Computational Intelligence - February 2017 - 38
TABle 3 The road traffic density distributions
in the 9 × 9 network.
rOAd
TyPe
NuMBer
TrAFFic
FlOw
cellS crOSSed By rOAdS
1. MAIn
rOAd
4
[200, 400]
{10, 11, 12, 21, 22, 23,
24, 25, 26, 35, 44, 54, 63}
{9, 8, 17, 25, 24, 34, 51,
59, 68, 76, 75, 74, 73}
{5, 14, 22, 31, 40,
41, 42, 51, 60, 70, 71, 80}
{3, 13, 21, 30, 39, 38, 47, 55, 64}
2. STreeT
7
[100, 250]
{2, 3, 4, 5, 6, 7, 8, 9}
{3, 12, 20, 19, 28, 29,
37, 46}
{6, 15, 23, 32, 40, 41}
{33, 34, 35, 44, 53, 61, 62}
{59, 51, 42, 34, 35, 36, 27, 18}
{77, 69, 60, 61, 53, 43, 44, 45}
{50, 49, 57, 66, 74, 75}
3. Alley
7
[50, 150]
{28, 37, 46, 47, 48, 56, 55,
64, 73}
{1, 2, 3, 12, 20, 30, 31, 32}
{16, 7, 8, 9}
{55, 56, 57, 58, 59, 60, 52}
{65, 66, 67, 76}
{78, 70, 71, 72}
{79, 80, 81}
solutions obtained by solving the multi-objective model is
calculated using the same weights of the single-objective
model. The solution with the minimal weighted sum value is
compared with the best solution found by the single-objec-
tive model. Tables 4, 5 and 6 present their location update
cost, paging cost and their weighted sum cost for the ten different groups of parameters. The results show that the multiobjective model can significantly outperfor m the
single-objective model in terms of the weighted sum. It lives
up to our expectation that multi-objective formulations for
the TA planning problem are more effective than the single-objective model.
A multi-objective optimization problem has a set of Pareto
optimal solutions, and their images in the objective space are
called Pareto Front (PF). The convergence of an EMO algorithm can be measured by the closeness of its obtained solutions to the PF. Since each element of PF represents a trade-off
among the objectives, the diversity along the PF is also important when measuring the quality of obtained solutions. In our
experiments, the quality of obtained solutions by EMO algorithms is measured by the HyperVolume (HV)-metric which
can measure the convergence to the PF and the diversity along
the PF at the same time [38]. Let y ) = (y )1, f, y m) ) be a reference point in the objective space which is dominated by any
point in the PF, and S be a set of obtained approximation to
the PF. Then the HV-metric value of S (with regard to the reference point y )) is the volume of the region which is dominated by S and dominates y ) . The reference point
y ) = (8 ) 10 4, 3 ) 10 5) is used in our study. The larger the HVmetric is, the better the algorithm performance is. Considering the randomness of EMO algor ithms, M2M and
MOEA/D both run 15 times for each test network. The best,
worst, median, mean and standard deviation of HV-metric values in the 15 independent runs for each network are shown in
Table 7. It indicates that the solutions obtained by M2M have
both better convergence and diversity than MOEA/D in
terms of the HV-metric. To investigate the sensitivity of solutions quality to the setting of maximum number of generations, we plot the HV-metric of solutions obtained by the
proposed algorithm with different number of generations for
network 1 in Fig. 7.
TABle 4 The Location Update Cost (LUC), Paging Cost (PC) and Weight Sum Cost (WSC) of network
1 for 10 groups of test parameters.
38
NeTwOrk 1
MulTi-OBjecTive MOdel
grOuP
luc
Pc
SiNgle-OBjecTive MOdel
wSc
luc
Pc
wSc
1
6550
16996
82496
6680
23579
90379
2
6220
19569
81769
5958
22583
82163
3
5526
25999
81259
8114
17573
98713
4
5682
21991
78811
6656
17112
83672
5
5024
21272
71512
5730
24587
81887
6
6004
18638
78678
7736
16697
94057
7
6214
23456
85596
5862
24360
82980
8
5462
22799
77419
6904
18388
87428
9
4802
26519
74539
5968
24574
84254
10
5554
19359
74899
6120
25043
86243
IEEE ComputatIonal IntEllIgEnCE magazInE | FEbruary 2017
Table of Contents for the Digital Edition of Computational Intelligence - February 2017
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