initialization until the iteration number reaches
the max iteration number. The latest global best
solution is the final solution. Evolution includes the
processes of customers clustering and sequencing.
Customer clustering is conducted by using CPSO
while the visiting order is conducted by using SA.
Then the algorithm computes objective function
value for every particle according to Eq. (3). After
that, we perform a local search on top three best
particles for solution improvement. The procedure is
to choose two routes randomly with considering
capacity constraints, to select a customer from each
of selected routes, and then to exchange their vehicle
numbers. After that, find new visiting orders of these
two routes by using SA. If new solution is better,
A HYBRID PSO ALGORITHM FOR THE CVRP PROBLEM
541

replace the current solutions with new solutions.
Before the iteration ends, particles update their Pbest
and Gbest solutions.
3.3.1 Customer Clustering
In every iteration, customer clustering solution is
found by using Eqs.(10)~(16). pop particles with
solutions XVs flying in a space of n dimensions take
place between two transition phases: from discrete
space to continuous space and from continuous
space back to discrete space.
Transition Phase 1:















otherwise0
) xv( if randomly,1or1
) (xv if1
) ( if1
111-iter
pd
11-iter
pd
1
1
iter
d
iter
pd
iter
pd
iter
d
iter
pd
iter
pd
GP
P
G
xv
y
(10)
Flying Phase:
y
d
iter
pd
 1 1
, distance between
1iter
pd
xv
and
1
P
iter
pd
(11)
y
d
iter
pd
1 2 
, distance between
1iter
pd
xv and
1iter
d
G
(12)
222 111
1
drcdrcvwv
iter
pd
iter
pd


(13)
iter
pd
iter
pd
iter
pd
v
y


(14)
Transition Phase 2:








otherwise0
) (if1
) (if1




iter
pd
iter
pd
iter
pd
y
(15)










otherwise eany vehicl
)1 ( if
)1 ( if
1
1
y
P
y
G
xv
iter
pd
iter
pd
iter
pd
iter
d
iter
pd
(16)
Where,
iter is the current number of iteration；p and
d represent particle number and dimension number;
iter
pd
y is a dummy variable for dimension d of particle
p in iteration iter；
iter
d
G is the value of dimension d
for the gbest solution in iteration
iter；
iter
pd
P is the
value of dimension
d of the pbest solution for
particle
p in iteration iter；
iter
pd
xv ：is the value of
dimension
d of particle p in iteration iter；c1 and c2
are weighting coefficients；
iter
pd
v is the velocity for
dimension
d of particle p in iteration iter；r1 and r2
are random numbers between 0 and 1；
iter
pd
 ：value
of dimension
d for particle p in iteration iter；

is a
threshold parameter.
3.3.2 Customer Sequencing
Because capacity constraints have been considered
in the first stage, this stage just needs to consider the
total distance traveled by vehicles. Thus the problem
of customer sequencing for a vehicle is equivalent to
a TSP problem. Initial solutions are generated by
using a greedy method and then SA algorithm, Eqs.
(17) and (18), is used to improve initial solutions.
P(S’) is the probability that SA accepts new solution
S’ which is worse than the current solution S.
)()( SfSf




(17)
)/exp()( tSp




(18)
4 EXPERIMENTS
To verify our approch, we compare our algorithm
with that proposed by Chen et al. in terms of
solution quality and CPU time. CPSO-SA was coded
in Java and executed on a PC with 3.5GB of RAM
and Intel Core 2 CPU E8400 3GHz. The CPSO-SA
parameters used for all CVRP problems are pop =
the amount of customers, w = 0.8,
1C = 1.1 , 2C =
1.4, max iteration number = 300,

0
t 3,
f
t = 0.01, L
=
2

n , 8.0


. Five CVRP test problems are
collected from the website (http://www.branchand
cut.org/VRP/data/).
Figure 1 presents the convergence trend of
running CPSO-SA for solving the first test problem.
The results of five test problems is listed in Table 1.
To compare computational times, Chen’s CPU times
have been properly converted based on an equation
(Patterson and Hennessy, 2005). The results show
that CPSO-SA’s solutions are equal to those
obtained by Chen’s approach with much less CPU
times.
Figure 1: Convergence trend when CPSO-SA solving
problem A33-K5.
FEC 2011 - Special Session on Future of Evolutionary Computation
542

Table 1: Experimental results.
No. Problem Cus. # Vehicle #
Objective function value CPU time (seconds)
BKS DPSO-SA CPSO-SA DPSO-SA CPSO-SA
1 A-n33-k5 32 5 661 661 661 19.4
0.7
2 A-n46-k7 45 7 914 914 917 77.3
2.4
3 A-n60-k9 59 9 1354 1354 1354 185.3
6.5
4 B-n35-k5 34 5 955 955 955 22.6
1.2
5 B-n45-k5 44 5 751 751 751 80.5
4.8
BKS: the best known solution in the literature
DPSO-SA used Intel Pentium IV CPU 1.8 GHz with 256M RAM
CPSO-SA uses Intel Core 2 CPU E8400 3GHz with 3.5G RAM
5 CONCLUSIONS
In our proposed approach, CPSO is first used to
cluster customers into different vehicles and then SA
is used to arrange the visiting order of each vehicle.
A new solution representation is also proposed in
order to reduce computing time which is necessary
for judging feasibility and repairing infeasible
solutions. Experimental results show that the CSPO-
SA algorithm can effectively solve the CVRP
problem within a more reasonable time period.
REFERENCES
Ai, T. J., Kachitvichyanukul, V., 2009. Particle swarm
optimization and two solution representations for
solving the capacitated vehicle routing problem.
Computers & Industrial Engineering (56), pp. 380-
387.
Baker, B. M. and Ayechew, M. A., 2003. A genetic
algorithm for the vehicle routing problem. Computers
& Operations Research (30), pp. 787-800.
Bell, J. E. and McMullen, P. R., 2004. Ant Colony
Optimization techinques for the vehicle routing
problem. Advanced Engineering Informatics (18), pp.
41-48.
Chen, A. L., Yang, G. K. and Wu, Z. M., 2006. Hybrid
discrete particle swarm optimization algorithm for
capacitated vehicle routing problem. Journal of
Zhejiang University SCIENCE A 7(4), pp. 607-614.
Cordeau, J. F., Laporte, G., Savelsbergh, M. W. P. and
Vigo, D., 2007. Chapter 6 Vehicle Routing. In:
Barnhart, C. and Laporte, G. eds. Handbook in
Operations Research and Management Science(14),
pp. 367-428.
Jarboui, B., Cheikh, M., Siarry, P. and Rebai, A., 2007.
Combinatorial particle swarm optimization (CPSO)
for partitional clustering problem. Applied
Mathematics and Computation (92), pp. 337-345.
Jozefowiez, N., Semet, F. and Talbi, E. G., 2008. Multi-
objective vehicle routing problems. European Journal
of Operational Research (189), pp. 293-309.
Kennedy, J. and Eberhart, R., 1995. Particle swarm
optimization. Proceedings of IEEE International
Conference on Neural Networks, pp. 1942–1948.
Marinakis, Y., Magdalene, M. and Dounias, G., 2010. A
hybrid particle swarm optimization algorithm for the
vehicle routing problem. Engineering Applications of
Artificial Intelligence (23:4), pp. 463-472.
Osman, I. H., 1993. Metastrategy simulated annealing and
tabu search algorithms for the vehicle routing problem.
Annals of Operations Research, pp. 421-451.
Patterson D. A., Hennessy J. L. (Eds.), 2009. Computer
organization and design: the hardware/software
interface. Burlington, MA: Morgan Kaufmann
Publishers.
Solomon, M. M., 1987. Algorithms for the Vehicle
Routing and Scheduling Problems with Time Window
constraints. Operations Research (35:2), pp. 254-265.
Zhang, X. and Tang, L., 2009. A new hybrid ant colony
optimization algorithm for the vehicle routing
problem. Pattern Recognition Letters (30), pp. 848-
855.
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