## 4 Methodology
For the non-differentiable and multimodal problems, one can notice low cardinality uMHs can be attributed to the SAHH solving procedure and other parameters worth studying in future works. Besides, \(\varpi\) distributions found for uMHs solving differentiable and multimodal functions are diverse and range almost every value for all dimensions. It is crucial to notice that for this category (DU = 10), our approach managed to find unfolded metaheuristics with up to 50 search operators with a remarkable frequency. The remaining category, non-differentiable and unimodal problems, renders quite interesting patterns, most of them showing compact distributions about the 100-cardinality uMHs. We may remark that this behaviour is chiefly led by the fact that such a characteristic combination is a bit unusual, so there are few problems within this category (DU = 10). In particular, our collection comprises 26, 45, 8, and 28 problems for the DU categories 11, 10, 01, and 00. Furthermore, Figure 5b exhibits the steps required by the hyper-heuristic to find a proper uMH for solving a problem from a given category and dimensionality. These distributions are, by far, smoother than those for the cardinality, which is somehow understandable because they belong to a higher level of abstraction (indirectly related to the problem domain). Except for the two-dimensional case, the steps required by SAHH to find the uMHs vary from a distribution that is highly concentrated in the lower limit into one akin to a uniform distribution. Again, we can explain the particular behaviour for 2D because most of the candidate heuristic sequences during the HH procedure were worth considering. Using an analogy with the Law of Supply and

Figure 4: Scatter plot and marginal distribution for SAHH Steps and HH Cardinality grouped by Problem Characteristics (DU) and Dimensionality (Dim). (**a**) Categories (DU). (**b**) Dimensionality. Demand, we could state that there is more supply than demand. For the intermediate categories (say, 10 and 01), a slight transition from left- to right-hand loaded distributions is observable. Thus, dimensionality seems to directly influence the number of steps required for designing a solver that deals with problems from these categorical features. Lastly, for the non-differentiable and multimodal problems, we notice a reversed pattern with respect to distributions from the differentiable and unimodal problems. It is quite interesting to see how our approach manages to find uMHs for these functions using few steps with more frequency. ### Nature of Search Operators

By this point, we have analysed the influence of categories and dimensionality in the unfolded metaheuristic cardinality and the required hyper-heuristic steps. Now, we delve into the nature of the search operators, particularly the perturbators, conforming these uMHs achieved via Experiment 1. We also disregard a detailed analysis from Experiment 2 due to their similarities with this case. Figure 6 presents bidimensional histograms involving the operator families employed in the metaheuristics for solving problem domains with different dimensionalities. We believe that Figure 6 is quite illustrative about the manner we perform this classification (see the thumbnail blocks in the upper centre). In the first block (upper-left corner), we observe the family distributions for 'All' the problems without considering their categorical features. We split all these results using the categorical characteristics, unimodality and differentiability, as we did above. The purple and magenta plots in the first row and column, respectively, correspond to the marginal cases where only one feature is considered, i.e., only differentiability (right blocks) or unimodality (bottom blocks). Greenish plots show the results for particular combinations of the differentiability and unimodality characteristics in a matrix-shape form.