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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION 1
Decomposition-Based Memetic Algorithm for
Multiobjective Capacitated Arc Routing Problem
Yi Mei, Student Member, IEEE, Ke Tang, Member, IEEE, and Xin Yao, Fellow, IEEE
Abstract—The capacitated arc routing problem (CARP) is a
challenging combinatorial optimization problem with many real-
world applications, e.g., salting route optimization and fleet man-
agement. There have been many attempts at solving CARP using
heuristic and meta-heuristic approaches, including evolutionary
algorithms. However, almost all such attempts formulate CARP
as a single-objective problem although it usually has more than
one objective, especially considering its real-world applications.
This paper studies multiobjective CARP (MO-CARP). A new
memetic algorithm (MA) called decomposition-based MA with
extended neighborhood search (D-MAENS) is proposed. The
new algorithm combines the advanced features from both the
MAENS approach for single-objective CARP and multiobjective
evolutionary optimization. Our experimental studies have shown
that such combination outperforms significantly an off-the-shelf
multiobjective evolutionary algorithm, namely nondominated
sorting genetic algorithm II, and the state-of-the-art multiob-
jective algorithm for MO-CARP (LMOGA). Our work has
also shown that a specifically designed multiobjective algorithm
by combining its single-objective version and multiobjective
features may lead to competitive multiobjective algorithms for
multiobjective combinatorial optimization problems.
Index Terms—Capacitated arc routing problem (CARP), local
search, memetic algorithms (MA), meta-heuristics, multiobjective
optimization.
I. Introduction
T
HE CAPACITATED arc routing problem (CARP) [1] is a
well-known combinatorial optimization problem. Due to
its wide applications in the real world, including winter gritting
[2]–[6], urban waste collection [7], [8], and snow removal [9],
[10], CARP has been intensively investigated in the past few
decades. Given a graph with some edges and arcs required to
be served (called tasks) and a number of vehicles with limited
Manuscript received October 6, 2009; revised February 7, 2010 and April
14, 2010. This work was partially supported by the National Natural Science
Foundation of China, under Grants 60802036, U0835002, and 61028009, by
the Fund for Foreign Scholars in University Research and Teaching Programs,
under Grant B07033, and by the ESPRC, under Grant EP/E058884/1, on “Evo-
lutionary Algorithms for Dynamic Optimization Problems: Design, Analysis,
and Applications.”
Y. Mei and K. Tang are with the Nature Inspired Computation and
Applications Laboratory, School of Computer Science and Technology, Uni-
versity of Science and Technology of China, Hefei 230027, China (e-mail:
meiyi@mail.ustc.edu.cn; ketang@ustc.edu.cn).
X. Yao is with the Nature Inspired Computation and Applications Labora-
tory, School of Computer Science and Technology, University of Science and
Technology of China, Hefei 230027, China and also with the Center of Excel-
lence for Research in Computational Intelligence and Applications, School of
Computer Science, University of Birmingham, Edgbaston, Birmingham B15
2TT, U.K. (e-mail: x.yao@cs.bham.ac.uk).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEVC.2010.2051446
capacity, a CARP is defined as seeking an optimal routing
plan for the vehicles under the following conditions.
1) Each vehicle starts and ends at a predefined vertex,
namely depot.
2) Each task is served by exactly one vehicle.
3) The total demand of the tasks served by each vehicle
does not exceed its capacity.
Since CARP is NP-hard [11], exact methods are only
applicable to the instances with small problem sizes. However,
many real-world applications involve large-size CARPs, and
routing plans must be made within a restricted time bud-
get. Therefore, heuristics and meta-heuristics are promising
approaches in such a situation in order to obtain acceptable
solutions in time. During the last century, constructive heuris-
tics were often adopted because of their ability to generate
relatively good solutions in a very short time period. To name
a few, the Augment-Merge heuristic proposed by Golden and
Wong [11], the path scanning heuristic proposed by Golden
et al. [12], and Ulusoy’s splitting heuristic proposed in [13] are
three typical heuristics for CARP. More recently, researchers
shifted their attentions to meta-heuristics, which can provide
much better solutions. Although meta-heuristics usually induce
higher computational cost, this additional cost is now afford-
able due to the rapid development of the computational power
of computers. The first meta-heuristic approach to CARP is
the tabu search proposed by Hertz et al. [14]. After that,
the variable neighborhood descent algorithm [15], the guided
local search [16], the tabu scatter search [17], the memetic
algorithm (MA) [18], and another tabu search algorithm [19]
have been proposed. A comprehensive survey of the recent
results on various arc routing problems is presented in [20].
We have also conducted intensive investigations on CARP
in our previous work. A global repair operator that can be
embedded in any search-based approach was proposed in
[21]. More importantly, we proposed a MA with extended
neighborhood search (MAENS) [22], which has been shown
to outperform most existing approaches in terms of solution
quality.
So far, CARP has been predominantly formulated as a
single-objective problem with the only objective of minimizing
the total cost of the service. However, there is a huge gap
between such a formulation and reality. Contributions are now
needed to fill this gap in literature. For this purpose, Lacomme
et al. [23] considered minimizing total cost and makespan
(i.e., the cost of the longest route) simultaneously. Specifically,
1089-778X/$26.00
c
2010 IEEE
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2 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
they formulated a multiobjective CARP (MO-CARP) and
developed a hybrid algorithm for it by combining an approach
for single-objective CARP (SO-CARP) [18] and a commonly
used multiobjective evolutionary algorithm (MOEA), namely
Nondominated Sorting Genetic Algorithm II (NSGA-II)
[24].
The two objectives considered by Lacomme et al. in [23]
are conflicting with each other. Thus, no unique global optimal
solution exists in this case. Instead, Lacomme et al. proposed
a multiobjective genetic algorithm (referred to as LMOGA
in this paper) to maintain a set of solutions, which are
good “tradeoffs” between the two objectives. Essentially, the
MO-CARP lies in the reign of multiobjective optimization.
Numerous previous publications have shown that MOEAs
are good approaches to this kind of problem. Nevertheless,
how to make the best use of MOEAs in the context of
MO-CARP has not been fully investigated. Motivated by
this, this paper aims to contribute from two aspects. First,
a number of important issues for evolutionary multiobjective
optimization (EMO) are discussed, and the utility of existing
EMO strategies in the context of MO-CARP is examined.
Second, an algorithm named decomposition-based MAENS
(D-MAENS) is proposed. The D-MAENS employs the frame-
work of the MOEA based on decomposition (MOEA/D), with
MAENS embedded in it. In addition, it adopts proper EMO
strategies based on domain-specific considerations of MO-
CARP. Comparative studies are also presented to evaluate the
efficacy of D-MAENS.
The rest of this paper is organized as follows. Section II
gives the background, including the detailed introduction to
MO-CARP and related work on EMO. Section III discusses
the important issues in solving MO-CARP with MOEAs
and evaluates the existing strategies for addressing them.
Section IV describes the proposed D-MAENS. Afterwards,
experimental studies are presented in Section V. Finally, the
paper is concluded in Section VI.
II. Background
A. Multiobjective CARP
CARP is defined on a graph G(V, E, A), where V , E, and A
stand for the set of vertices, edges, and arcs (directed edges),
respectively. For each edge (v
i
,v
j
) ∈ E and arc v
i
,v
j
∈
A, three nonnegative features are associated, i.e., the traversal
cost c
trav
(v
i
,v
j
), the serving cost c
serv
(v
i
,v
j
), and the demand
d(v
i
,v
j
). An edge or arc with a positive demand is called a
task, and is required to be served by vehicles at the cost of its
serving cost. We denote the edge task set as E
R
= {(v
i
,v
j
) ∈
E|d(v
i
,v
j
) > 0} and the arc task set as A
R
= {(v
i
,v
j
) ∈
A|d(v
i
,v
j
) > 0}. Then, the task set is T = E
R
∪ A
R
. Note that
the serving cost is only induced by serving a task, we have
c
serv
(v
i
,v
j
) > 0 ⇐⇒ d(v
i
,v
j
) > 0 and c
serv
(v
i
,v
j
)=0⇐⇒
d(v
i
,v
j
)=0.m vehicles with an identical capacity Q are based
at the depot v
s
∈ V to serve the tasks. For an edge task (v
i
,v
j
),
service in either direction is acceptable. In order to facilitate
the problem definition, each edge task is assigned two IDs
(say t
1
and t
2
), one for each direction, and each arc task is
assigned one ID. All the IDs are unique positive integers. For
Fig. 1. Example of a CARP solution
an ID t ∈ N
+
, the following six features are associated: the tail
vertex tv(t), the head vertex hv(t), the traversal cost c
trav
(t),
the serving cost c
serv
(t), the demand d(t), and the inverse ID
inv(t). For an edge task (v
i
,v
j
), these features are defined as
follows:
1) hv(t
1
)=tv(t
2
)=v
i
;
2) tv(t
1
)=hv(t
2
)=v
j
;
3) c
trav
(t
1
)=c
trav
(t
2
)=c
trav
(v
i
,v
j
);
4) c
serv
(t
1
)=c
serv
(t
2
)=c
serv
(v
i
,v
j
);
5) d(t
1
)=d(t
2
)=d(v
i
,v
j
);
6) inv(t
1
)=t
2
, inv(t
2
)=t
1
.
For an arc task v
i
,v
j
and its ID t, the features are defined
as:
1) hv(t)=v
i
, tv(t)=v
j
;
2) c
trav
(t)=c
trav
(v
i
,v
j
);
3) c
serv
(t)=c
serv
(v
i
,v
j
);
4) d(t)=d(v
i
,v
j
);
5) inv(t)=−1.
Since all the IDs are positive, inv(t)=−1 indicates the
inverse ID of t does not exist. In addition, zero is defined as
the ID of the depot loop with the following definitions:
1) tv(0) = hv(0) = v
s
;
2) c
trav
(0) = c
serv
(0) = d(0)=0;
3) inv(0)=0.
Using the above notations, a CARP solution can be repre-
sented as a set of routes S =(R
1
,R
2
,... ,R
m
). Each route R
k
is a sequence of the IDs, i.e., R
k
=(t
k
1
,t
k
2
,... ,t
k
l
k
), where t
k
p
(1
p l
k
) are the IDs. In order to ensure that each route starts and
ends at the depot, R
k
starts and ends at the depot loop 0, i.e.,
t
k
1
= t
k
l
k
= 0. An example is illustrated in Fig. 1. In the graph,
the edge task set E
R
= {(v
1
,v
5
), (v
2
,v
6
), (v
3
,v
7
), (v
4
,v
8
)}, and
the depot is v
0
. There is no arc task in this case. The task IDs
1, 2, 3, and 4 are assigned to v
1
,v
5
, v
2
,v
6
, v
3
,v
7
, and
v
4
,v
8
, respectively, while 5, 6, 7, and 8 are assigned to their
inversions. There are two numbers associated with each edge
task, the one out of the parenthesis denotes the task ID of the
direction traversed by the route, while the other one denotes
its inversion. The dashed arrows between adjacent task IDs
(e.g., v
0
,v
1
and v
7
,v
6
in Fig. 1) stand for the intermediate
paths.
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MEI et al.: DECOMPOSITION-BASED MEMETIC ALGORITHM FOR MULTIOBJECTIVE CAPACITATED ARC ROUTING PROBLEM 3
For each route R
k
=(t
k
1
,t
k
2
,... ,t
k
l
k
), its total cost
c
tot
(R
k
) and total demand d(R
k
) can be calculated as
c
tot
(R
k
)=
l
k
−1
p=1
[c
serv
(t
k
p
)+dist(tv(t
k
p
),hv(t
k
p+1
))]
d(R
k
)=
l
k
p=1
d(t
k
p
)
where the function dist(v
1
,v
2
) is the distance from vertex v
1
to vertex v
2
, which is equal to the length of the shortest path
from v
1
to v
2
.
Under such a solution representation scheme, the MO-
CARP can be represented as follows:
min c
tot
(S)=
m
k=1
c
tot
(R
k
) (1)
min c
max
(S) = max
k
c
tot
(R
k
)(2)
s.t. :
m
k=1
(l
k
− 2) = |T | (3)
t
k
1
p
1
= t
k
2
p
2
, ∀(k
1
,p
1
) =(k
2
,p
2
)(4)
t
k
1
p
1
= inv(t
k
2
p
2
), ∀(k
1
,p
1
) =(k
2
,p
2
) (5)
d(R
k
) Q, ∀1 k m (6)
where the inequation (k
1
,p
1
) =(k
2
,p
2
) between the two pairs
(k
1
,p
1
) and (k
2
,p
2
) indicates that at least one of the two
inequations k
1
= k
2
and p
1
= p
2
is satisfied. Equation (1)
is the total cost of all the routes and (2) is the makespan.
Constraints (3)–(5) guarantee that each task is served exactly
once by one vehicle. Constraints (6), which are also called the
capacity constraints, indicate that the total demand served by
each vehicle does not exceed its capacity.
B. Evolutionary Multiobjective Optimization Revisited
A multiobjective optimization problem can be briefly stated
as follows:
min F(x)=(f
1
(x),... ,f
n
(x))
s.t. : x ∈
where is the decision variable space. F:→R
n
consists of
n objective functions that are conflicting with each other. For
a multiobjective optimization problem, one aims to seek a set
of solutions that have good tradeoffs among the objectives. In
order to make a clear notion of optimality in this scenario,
Pareto defined domination relationship and Pareto optimality
[25]. Let u, v ∈ R
n
, u dominates v if and only if u
i
v
i
for each i ∈{1,... ,n} and u
j
<v
j
for at least one
j ∈{1,... ,n}. Then, a decision variable x
∗
∈ is said
to be Pareto optimal if there is no other x ∈ so that F (x)
dominates F (x
∗
). With the above definitions, a multiobjective
optimization problem requires finding or approximating the set
of Pareto optimal solutions and their corresponding objective
vectors (called Pareto front). Hence, a MOEA should return a
set of nondominated solutions that can well approximate the
Pareto optimal solutions [26].
There are three important issues that must be addressed
in EMO, i.e., fitness assignment, diversity preservation and
elitism. Unlike in single-objective optimization problems, the
fitness of a solution needs to be assigned according to multiple
criteria in a multiobjective optimization problem. Diversity
preservation is important for MOEAs to obtain solutions
that are uniformly distributed on the Pareto front. Elitism is
implemented in MOEAs to keep the nondominated solutions in
the population during the search. These three issues have been
addressed in various ways, and thereby numerous MOEAs
have been proposed (see [24]–[29]).
Despite the lack of research on MO-CARP, evolutionary
multiobjective combinatorial optimization has attracted a lot
of interest. Ehrgott et al. gave a survey of multiobjective
combinatorial optimization problems [30], and introduced the
characteristics of the problems and nominated MOEAs as one
available methodology. Some examples of the approximative
solution methods in multiobjective combinatorial optimization
problems were presented in [31], including various MOEAs,
simulated annealing and tabu search. In addition to directly
using the traditional MOEAs, some researchers considered
combining the MOEA framework with local search to pursue
enhanced performance. For example, Ishibuchi et al. proposed
a genetic local search for solving the flowshop scheduling
problem [32]. Jaszkiewicz proposed a genetic local search
framework for multiobjective combinatorial optimization prob-
lems to determine the weight vectors used in the weighted
sum approach for aggregating the objective functions during
the local search, and successfully applied it to the traveling
salesman problem [33] and the 0/1 knapsack problem [34]. Tan
et al. developed a MOEA for solving a multiobjective vehicle
routing problem in [35]. The algorithm incorporates two
problem-specific heuristics for local exploitation. However,
due to different structures of combinatorial optimization prob-
lems, it is often difficult to directly apply a MOEA developed
for one problem to another. For the same reason, although
traditional MOEAs have shown satisfactory performance on
numerical optimization benchmark test functions, they do not
necessarily guarantee good performance on MO-CARP. First,
our preliminary studies showed that the problem natures of
SO-CARP such as the discrete search space, the lack of a nat-
ural definition of neighborhood and various constraints made
successful algorithms for numerical optimization problems
failed on SO-CARP. This phenomenon may also occur in the
multiobjective case. Second, the shape of the Pareto front can
directly influence the performance of MOEAs [36]. Therefore,
the difference between the shape of the Pareto fronts of MO-
CARP and the numerical test functions makes the performance
of an existing MOEA in the case of MO-CARP unpredictable.
In general, MO-CARP can be solved from two different
directions. One is to extend an approach for SO-CARP to a
multiobjective one, and the other is to directly use an existing
MOEA by employing the problem-specific solution represen-
tation and operators. Lacomme et al. followed the former
direction in [23], while the latter direction has been overlooked
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4 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
so far. Both methods have their advantages and disadvantages,
and are actually complementary to each other. When extending
a SO-CARP approach, the strength of searching in the com-
plicated solution space can be inherited. However, the EMO
issues need to be addressed appropriately. On the other hand,
when applying an existing MOEA, it is difficult to search
effectively in the solution space of MO-CARP, although the
EMO issues are deeply considered. Therefore, it is reasonable
to increase the synergy between the two directions so that
both of their drawbacks can be overcome. In this paper, we
consider incorporating an existing SO-CARP approach and
various strategies proposed for EMO issues. By this means, the
hybridized algorithm will be strong in both searching within
the solution space and addressing the EMO issues. In order to
accomplish this, it is necessary to evaluate the EMO strategies
in the context of MO-CARP, as will be presented in the next
section.
III. EMO Issues in MO-CARP
MO-CARP is a combinatorial problem that tries to find
a set of Pareto optimal feasible solutions in a discrete and
finite solution space subject to a number of constraints. The
hybridization of EA with local search has been reported to be
quite efficient for solving combinatorial problems including
SO-CARP (see [18], [22]). This hybridized approach is also
called MA. When combining a MOEA with local search,
a new important issue arises. That is, how to identify a
solution in the neighborhood to replace the current solution.
Usually, the best solution in the neighborhood is selected, and
thus the issue can be seen as identifying the best solution in
the neighborhood. Therefore, when solving MO-CARP with
evolutionary algorithms (EAs), one may have to consider
the following four important issues: 1) fitness assignment;
2) diversity preservation; 3) elitism; and 4) identifying the
best neighboring solution during local search. The first three
issues are commonly considered in EMO, while the last one
must be addressed when local search is employed. The existing
strategies for addressing these issues are evaluated in the case
of MO-CARP one by one.
A. Fitness Assignment in MO-CARP
The existing strategies for fitness assignment in EMO can
be categorized into three types: 1) the criterion-based (see
[27]); 2) domination-based (see [24]); and 3) decomposition-
based (see [29]) methods. Previous studies on numerical
test functions showed that the criterion-based methods will
overlook the intermediate regions of the Pareto front, while the
domination-based methods will not. Since in criterion-based
and domination-based methods, the fitness of a solution only
depends on the values of objective functions, the conclusions
drawn from numerical test functions should still hold in the
case of MO-CARP. Hence, domination-based methods are
more appropriate than criterion-based methods in this context.
On the other hand, the decomposition-based methods are
based on the assumption that each Pareto optimal solution
can be seen as the optimal solution to a scalar optimization
subproblem. However, this is not true in the case of MO-
CARP. In fact, there usually exist solutions which are not
optimal for any weighted sum of the objectives in MO-CARP
[37]. Furthermore, due to the discreteness of the Pareto front
in MO-CARP, one Pareto optimal solution may be the optimal
solution of multiple decomposed subproblems. The search
process may be hindered since most of computing resources
may be wasted to find the same Pareto optimal solution.
Therefore, the decomposition-based methods may not perform
well in MO-CARP.
B. Diversity Preservation in MO-CARP
The niching technique, cell-based methods and crowding
distance method are three typical existing strategies for di-
versity preservation. They are mainly based on preventing
solutions close to each other from appearing simultaneously
in the population. Therefore, they are expected to be able
to maintain diversity in MO-CARP as well. Among them,
the performance of the niching technique and cell-based
methods are parameter-dependent, i.e., they largely depend on
the parameters such as the sharing parameter in the niching
technique and the cell size in the cell-based methods. The
performance of the crowding distance method is expected to
have a small variance since it has no user-defined parameter. In
addition to the above three strategies, algorithms like MOEA/D
utilize an implicit strategy to maintain diversity. That is, the
diversity is naturally preserved by the “diversity” among sub-
problems [29]. However, this is based on the assumption that
different subproblems can reach different optimal solutions.
In MO-CARP, one Pareto optimal solution can be the optimal
solutions to multiple subproblems. As a result, the diversity
can no longer be maintained in this way.
C. Elitism in MO-CARP
The elitism mechanism can be implemented by either stor-
ing the nondominated solutions in an archive or combining the
parents and offsprings for selection. The archive strategy can
be further divided into two types: the solutions stored in the
archive do or do not influence the search process. There is no
big difference when they are adopted in the test functions and
MO-CARP. Therefore, they can be applied to MO-CARP in
exactly the same way as to the test functions.
D. Evaluating Solutions During Local Search in MO-CARP
Identifying the best neighboring solution is essentially
equivalent to assigning fitness to each solution and then
selecting the one with the best fitness. Therefore, this issue
can be examined from the perspective of fitness assignment.
The criterion-based and domination-based strategies divide
the solutions into different fronts, each of which consists of
solutions with the same fitness. In this way, one can hardly tell
which solution in each front is the best one. The only available
strategy is to use decomposition-based methods. When solving
each decomposed scalar subproblem, the best solution can
be easily identified during local search. In addition to the
decomposition-based methods, aggregating objective functions
into a single one has been a commonly used idea in the
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MEI et al.: DECOMPOSITION-BASED MEMETIC ALGORITHM FOR MULTIOBJECTIVE CAPACITATED ARC ROUTING PROBLEM 5
literature (see [32], [33]). This method is usually faster than
the domination-based methods, but its performance largely
depends on the weight vector.
Based on the evaluations of the existing strategies, one can
either select an existing MOEA for solving MO-CARP or
design a specialized algorithm according to practical require-
ments. Based on the previous discussions, it can be seen that
the existing MOEAs other than MOEA/D are able to address
the first three issues well. MOEA/D is the only algorithm that
can address the last issue in MO-CARP because its distinctive
decomposition-based framework provides a natural way to em-
ploy local search. Therefore, we propose a multiobjective MA,
named D-MAENS, to address all four issues properly. In order
to keep the algorithm simple, among the available strategies,
the ones with the least parameters are employed. Specifically,
D-MAENS employs the fast nondominated sorting procedure
of NSGA-II for fitness assignment, the crowding distance
method of NSGA-II for diversity preservation, both of the two
existing strategies for elitism and the decomposition strategy
of MOEA/D for identifying the best solutions during local
search. Next section describes the full details of D-MAENS.
IV. D-MAENS
D-MAENS adopts a decomposition-based framework which
is analogous with that of MOEA/D. It decomposes the original
MO-CARP into a number of scalar subproblems using the
weighted sum approach with a set of uniformly distributed
weight vectors. A population of individuals (solutions) whose
size equals the number of the subproblems is maintained. Each
individual in the population corresponds to a unique subprob-
lem. When solving each subproblem, the evolutionary opera-
tors and local search are applied to the individuals correspond-
ing to the neighboring subproblems of the current one. The
crossover operator and local search process are exactly those
employed in MAENS. We will describe the decomposition-
based framework of D-MAENS in Section IV-A. Then, the
MAENS ingredients will be briefly introduced in Section IV-B.
A. Decomposition-Based Framework
In the decomposition framework, the original MO-CARP
is first decomposed into a number of SO-CARPs by the
weighted sum approach. To be specific, given the objective
vector F(x)=(f
1
(x),... ,f
n
(x)) and a weight vector
λ =(λ
1
,... ,λ
n
), the objective function of a subproblem is
stated as
g
ws
(x|λ)=
n
i=1
λ
i
f
i
(x).
Suppose there are N weight vectors λ
1
,... ,λ
N
, the orig-
inal MO-CARP is thus decomposed into N SO-CARPs.
The objective function of the ith subproblem is g
ws
(x|λ
i
).
D-MAENS maintains a population X = {x
1
,... ,x
N
} through-
out the optimization process. At each generation, the popula-
tion is evolved in the following steps. First, each subproblem
is assigned a unique solution x ∈ X, which is called its rep-
resentative. Then, N subpopulations are constructed, each for
a subproblem. The subpopulation of a subproblem associated
with weight vector λ
i
is composed of the representatives of
the T subproblems whose associated weight vectors are the
T closest (in term of Euclidean distance) weight vectors to
λ
i
, where T is the size of subpopulation. As stated in [29],
the optimal solution of the ith subproblem should be close
to that of the jth subproblem if λ
i
is close to λ
j
. Thus, the
information of the subproblems whose weight vectors are close
to that of the current subproblem should be helpful for solving
the current subproblem. Second, one new solution is generated
for each subproblem. For the ith subproblem, two parents
are selected from the subpopulation associated with it, and
then crossover and local search of MAENS are applied to the
parents to generate an offspring y
i
. By repeating this procedure
for all subproblems, an offspring population Y = {y
1
,... ,y
N
}
is generated. Finally, the solutions in both X and Y are
combined together and then sorted by the fast nondominated
sorting procedure and the crowding distance method. The
best N solutions are kept to form the population X in the
next generation. The detailed steps of the decomposition-based
framework are given as below.
Input:
1) a MO-CARP instance P ;
2) a stopping criterion;
3) the number of decomposed subproblems, denoted as N;
4) a number of uniformly distributed weight vectors
λ
1
,... ,λ
N
;
5) the size of the neighborhood of each subproblem,
denoted as T .
Output: A set of nondominated solutions X
∗
.
Step 1: Initialization.
a) Set X
∗
= ∅.
b) Decompose the original MO-CARP P into a set of SO-
CARPs {P
1
,P
2
,... ,P
N
} with λ
1
,... ,λ
N
.
c) Initialize a population X = {x
1
,... ,x
N
} randomly or by
problem-specific methods.
d) Compute the Euclidean distance between each pair of
weight vectors. Then, get the neighborhood B(i)=
{i
1
,... ,i
T
} for each P
i
, so that λ
i
1
,... ,λ
i
T
are the T
closest weight vectors to λ
i
(including λ
i
itself).
Step 2: Search for new solutions.
a) Assign each subproblem a unique representative x
r
i
∈ X.
b) Construct a subpopulation X
i
= {x
r
i
1
,... ,x
r
i
T
} for each
subproblem.
c) Set i =1.
d) Randomly select two solutions x
r
k
and x
r
l
from X
i
.
e) Apply the crossover and local search operators of
MAENS to x
r
k
and x
r
l
to generate y
i
for P
i
.
f) Remove from X
∗
all the vectors dominated by F (y
i
).
Insert F (y
i
)inX
∗
if no vector in X
∗
dominates it.
g) Set i → i +1. Ifi N, go back to Step 2d.
h) Sort the solutions in the set Z = X ∪ Y by the fast
nondominated sorting procedure and crowding distance
approach of NSGA-II [24]. Then, let X be the set of the
best N solutions in the sorted Z.
Step 3: Termination. If stopping criteria are satisfied,
terminate the algorithm. Otherwise, go to Step 2.
