## 4 Methodology
Furthermore, it is worth noticing that there are some cases where quite similar performances can be rendered from dramatically different heuristic sequences, for example, those achieved for the Schaffer N3 problem with \(D\geq 10\). In that case, it is preferable to use the setup from Experiment 1 or to enhance the HH solver. Plus, we can observe with ease that, on average, Experiment 1 and 2 obtained 92- and 460-cardinality uMHs, in HH procedures with an average of 121 and 283 steps, respectively. In terms of search operators, both implementations achieve methods close to the upper limit, but that was not the case of the required hyper-heuristic steps, at least in this illustrative example. A relevant fact to highlight from these data is the apparent absence of a cardinality (\(\alpha\)) and the increasing trend of heuristic steps (\(s\)) related to the dimensionality. It seems that our approach somehow deals with the dimensionality curse when solving the MCOP. However, this occurs at the high level, where this parameter can be considered as a design condition. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**DU**} & \multirow{2}{*}{**Dim.**} & \multicolumn{2}{c}{**Experiment 1**} & \multicolumn{3}{c}{**Experiment 2**} \\  & & & **Perf.** & \(\alpha\) & \(s\) & **Perf.** & \(\alpha\) & \(s\) \\ \hline \multirow{6}{*}{Schwefel} & \multirow{3}{*}{11} & 2 & 0.001 & 67 & 129 & **6.2 \(\times\) 10\({}^{-4}\)** & 494 & 323 \\  & & 5 & 0.402 & 99 & 189 & **0.050** & 226 & 421 \\ \cline{1-1}  & & 10 & 11.025 & 100 & 187 & **0.669** & 483 & 216 \\ \cline{1-1}  & & 20 & 38.686 & 99 & 158 & **8.475** & 498 & 465 \\ \cline{1-1}  & & 30 & 74.324 & 94 & 180 & **37.957** & 499 & 273 \\ \cline{1-1}  & & 40 & 89.752 & 99 & 164 & **55.137** & 498 & 288 \\ \cline{1-1}  & & 50 & 137.846 & 90 & 99 & **70.506** & 499 & 441 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Illustrative results from the selected problems in Figure 3. Columns correspond to the problem name, its binary-encoded category (DU), the achieved performance metric value (Perf. ), unfolded metaheuristic cardinality (\(\alpha\)), and hyper-heuristic steps (\(s\)). Better performances are in bold face. Figure 3: Fitness order variation obtained at the first (light-gray) and last (coloured) SAHH step for an illustrative problem for each category (DU) and considering different dimensionalities. This order is calculated by using the logarithm base 10 to each fitness value achieved per run, \(\log f(\vec{x}_{*})\). uMH stands for unfolded metaheuristic. (**a**) [PERSON] (11). (**b**) [PERSON] (10). (**c**) Type I (01). (**d**) Schaffer N3 (00). ### Low- and High-Level Iterations

After taking a smooth dive into the results using some punctual examples, we now analyse them from different perspectives. Figure 4 shows the normalised distribution of steps and cardinality when grouping this information by using the categories (DU) and the number of dimensions. It is pretty evident that for both parameters, \(\varpi\) and \(s\), the maximal concurrence is near the upper limit (see the distributions in the margins of the diagonal plots). In this case, we used all the results obtained from Experiment 1, but one can notice a somewhat similar pattern from Experiment 2. From Figure 4a, we observe that uMHs with \(\varpi\) of about 30 (followed by 70) were more frequently designed for non-differentiable problems. At the same time, for the differentiable one, the cardinalities were more scattered along with the domain. Particularly, it is interesting that our approach found several'short' heuristic sequences with up to 20 search operators for differentiable and multimodal (DU = 10) domains. This is remarkable because short procedures would be preferred in many practical implementations. Now, concerning the steps spent finding these uMHs, the distributions show relatively uniform shapes along with the limits.