Computational Intelligence - February 2014 - 51

high traffic demand had been selected,
the m m TPs with medium traffic
demand and the m l TPs with low traffic
demand were selected with the same
procedure.
The positions of CSs were also randomly selected in the set of crossing
points of the regular grid. The adopted
procedure was similar to the previous
one. That is, each grid point was
assigned a weight that equals to the sum
of the traffic demand of neighboring
TPs. After each iteration, the weight of
the selected grid point was set at zero.
To make the numerical results easily
readable, we adopted the same cost for
all BSs and separately reported the system capacity (i.e. the number of the user
served) and the installation cost (i.e. the
number of the active BSs). In the following sections, we present the numeri-

2

1
1.
2
1.
4
1.
6
1.
8

0
0.
2
0.
4
0.
6
0.
8

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

FIgure 5 Synthetic but realistic instances,
where : High; : Medium; : Low; *: CS.

2,800
#TD

Input: Q: a set of individual solutions
their objective values.
Output: P1, f, PT .
for k ! 1 to T do
Initialize Pk as the solutions in Q
whose normal objective values are
in X k;
if Pk 1 N then
randomly select N - Pk solutions
from Q and add them to Pk .
end if
if Pk 2 N then
rank the solutions in Pk using the
constraint nondominated sorting
method [31] and remove from Pk
the S - Pk lowest ranked solutions.
end if
end for

2,750
2,700
2,650
2,600
12

14

16

18 20
#BS

22

24

FIgure 6 The result obtained by the algorithm on a medium size instance, where
#TD denotes the number of covered the
traffic demand, while #BS is the number of
the active BSs.

cal results obtained with the medium
and large size instances.
B. The Parameters of the
Instances and Algorithm

A medium size and a large size instances
were pseudo-randomly generated by the
tool described above. Specifically, a
2000 #  2000 meter service area was
divided into 400 square regions of
100 # 100 meter in the medium size
instance. The numbers of the regions
with high, medium and low traffic
demand were m h = 50, m m = 150, m l =
200, respectively, and the corresponding
numbers of the traffic demand per
region were 12, 6 and 2 in the uplink
and 7, 3, 1 in the downlink. Then, the
traffic demand was 1900 in the uplink
and 1000 in the downlink. The total
traffic demand was 2900 in the system.
Also, 100CSs, i.e. n = 100 were pseudorandomly generated by using the above
described method.
For the large size instance, a 3000 #
3000 meter service area was divided into
900 square regions of 100 # 100 meter.
The numbers of the regions with high,
medium and low traffic demand were
m h = 100, m m = 400, m l = 400, respectively, and the corresponding numbers of
the traffic demand per region were 12, 5
and 2 in the uplink and 6, 3, 1 in the
downlink. Then, the traffic demand was
4000 in the uplink and 2200 in the
downlink. The total traffic demand was
6200 in the system. 150 CSs were
pseudo-randomly generated by using the
above described method.

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2

2,850

1
1.
2
1.
4
1.
6
1.
8

2,900

0
0.
2
0.
4
0.
6
0.
8

Algorithm 4: Allocation of individuals
to subpopulations.

FIgure 7 The distribution of the active BSs
of the fifth solution from the left in Fig. 6,
where represents the active BSs, and *
represents the CSs.

We conducted the algorithm on
these instances with the number of the
subproblems T = 10 and the size of the
subpopulation N = 10. For the medium
size instance, the maximum number of
generations was set at max _gen = 200,
while max _gen = 300 for the large
size instance.
C. Computational Results

1) experimental results on
the medium Size Instance
Fig. 6 plots the final solutions obtained
by the proposed algorithm. There are a
set of solutions obtained in a single run.
These solutions can be roughly divided
into two parts. The first part of solutions,
represented with red point, require less
BSs, while the other solutions, represented with blue point, have a higher
coverage rate. Fig. 7 plots the distribution of the active BS of the fifth solution
from the left in Fig. 6.
2) experimental results
on the large Size Instance
Fig. 8 plots the final solutions obtained
by the proposed algorithm. Similar with
the medium size instance, there are also
a set of solutions obtained in a single
run. These solutions can be roughly
divided into two parts. The first part of
solutions, represented with red point,
require less BSs, while the other solutions, represented with blue point, have
a higher coverage rate. Fig. 9 plots the
distribution of the active BS of the
eighth solution from the left in Fig. 8.

February 2014 | Ieee ComputatIonal IntellIgenCe magazIne

51



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