Computational Intelligence - February 2014 - 50
Algorithm 3: MOEA/D-M2M for
WCDMA network planning problem.
Input: The WCDMA planning problem;
max_gen: the maximum number of
generations;
T : the number of the subproblems;
N : the size of the subpopulation.
Output: All the nondominant solutions in
T
jk = 1 P k .
Initialization: Generate 2NT initial individuals " y 1, g, y 2NT , by the problem specific method described above, evaluate
their objective values and then use these
initial individuals to set P1, f, P T . The current generation set: gen = 1.
while gen 1 max_gen do
Generation of New Solutions:
Set P = 0 ;
for k ! 1 to T do
for each y d Pk do
Randomly choose yl from Pk ;
Apply crossover and mutation
operators on y and yl to generate a new solution z ;
Compute the objective values of z ;
P: = P j " z ,
end for
end for
Updating the subpopulations:
Q: = P , ^jTk = 1 Pkh;
Use Q to set P1, f, PT .
end while
Find all the nondominated solutions in
jTk =1 Pk and output them.
in CS 2 with configuration combination
level 10. The 10th row of the orthogonal
array L 36 (6 6) is (2,4,5,6,1,2). The information of the configuration parameters
of BS 2 is available. That is, h = 1,
b 1 = 9, b 2 = 12, b 3 = 15, c = 0 and
tp = 1.2. Moreover, if there is no BS
installed in CS j, y j = 0. We have that
y j ! {0, 1, g, l} for j ! J.
component by selecting a random number between 1 and n. If the hth component of y c is selected to be mutated, then
y ch = k ,
where k is randomly selected from
{0, 1, g, l} .
C. MOEA/D-M2M
Multiobjective evolutionary algorithm,
as one of the most powerful tool to
solve the complex optimization problems, especially the combinatorial optimization problems, has been widely
used in the various fields [23]-[26]. A
number of multiobjective evolutionary
algorithms have been proposed in the
past few years, such as NSGA-II [31],
MOEA/D [32], MOEA/D-M2M [27]
and so on. MOEA/D-M2M has strong
ability to maintain the diversity of the
population [27]. There are a few of
Pareto solutions in this optimization
problem, the diversity of the population
is more important, we therefore take
MOEA/D-M2M to solve the problem
described above.
In MOEA/D-M2M, the multiobjective optimization problem is decomposed
into a number of simple multiobjective
optimization subproblems, which are
then solved in a collaborative manner. T
direction vectors v 1, f, v T are uniformly chosen from the unit circle in the
first quadrant, where T is the number of
the subproblems. Then, the first quadrant
R 2+ is divided into T subregions
X 1, f, X T , w h e re X k (k = 1, f, T)
defined as
B. Crossover and Mutation
We choose the single-point crossover to
create the offsprings. Given two solutions, this crossover operator creates a
random integer number u between 1
and n - 1 and swaps u leftmost bits of
each solution. For example, given parents y 1 = (0, 10, 30, 0, 5) and y 2 =
(0, 0, 0, 11, 12), the children will be
y c = (0, 10, 0, 11, 12) if u = 2.
The mutation operator is used to
escape a possible local optimum and find
a new neighborhood with a potentially
more promising solution. For each solution y c generated by the crossover operation, our implementation alters each
50
X k = {u ! R 2+ | G u, v k H # G u, v j H}
for any j = 1, f, T. G u, v j H is the acute
angle between u and v j . In other words,
u is in X k if and only if v i has the
smallest angle to u among all the T
direction vectors. For each subregion,
there is a subpopulation with N individuals to optimize it.
The initial population is generated as
f o l l ow s : Fo r e a c h e l e m e n t y j,
(j = 1, 2, g, n), of individual solution y,
if rand < 0.4 then y j = 0, where rand
is a random number generated by a uniform random number generator in [0,
IEEE ComputatIonal IntEllIgEnCE magazInE | FEbruary 2014
1]. That is, there is no BS installed in CS
j; Otherwise, y j is set at a random integer number within the range of
{1, 2, g, l} .
In each generation, MOEA/DM2M maintains T subpopulations:
P1, f, PT . Each subpopulation is utilized to optimize one subregion in the
objective space. Specifically, the MOEA/
D-M2M works as Algor ithm 3.
MOEA/D-M2M utilizes a number of
individual solutions to initialize and
update P1, f, PT . This procedure can be
realized by Algorithm 4, where Pk is
the number of the individuals in Pk .
IV. Experimental Results
This section will present some numerical results obtained from the application
of the MOEA/D-M2M to some synthetic but realistic instances. The aim of
these numerical experiments is not only
to demonstrate the effectiveness and
rationality of the proposed model, but
also to show the effectiveness of the
MOEA/D-M2M to solve the combinatorial optimization problems.
A. Instance Generator
These instances were generated by the
instance generator tool described in
[12]. It simulates the traffic demand distribution of service area and the CSs'
location which is usually selected by service providers considering traffic intensity in the area. A square service area
D # D was considered in each instance.
As shown in Fig. 5, we divided the service area into d # d smaller regions and
regarded the center of each region as a
TP. We assumed that the numbers of the
TP with high, medium and low traffic
demand is m h, m m and m l, respectively.
The traffic demand of each TP was
given as follows: We first selected the
TPs which were assigned with the high
traffic demand. A weight ~ i with initial
value 1 was associated to each TP i ,
i = 1, 2, g, m. TPs were iteratively
picked with a probability p i =
m
~ i / i 1 ~ . After each iteration, the
=
i
weight ~ i of the selected TP was set at
zero, and the weights of the neighboring
TPs which had not been selected were
increased by 1. Once all m h TPs with
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