Computational Intelligence - May 2017 - 42
TaBLe 4 HV and I e + Results for three-objective and four-objective formulations.
inSTance
FOrMuLaTiOn
inDicaTOr
nSga-ii-HH
SPea2-HH
iBea-HH
MOea/D-Dra-HH
JaMES
3o
hV
0.9637 (6.0e-5)
0.9637 (8.0E-5)
0.9470 (5.4E-3)
0.9633 (3.2E-4)
Ie +
0.0034 (4.6e-3)
0.0050 (4.8E-3)
0.0352 (7.3E-3)
0.0136 (1.7E-3)
4o
hV
0.6269 (1.3E-4)
0.6270 (8.0e-5)
0.6236 (3.8E-4)
0.6255 (4.0E-4)
Ie +
0.0106 (3.0e-3)
0.0106 (3.3E-3)
0.0294 (0.0000)
0.0242 (5.2E-3)
3o
hV
0.9938 (6.0e-5)
0.9938 (6.0e-5)
0.5700 (9.1E-3)
0.9928 (2.9E-4)
Ie +
0.0039 (8.9E-4)
0.0037 (1.0e-3)
0.4320 (9.0E-3)
0.0050 (5.9E-4)
4o
hV
0.6368 (2.8E-3)
0.6371 (1.7e-3)
0.1894 (3.1E-3)
0.6325 (4.4E-3)
Ie +
0.0408 (1.4E-2)
0.0390 (1.3e-2)
0.6754 (1.0E-3)
0.0498 (1.1E-2)
caS
wEathEr
StatIon
3o
4o
E-ShoP
3o
4o
hV
0.9922 (5.0e-5)
0.9922 (6.0E-5)
0.5658 (1.0E-2)
0.9896 (6.4E-4)
Ie +
0.0026 (7.1e-4)
0.0027 (7.6E-4)
0.4365 (1.0E-3)
0.0068 (1.3E-3)
hV
0.6587 (7.5e-4)
0.6577 (1.1E-3)
0.2749 (4.9E-3)
0.6539 (1.1E-2)
Ie +
0.0163 (3.8e-3)
0.0218 (6.9E-3)
0.5198 (8.1E-4)
0.0368 (6.2E-2)
hV
0.9974 (4.0e-5)
0.9973 (3.0E-5)
0.5452 (5.3E-3)
0.9957 (5.6E-4)
Ie +
0.0014 (4.2e-4)
0.0018 (4.5E-4)
0.4584 (5.4E-3)
0.0032 (7.4E-4)
hV
0.6860 (1.8e-2)
0.5890 (2.9E-2)
0.2396 (2.2E-3)
0.6055 (4.1E-2)
Ie +
0.1187 (6.1e-2)
0.3838 (7.0E-2)
0.5656 (1.0E-3)
0.3442 (1.1E-1)
products increases (case of E-Shop, which has the greatest
number of products) and has similar performance to the other
algorithms for instances with few products (for example, James).
These findings hold true for both formulations. Thus, NSGAII-HH was chosen as the best algorithm for this problem and
was used to answer the next research questions. It is important
to note that IBEA-HH has the worst performance for all
instances and both formulations.
B. RQ2-UCB vs Random Selection Methods
As described in the previous section, NSGA-II-HH outperforms other algorithms by obtaining the best results. So, the
results reached by this algorithm were compared to those generated by its version where the LLHs are randomly selected
(called NSGA-II-RAND here). Table 5 shows the mean values, standard deviations, p-values for Mann-Whitney test and
t 12 for HV, I e + and execution time (in seconds), considering
A
three-objective (3O) and four-objective (4O) formulations.
Analyzing the results regarding the three-objective formulation, we observe that NSGA-II-HH outperforms NSGA-IIRAND on the James and E-Shop instances. On the other
hand, NSGA-II-RAND generates better results for CAS and
Weather Station. However, the results show that there is no statistical difference between them on the James, CAS and Weather Station instances when considering a 0.05 significance level.
It is important to notice that NSGA-II-HH produces the best
results for E-Shop with statistical difference (p-value = 0.01)
t 12 = 0.68 h for HV, but it produces
and medium effect size ^A
results with no statistical difference for I e + (p-value = 0.053).
In regards to the execution time, NSGA-II-HH outperforms
NSGA-II-RAND in the instances with a large number of
products (Weather Station and E-Shop).
42
IEEE ComputatIonal IntEllIgEnCE magazInE | may 2017
Regarding the four-objective formulation, we can see that
NSGA-II-RAND outperforms NSGA-II-HH for James and
CAS. On the other hand, NSGA-II-HH outperforms NSGAII-RAND for Weather Station and E-Shop. However, the
results show that there is no statistical difference between them
for James, CAS and Weather Station when considering a 0.05
significance level. Again, NSGA-II-HH obtains the best results
for E-Shop with statistical difference (p-value = 0.01) and
t 12 = 0.69 h for both HV and I e + indicamedium effect size ^A
tors. In regards to the execution time, NSGA-II-RAND outperforms NSGA-II-HH in all instances, but for the E-Shop
instance there is no statistical difference (p-value = 0.8) and the
t 12 = 0.51 h .
effect size is negligible ^A
To sum up, the results show that NSGA-II-HH reaches, on
average, the best results for both formulations, mainly when the
number of products increases. NSGA-II-RAND presents lower
execution time for instances with a small number of products.
However, the NSGA-II-HH execution time is better for the
large instance (E-Shop) and three-objective formulation, and
there is a statistical equivalence with NSGA-II-RAND when
the four-objective formulation is used. An explanation about the
difference between the times is not evident.This behavior should
be investigated in future work by using larger instances. Hence,
for this problem, it is possible to conclude that the random LLH
selection method can be sufficient to achieve good results when
using simple instances, that is, instances with a small number of
products. On the another hand, when the number of products
increases, NSGA-II-HH should be considered.
C. RQ3-Analyzing the Solutions
To analyze the impact of using three and four objectives, we
consider solutions of the best algorithm, NSGA-II-HH, with a
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