IEEE Computational Intelligence Magazine - November 2021 - 32

can attain better optimization results than MOMFEA-II.
This is because the proposed GRA can effectively identify
the task to enjoy additional computational resources for
better enhancing the overall performance. As for the complex
problems and many-task problems listed in Table VIII
and Table IX, the proposed GRA can generally outperform
the other algorithms, since it can identify the optimization
difficulties of the tasks and intelligently allocate
the resources.
E. Sensitivity Analysis of Resource Allocation Intensity
One of the major contributions of GRA lies in the adjustable
resource allocation intensity. Recall that, in Algorithm 2 the
allocation intensity RAI is equivalent to the parameter RAP
left for users to tune, it is necessary to investigate the behavior
of GRA with various settings of RAP herein.
To be specific, the MSS comparison on 10 complex multitasking
MOO problems in 30 independent runs has been
shown in Table VII, with RAP spanning from 0.35 to 0.65.
Meanwhile, the rankings of each parameter choice across
the seven settings have also been depicted in Fig.4. It is
interesting to note that the performance of different parameters
on various problems can be unstable. As demonstrated
in Table VII, RAP of 0.35, which is the smallest choice,
ranks highest in CPLX4, CPLX6, and RAP of 0.65, ranks
highest in CPLX1, CPLX2, and CPLX10. Based on the
observations above, it can be indicated that for different
EMTO problems different resource allocation intensities are
favored, and hence it naturally implies that a superior
resource allocation mechanism should be able to adjust the
resource allocation intensity flexibly, which in turn justifies
the contribution of our controllable allocation intensity as
introduced in Section III.B7.
In spite of the various performance levels of different param7
6
5
4
3
2
1
0.35
0.4 0.45 0.5 0.55 0.6 0.65
FIGURE 4 Rankings of GRA with varying RAP parameters.
eters choices, some parameters are still recommended for the
proposed GRA. As demonstrated in Table VII and Fig.4, RAP
of 0.5 and 0.55 generally can have relatively stable performances
compared with others. In Table VII, RAP of 0.55 and RAP of
0.5 can achieve the best sum MSS score, -6.70, and the second
best score, -0.86. Also, as shown in Fig.4, the deviation magnitude
in rankings for RAP of 0.55 is the lowest, and that of RAP
in 0.5 is also lower compared with others. Therefore, these two
parameter values are recommended, and RAP of 0.55 is adopted
in this paper.
TABLE VIII Mean standard score comparison with adaptive
methods on multitasking MOO complex problems over 30
independent runs.
PROBLEMS
CPLX1
CPLX2
CPLX3
CPLX4
CPLX5
CPLX6
CPLX7
CPLX8
CPLX9
CPLX10
SUM
RANK
GRA
−7.4769E-01
−1.3199E+00
−8.0401E-01
−1.0158E+00
−1.4135E+00
−7.7649E-01
−7.9105E-01
−9.1896E-01
−7.5971E-01
−4.4113E-01
−8.9882E+00
1
MOMFEA-II
1.4134E+00
1.0990E+00
1.4094E+00
1.3600E+00
7.4452E-01
1.4117E+00
1.4104E+00
1.3675E+00
1.4125E+00
6.9916E-01
1.2327E+01
3
EMTIL
−6.6567E-01
2.2089E-01
−6.0536E-01
−3.4422E-01
6.6900E-01
−6.3519E-01
−6.1933E-01
−4.4850E-01
−6.5278E-01
−2.5803E-01
−3.3392E+00
2
F. Another Implementation: GRA based on MP-MFEA
To clarify the generality of the proposed GRA on multi-population
based algorithms, herein GRA and MTO-DRA are
integrated with MFEA under the multi-population settings
(MP-MFEA). To mimic the performance of the single-population
MFEA, according to [47], the rmp value for MP-MFEA
is set to 0.15, and other parameter settings of MP-MFEA are
the same as those of MFEA introduced in Section IV.C. For
more implementation details of multi-population based
MFEA, one can refer to [48, 49]. Since the base optimizer,
MP-MFEA, has a weaker searching ability than MMDE [31],
the resultant convergence trends across various tasks can also
be different, which makes it improper to apply the same
parameters in MMDE/GRA into MP-MFEA/GRA. Therefore,
herein we set the RAI of GRA to 0.40, since in twotask
problems where both tasks have different complexities, a
large RAI value can lead to unequal resource allocation for
the more complex task, considering the fact that the base
optimizer has the relative weak searching ability. Furthermore,
MTO-DRA is also extended to multi-objective settings using
our attainment function based method, and it is notable that
the RAI of MTO-DRA is fixed, which means it cannot adapt
to different base optimizers.
7 Recall that, MTO-DRA has an unadjustable resource allocation intensity.
32 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
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