## 4 Methodology
We first discuss some punctual implementations to detail the unfolded metaheuristics designed by the SAHH approach. Then, we study the number of iterations from both low- and high-level approaches. They are the metaheuristic cardinality \(\varpi\) (or the number of search operators) and the hyper-heuristic steps \(s\). Later, we discuss the nature of those operators selected by SAHH to be part of the customised uMHs. Right after analysing these characteristics, we evaluate the performance of the overall approach and the individual unfolded metaheuristics. Finally, we focus our attention on the time complexity of the methodologies that comprise this research. ### Illustrative Selected Problems

First of all, we selected a problem from each category to illustrate the implementation of the proposed approach. Figure 3 shows how the SAHH procedure improves the heuristic sequence that it initially picked at random. To enhance the readability of these data, we plotted fitness order values, which were calculated using logarithm base 10 \(\log f(\vec{x}_{*})\), instead of the bare fitness values. Note that each violin in this figure corresponds to a heuristic sequence, where light-grey violins stand for the initial candidate sequence. In contrast, coloured ones correspond to those obtained from each experiment. In all the plots, the influence of the dimensionality over the problem domain is quite observable. This is expected, considering the established conditions (same population size and maximal cardinality). Regarding the maximal cardinality, it is straightforward to infer that unfolded metaheuristics (uMH) obtained from Experiment 2, which has a loose cardinality range, generally improve upon the corresponding uMHs from Experiment 1. An exception of this general improvement, which may not be the only one, can be seen when solving the problem Type I (Figure 3) with 20 dimensions. However, when the number of dimensions equals 50 for the same problem, it is easy to notice a remarkable difference between the two resulting metaheuristics. However, this does not imply that one achieved uMH is better than the others. Recall that each experiment has different conditions, and the resulting methodologies have distinctive properties, e.g., the type and the number of search operators (SOs) composing them. Nevertheless, an objective fact noticed at a glance from Figure 3 is the relatively compact shape of uMHs from Experiment 2. This is to be expected, chiefly because they were designed to have more SOs than those from Experiment 1. Thus, in raw words, they had more opportunities to'refine' their search procedure. It is also worth remarking that even considering one design specification or the other, employing a particular approach may not grant a relevant benefit when dealing with some problems with exceptional 'hardness'. An example of that can be observed in Figure 3d, where the uMHs tailored for the non-differentiable and multimodal (DU = 00) Schaffer N3 problem exhibit similar performances, close to that of the first uMH guessed by SAHH. Figure 3: Cont. Table 3 complements the results from Figure 3. Concerning the performance value, it is evident that Experiment 2 promotes reaching better performing unfolded metaheuristics for most cases. However, this is not a general rule. An exception is observed in the performance value of the uMH designed for the 20-dimensional Type I problem (Figure 3). This is self-explained by remembering that SA is just a single-agent metaheuristic implemented as an HH. We have not run several implementations of the same HH problem as we did for evaluating the heuristic sequences because it exhibits a high computing cost. For this work, we employed SAHH as a well-established (stochastic) approach to design uMHs by solving the MCOP, which is considered ill-posed due to the presence of multiple solutions. Bear in mind that if there exists a unique solution, the NFL theorem is violated. This is an interesting discussion that is beyond our scope.