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In this section, we experimentally analyze the effect of the multiple sampling strategy on the performance of IMMA. To carry out this study, we apply the IMMA with different sampling number L on the functions. We are interested in whether the multiple sampling strategy is beneficial to the exploration capability of IMMA. The algorithmic parameter settings in the experiments are M = 60. L is varied from 1 to 5. Table 4 lists the experimental results obtained by the IMMA with different L settings in terms of the best error values and NFEs. In the table, the best results are shown in bold type. The p values of the rank sum test on the results of L=2 and L=3 against the other L values are listed in Table 5. From Table 5 we see that:

  1. On the functions that cannot be solved successfully, the IMMA with sampling size L = 1 has the best performance in terms of error value on only one function (f27), but this performance difference is not significant. Moreover, it can be seen that the IMMA with a large sampling size (L=4, 5) does not necessarily result in better FEVs.

  2. In terms of efficiency (NFEs), the IMMAs with L=2, 3 require fewer fitness evaluations to reach the accuracy level on all the multimodal functions except f6,31 (functions f1 through f3 and f28 are unimodal).

Table 4:
NFEs obtained for the selected functions with n = 30 that can be successfully solved and the best error values for the functions that cannot be solved by the IMMA with varied sampling size L.
FunctionL=1L = 2L=3L=4L=5
AVGSDN      
f1 3.08e+03 5.35e+02 5.15e+03 3.28e+02 6.55e+03 6.27e+02 8.00e+03 5.43e+02 9.31e+03 5.46e+02 
f2 1.28e+03 8.01e+02 1.38e+03 1.17e+03 1.01e+04 1.03e+03 1.16e+04 2.38e+03 1.26e+04 3.71e+03 
f3 3.21e+04 2.63e+03 3.04e+04 2.11e+03 3.13e+04 2.57e+03 3.27e+04 3.48e+03 3.29e+04 2.88e+03 
f6 1.43e+04 8.91e+03 2.18e+04 1.60e+04 3.11e+04 3.03e+04 4.74e+04 4.94e+04 3.27e+04 2.48e+04 
f7 1.71e+04 5.88e+03 1.77e+04 1.49e+04 1.02e+04 6.28e+03 6.19e+04 7.96e+04 4.64e+04 5.82e+04 
f9 4.11e+04 1.38e+04 3.51e+04 1.07e+04 3.94e+04 1.46e+04 3.99e+04 1.12e+04 4.25e+04 1.67e+04 
f10 4.96e+04 1.61e+04 4.66e+04 1.39e+04 4.20e+04 1.61e+04 4.63e+04 1.57e+04 4.42e+04 1.33e+04 
f28 3.09e+03 6.15e+02 4.83e+03 3.64e+02 6.30e+03 3.56e+02 7.93e+03 4.23e+02 9.17e+03 6.05e+02 
f29 2.69e+04 6.39e+03 2.69e+04 6.25e+03 2.67e+04 6.21e+03 2.64e+04 5.91e+03 2.70e+04 6.63e+03 
f30 2.65e+04 9.18e+03 2.11e+04 5.24e+04 7.68e+03 3.28e+03 3.57e+04 6.89e+04 3.48e+04 6.37e+04 
f31 4.07e+04 1.42e+04 4.40e+04 9.10e+03 4.69e+04 1.46e+04 5.24e+04 1.83e+04 5.38e+04 1.26e+04 
f32 1.48e+04 3.64e+03 1.37e+04 2.56e+03 1.47e+04 3.74e+03 1.72e+04 3.56e+03 1.95e+04 4.06e+03 
f33 3.02e+04 1.60e+04 2.43e+04 1.25e+04 2.02e+04 9.15e+03 2.01e+04 9.74e+03 2.38e+04 1.37e+04 
f34 3.91e+04 2.00e+04 3.79e+04 2.67e+04 4.02e+04 2.72e+04 3.79e+04 1.49e+04 4.16e+04 2.11e+04 
f37 2.04e+04 4.34e+03 2.53e+04 9.95e+03 1.93e+04 1.65e+04 2.12e+04 1.41e+04 2.39e+04 2.32e+04 
AVGSDE      
f4 2.10e+05 1.94e+04 1.11e+05 1.63e+04 2.94e+04 6.16e+03 2.80e+05 3.18e+04 2.52e+05 2.18e+04 
f5 2.47e+03 5.46e+02 1.74e+03 3.19e+02 1.68e+03 3.27e+02 1.77e+03 3.17e+02 1.69e+03 2.37e+02 
f8 2.00e+01 1.18e–10 2.00e+01 1.29e–10 2.00e+01 1.12e–10 2.00e+01 1.20e–10 2.00e+01 1.86e–10 
f26 3.92e–01 5.34e–02 3.40e–01 5.00e–02 2.84e–01 4.73e–02 2.94e–01 4.90e–02 2.96e–01 4.36e–02 
f27 3.75e+01 4.11e+01 3.79e+01 4.57e+01 4.13e+01 4.97e+01 4.93e+01 5.11e+01 3.92e+01 4.72e+01 
FunctionL=1L = 2L=3L=4L=5
AVGSDN      
f1 3.08e+03 5.35e+02 5.15e+03 3.28e+02 6.55e+03 6.27e+02 8.00e+03 5.43e+02 9.31e+03 5.46e+02 
f2 1.28e+03 8.01e+02 1.38e+03 1.17e+03 1.01e+04 1.03e+03 1.16e+04 2.38e+03 1.26e+04 3.71e+03 
f3 3.21e+04 2.63e+03 3.04e+04 2.11e+03 3.13e+04 2.57e+03 3.27e+04 3.48e+03 3.29e+04 2.88e+03 
f6 1.43e+04 8.91e+03 2.18e+04 1.60e+04 3.11e+04 3.03e+04 4.74e+04 4.94e+04 3.27e+04 2.48e+04 
f7 1.71e+04 5.88e+03 1.77e+04 1.49e+04 1.02e+04 6.28e+03 6.19e+04 7.96e+04 4.64e+04 5.82e+04 
f9 4.11e+04 1.38e+04 3.51e+04 1.07e+04 3.94e+04 1.46e+04 3.99e+04 1.12e+04 4.25e+04 1.67e+04 
f10 4.96e+04 1.61e+04 4.66e+04 1.39e+04 4.20e+04 1.61e+04 4.63e+04 1.57e+04 4.42e+04 1.33e+04 
f28 3.09e+03 6.15e+02 4.83e+03 3.64e+02 6.30e+03 3.56e+02 7.93e+03 4.23e+02 9.17e+03 6.05e+02 
f29 2.69e+04 6.39e+03 2.69e+04 6.25e+03 2.67e+04 6.21e+03 2.64e+04 5.91e+03 2.70e+04 6.63e+03 
f30 2.65e+04 9.18e+03 2.11e+04 5.24e+04 7.68e+03 3.28e+03 3.57e+04 6.89e+04 3.48e+04 6.37e+04 
f31 4.07e+04 1.42e+04 4.40e+04 9.10e+03 4.69e+04 1.46e+04 5.24e+04 1.83e+04 5.38e+04 1.26e+04 
f32 1.48e+04 3.64e+03 1.37e+04 2.56e+03 1.47e+04 3.74e+03 1.72e+04 3.56e+03 1.95e+04 4.06e+03 
f33 3.02e+04 1.60e+04 2.43e+04 1.25e+04 2.02e+04 9.15e+03 2.01e+04 9.74e+03 2.38e+04 1.37e+04 
f34 3.91e+04 2.00e+04 3.79e+04 2.67e+04 4.02e+04 2.72e+04 3.79e+04 1.49e+04 4.16e+04 2.11e+04 
f37 2.04e+04 4.34e+03 2.53e+04 9.95e+03 1.93e+04 1.65e+04 2.12e+04 1.41e+04 2.39e+04 2.32e+04 
AVGSDE      
f4 2.10e+05 1.94e+04 1.11e+05 1.63e+04 2.94e+04 6.16e+03 2.80e+05 3.18e+04 2.52e+05 2.18e+04 
f5 2.47e+03 5.46e+02 1.74e+03 3.19e+02 1.68e+03 3.27e+02 1.77e+03 3.17e+02 1.69e+03 2.37e+02 
f8 2.00e+01 1.18e–10 2.00e+01 1.29e–10 2.00e+01 1.12e–10 2.00e+01 1.20e–10 2.00e+01 1.86e–10 
f26 3.92e–01 5.34e–02 3.40e–01 5.00e–02 2.84e–01 4.73e–02 2.94e–01 4.90e–02 2.96e–01 4.36e–02 
f27 3.75e+01 4.11e+01 3.79e+01 4.57e+01 4.13e+01 4.97e+01 4.93e+01 5.11e+01 3.92e+01 4.72e+01 
Table 5:
The p values obtained through the rank sum test on the experimental results obtained by L=2 and L=3 with the other L values.
pL=2pL=3
FunctionL=1L=3L=4L=5L=1L=2L=4L=5
f1 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 0.000 <0.001 
f2 0.727 <0.001 <0.001 <0.001 <0.001 <0.001 0.008 0.003 
f3 0.019 0.189 0.009 0.002 0.287 0.189 0.119 0.049 
f6 0.049 0.190 0.022 0.079 0.014 0.190 0.173 0.842 
f7 0.863 0.028 0.012 0.025 <0.001 0.028 0.004 0.005 
f9 0.098 0.242 0.134 0.074 0.682 0.242 0.899 0.497 
f10 0.493 0.291 0.932 0.539 0.110 0.291 0.356 0.605 
f28 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 
f29 0.982 0.947 0.805 0.926 0.929 0.947 0.858 0.875 
f30 0.614 0.215 0.405 0.412 <0.001 0.215 0.053 0.044 
f31 0.330 0.418 0.052 0.004 0.142 0.418 0.251 0.084 
f32 0.257 0.292 0.001 <0.001 0.956 0.292 0.025 <0.001 
f33 0.160 0.195 0.200 0.898 0.012 0.195 0.987 0.279 
f34 0.858 0.769 0.997 0.587 0.877 0.769 0.721 0.831 
f37 0.034 0.135 0.253 0.782 0.752 0.135 0.662 0.433 
f4 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 
f5 <0.001 0.518 0.742 0.535 <0.001 0.518 0.333 0.902 
f8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 
f26 0.002 <0.001 0.003 0.003 <0.001 <0.001 0.470 0.360 
f27 0.974 0.803 0.414 0.922 0.771 0.803 0.580 0.880 
pL=2pL=3
FunctionL=1L=3L=4L=5L=1L=2L=4L=5
f1 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 0.000 <0.001 
f2 0.727 <0.001 <0.001 <0.001 <0.001 <0.001 0.008 0.003 
f3 0.019 0.189 0.009 0.002 0.287 0.189 0.119 0.049 
f6 0.049 0.190 0.022 0.079 0.014 0.190 0.173 0.842 
f7 0.863 0.028 0.012 0.025 <0.001 0.028 0.004 0.005 
f9 0.098 0.242 0.134 0.074 0.682 0.242 0.899 0.497 
f10 0.493 0.291 0.932 0.539 0.110 0.291 0.356 0.605 
f28 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 
f29 0.982 0.947 0.805 0.926 0.929 0.947 0.858 0.875 
f30 0.614 0.215 0.405 0.412 <0.001 0.215 0.053 0.044 
f31 0.330 0.418 0.052 0.004 0.142 0.418 0.251 0.084 
f32 0.257 0.292 0.001 <0.001 0.956 0.292 0.025 <0.001 
f33 0.160 0.195 0.200 0.898 0.012 0.195 0.987 0.279 
f34 0.858 0.769 0.997 0.587 0.877 0.769 0.721 0.831 
f37 0.034 0.135 0.253 0.782 0.752 0.135 0.662 0.433 
f4 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 
f5 <0.001 0.518 0.742 0.535 <0.001 0.518 0.333 0.902 
f8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 
f26 0.002 <0.001 0.003 0.003 <0.001 <0.001 0.470 0.360 
f27 0.974 0.803 0.414 0.922 0.771 0.803 0.580 0.880 

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