IEEE Computational Intelligence Magazine - November 2021 - 29

point p, and the cardinality of a given set, respectively. Thereafter,
to integrate IGD values of all the tasks, mean standard score
(MSS) is utilized for comparison and ranking, as introduced in
[44]. Given K tasks, T1, T2,..., TK, the average IGD value and
corresponding standard deviation for task k across all the algorithms
are μk and
vk , respectively. Then, the MSS of a certain
algorithm can computed as (17):
MSS
= 1 / vk
K k=1
K Ikk
- n
(17)
where Ik is the IGD value of the given algorithm on task k. It is
notable that smaller MSS values indicate better performance of
the given algorithms.
C. Comparison Algorithms and Parameter Settings
This section presents the detailed parameter settings for
comparative studies. To guarantee a fair comparison, the population
size and the maximum evaluation times are set to
65 K)
and Ke16)
tasks. The other parameter settings for each algorithm are
listed as follows.
1) GRA
❏ Allocation interval, GD = 20
❏ Sigmoid parameter, S = 1
❏ Transfer rate, ktr = 0.8
❏ Initial rmp,
rmp 005
ij, = .
❏ Nonlinear regression step size, M = 5
❏ Resource allocation probability, RAP = 0.55
❏ Base optimizer, MMDE [31]
2) MTO-DRA [40]
❏ Allocation interval, GD = 20
❏ Sigmoid parameter, S = 1
❏ Initial rmp,
rmp 005
ij, = .
❏ Nonlinear regression step size, M = 2
❏ Base optimizer, MMDE [31]
Remark 3. Since MTO-DRA is not applicable to MOO problems
due to the original IoI vector, we implement MTO-DRA using
our proposed normalized attainment function in this paper. Within the
MOO extension in MTO-DRA, the regression step size is 2,
because the original work [40] did not consider long-term tendency as
GRA does.
3) MMDE [31]
❏ Initial rmp,
rmp 005
ij, = .
❏ Local search frequency, every 3 generations
❏ Local search reward rate, m = 0.5 (0.8 for many-task)
❏ Constant c in JADE, c = 0.1
4) MOMFEA [43]
❏ Random mating probability, rmp = 0.3 [43]
❏ Crossover probability, Cr = 0.9
❏ Mutation probability, Mr = 0.1
1) Comparison with Baseline Methods
To illustrate the overall performance of GRA, baseline methods
MOMFEA and MFEARR are employed for comparison, in
the datasets of benchmark problems, complex problems, manytask
problems, of multitasking MOO, as depicted in Table I-III.
It can be shown in Table I that GRA can generally attain
better results in benchmark problems due to the searching
ability of MMDE and the allocation component in GRA.
According to Table I, the performance advantage of GRA can
be found in all the benchmark problems with various properties.
Moreover, in such problems, MFEARR can achieve
slightly worse IGD performance compared with MOMFEA,
which is inconsistent with the results indicated in the work
[35]. The reason for this is that in this paper the population
size is set to 65 for each task in all the methods, and thus the
related parameter settings may not be optimal, such as the parting
way threshold,
the complex problems illustrated in Table II and many-task
problems illustrated in Table III, GRA can outperform the
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 29
5) MFEARR [35]
❏ Random mating probability, rmp = 0.3
❏ Crossover probability, Cr = 0.9
❏ Mutation probability, Mr = 0.1
❏ Survival detecting window size, t = 153 [35]
❏ Parting way threshold, e = 0.01
6) MOMFEA-II [46]
❏ Crossover index, ch = 15
❏ Mutation index, mh = 15
7) EMTIL [27]
❏ Random mating probability, rmp = 0.3
❏ Crossover index, ch = 15
❏ Mutation index, mh = 20
❏ Local search probability, p = 0.3
, respectively, where K is the number of
D. Comparative Study
The comparative study is composed of three sections. The first is
to verify the superiority of GRA by comparison with a baseline
method MOMFEA and a baseline resource allocation method
MFEARR. The second is to clarify the effectiveness of GRA by
comparison with a state-of-the-art EMTO algorithm MMDE
and a state-of-the-art resource allocation algorithm MTODRA
that is implemented based on MMDE. Since GRA is
also designed on MMDE, the second comparison can serve as
component analysis to indicate the advantages of GRA over
the allocation component in MTO-DRA, and to justify the
advantage of the resource allocation component over the
MMDE without allocation components. Herein, it is notable
that MTO-DRA cannot be applicable in the context of MOO
but can be extended with our proposed MOO extension with
the normalized attainment function introduced in Section
III.D. The third is to compare the proposed GRA with the
recently proposed adaptive multitasking MOO algorithms.
e, in MFEARR. Furthermore, in terms of
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