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Although the results of PAIA for ZDT4 are much better than for other algorithms', it
has not fully converged to the true Pareto front. This result can be further improved
by using more iteration steps and such results can be found in experiment 2.
Δ
Table 2.
Mean and variance values relating to the diversity measure
ZDT1
ZDT2
ZDT3
ZDT4
Algorithm
σ
2
σ
2
σ
2
σ
2
Δ
Δ
Δ
Δ
NSGA II
0.4633
4.16e-2
0.4351
2.46e-2
0.5756
5.08e-3
0.4795
9.84e-3
SPEA
0.7302
9.07e-3
0.6781
4.48e-3
0.6657
6.66e-4
0.7321
1.13e-2
VIS
0.5420
8.25e-3
0.6625
2.58e-2
0.6274
1.60e-2
0.1011
1.37e-3
PAIA
0.3368
1.10e-3
0.3023
7.07e-4
0.4381
1.50e-3
0.3316
1.20e-3
Table 3.
Final population size and evaluation times of PAIA
Test
suite
Final Population
Evaluation Times
Mean Max/min
Mean Max/min
ZDT1
96
101/87
25372
26467/24494
ZDT2
101
106/96
25950
26649/25371
ZDT3
94
102/89
25365
26155/24587
ZDT4
96
103/85
25910
26654/25203
4.2 Experiment 2 (Full Convergence)
In this experiment, the number of iterations was set to 180 for ZDT1 and ZDT2, to
280 for ZDT3 and to 500 for ZDT4. Other parameters remained unchanged.
Iteration= 280 (ZDT3)
It
e
rati
on
= 1
8
0 (Z
D
T1)
Iteration= 180 (ZDT2)
Iteration= 500 (ZDT4)
1
1
1
1.4
0.9
0.9
0.8
1.2
0.8
0.8
0.6
1
0.7
0.7
0.4
0.6
0.6
0.8
0.2
0.5
0.5
0
0.6
0.4
0.4
-0.2
0.3
0.3
0.4
-0.4
0.2
0.2
0.2
0.1
0.1
-0.6
-0.8
0
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0. 5
0.6
0. 7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 3.
Pareto solutions obtained by PAIA on ZDT1~ZDT4
Through this experiment, it was found that PAIA possesses very fast convergence
properties. For ZDT1 and ZDT2, 180 iterations were enough for convergence, and for
ZDT4 500 iterations were sufficient. For all the four test problems, both algorithms
obtained good performances (except ZDT4 in VIS) in terms of both metrics. From
Table 5, one can see that PAIA generally uses fewer evaluations to achieve good
results. Although it used 46899 evaluations to fully converge, it only used 25910 (see
Table 3) evaluations to obtain similar results as those produced by VIS (see Table 4).
This is due to two reasons: 1) PAIA only preserves necessary Abs during each itera-
tion step so that only the necessary evaluations are carried out; 2) PAIA uses adaptive
clone size so that only the necessary clone size is assigned to each selected Ab. One
