IEEE Computational Intelligence Magazine - May 2018 - 39
checks. Alas, the Linear and the previously proposed transformations (identified as 'Exponential') led to a more computationally
expensive search path, increasing the median number of consistency checks. However, this number corresponds to the number
of validations that must be performed in those instances where
the solver could find a solution within the time limit. Therefore,
this increase in cost could be derived from the additional instances that were solved. Figure 13 also includes data for the combinations of explicit and implicit transformations. We did so
striving to analyze whether merging them led to a better performance. As shown, even if the performance improves, only the
Linear transformation yields a behavior similar to that of the kernel. Though seemingly enhancing the success rate, it hinders the
search cost. For the S-shaped transformation, however, including
the kernel-based distance hampers performance, leading to less
successful and more costly selection hyper-heuristics.
A Wilcoxon statistical test yielded the p-values shown in
Table 1. The S-shaped and both mixed (i.e., K+L and K+S)
transformations had a higher success rate than the original
approach (p-values below 0.05). Even so, the p-value of the
pure kernel (0.0548) was not too high, which makes it an
alternative worth keeping in mind. Analyzing the search cost
transformed using the S-shaped (middle) and the previously
reported (right) approaches. As shown, the region beyond
p 1 = 0.6 and p 2 = 0.6 is unused, and thus wasted. On the contrary, there is a region at 0.3 # p 1 # 0.5 and p 2 = 0.2 (approximately) where different kinds of instances are mixed up. By using
both transformations, the wasted space is reduced and distributed
throughout the remaining regions, broadening the zones in conflict and thus making them easier to separate.
B. Initial Testing
Success Rate (SR) (%)
HH (K+S)
HH (K+L)
HH (K)
HH (S)
HH (E)
HH (L)
HH (O)
Oracle
KAPPA
DEG
WDEG
Adjusted CC (Thousands)
Success Rate (%)
DOM
Adjusted Consistency
Checks (ACC)
A comparison of each base heuristic against a synthetic oracle
reveals that the latter performs a lot better (Fig. 12). Even so, this
only implies that there is a latent benefit derived from an appropriate combination of each heuristic. Nonetheless, Oracle data
were generated in a synthetic fashion by analyzing the performance of each standalone heuristic at each instance and selecting
the best one. Thus, it represents a Utopian scenario where a
perfect selection was carried out. In spite of this, and as it was
expected, all selection hyper-heuristics (including those with no
transformation) performed better than standalone heuristics.
Moreover, all transformations exhibited an average performance
quite close to that of the synthetic oracle (highlighted bar in
green) in terms of both, number of
adjusted consistency checks and success
100
1,000,000
rate. Even so, the S-shaped transformaACC
SR
tion completed, on average, a bit more
750,000
75
instances than the other approaches
500,000
50
(highlighted bar in blue). The kernel250,000
based approach was computationally
25
cheapest (highlighted bar in red). It is
0
0
also important to remark that, even the
worst transformations were not so bad.
In fact, the Linear transformation yielded
a success rate 3% higher than the best
performing heuristic (i.e., DOM) while FigurE 12 Average number of adjusted consistency checks (left bars) and success rate (right bars)
requiring about 60% less ACC. Besides, for all base heuristics, for a synthetic oracle, and for all hyper-heuristics (average of 15 runs each)
with two features: Original (O), Linear (L), Exponential (E), S-shaped (S), Kernel (K), Kernel+Linear
the combination of kernel with S-shaped (K+L), and Kernel+S-shaped (K+S). Highlighted columns: Synthetic Oracle (green), best adjusted
transformations required only 7% more consistency checks (red) and success rate (blue). Data distribution is shown in Fig. 13.
ACC than the best heuristic (i.e.,
KAPPA) but increased the success rate in
100
500
22%. A comparison of the average behavior of selection hyper-heuristics with no
90
400
transformation is also interesting. For this,
we focus on search cost and on success
80
300
rate. The best transformations shifted the
70
success rate by 8% and by 10% (respec200
tively), while requiring about 20% less
60
ACC (in both cases).
Selection hyper-heuristics were dis50
100
O L E S K K+L K+S
O L E S K K+L K+S
tributed in an interesting fashion (see
(a)
(b)
Fig. 13). For starters, all transformations
increased the median success rate in over
10%. Two of them (S-shaped and kernel) FigurE 13 Success rate (a) and search cost (b) for 15 runs of hyper-heuristics operating with
two features. Original values (O) and transformations: Linear (L), S-shaped (S), Exponential (E),
also reduced the median cost of the Kernel (K), Kernel+Linear (K+L), and Kernel+S-shaped (K+S). Crosses: outliers between
search in about 20,000 consistency 1.5 * IQR and 3 * IQR. Circles: outliers beyond 3 * IQR. IQR: Interquartile range.
may 2018 | IEEE ComputatIonal IntEllIgEnCE magazInE
39
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