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Suppression regulates the population so that it is adaptive to the search process. New-
comers are used to further increase the diversity of Abs (it is not used in PAIA and is
included here for completion). It is argued here that each part of the framework can be
implemented in various means; while the basic structure remains unchanged.
4 Experiments
The proposed approach is compared to two well-known algorithms-NSGA II [16] and
SPEA [21], and another immune algorithm-VIS proposed by Freschi
et al.
[7]. By
comparing with NSGA II and SPEA, it is shown that PAIA is a valuable alternative to
standard algorithms; by comparing with VIS, the difference between these two im-
mune algorithms is identified. ZDT1~ZDT4 test suite [16] is used for such a compari-
son. These test functions have two objectives and represent the same type of problems
with a large decision variable space, a concave and discrete Pareto front, and many
local optima. Results of NSGA II and SPEA are taken from [16] with a population
size of 100 and a maximum of 250 generations. This gives a total number of 25000
evaluation times. To make the comparison fair, VIS is also run using the same setting
(26000 for ZDT4). For PAIA, although the population is adaptive the final population
can be controlled by
so that the final
population size and evaluation times are around 100 and 25000 respectively. NSGA II
failed to converge for ZDT4 even with a larger number of evaluation times, while on
the other hand, although some algorithms may not fully converge within 25000
evaluations they have no difficulty to converge using larger evaluations. For this rea-
son, one can also compare PAIA and VIS when both have fully converged (otherwise,
it is only the best results to be used). Two performance metrics, namely the Genera-
tional Distance (GD) and the Spread
. Hence, one can set an adequate value for
σ
σ
Δ
[16], are used and are defined as follows [11]:
•
Generational Distance:
GD measures the closeness of the obtained Pareto solu-
tion set
Q
from a known set of Pareto-optimal set P
*
.
∑
=
Q
m
1
m
.
(
d
)
i
(6)
GD
=
i
1
Q
For a two-objective problem (m=2),
d
i
is the
Euclidean
distance between the solu-
tion
i
∈
Q
and the nearest member of P
*
.
Δ
•
Spread:
measures the diversity of the solutions along the Pareto front in the
final population.
__
∑
M
m
∑
Q
e
m
d
+
d
−
d
.
i
=
1
i
=
1
(7)
Δ
=
__
∑
M
m
d
e
m
+
Q
d
=
1
where
d
i
is the distance between the neighbouring solutions in the Pareto solution
set
Q
.
__
d
is the distance between the extreme solu-
tions of P
*
and
Q
along the
m
th objective.
d
is the mean value of all
d
i
.
e
