## 4 Methodology
In general, we notice that our approach manages to deal with a given problem with certain conditions (characteristics and dimensionality) by arranging operators from one family or another to build its solver. There are some situations where a family is often preferred from others, as we discuss below. Nevertheless, it does not mean that a particular set of operators is the best for such a situation. We are just showing the nature of those operators selected for conforming an unfolded metaheuristic found throughout a hyper-heuristic search. Figure 5: Cardinality and steps distributions varying the DU (differentiability and unimodality) characteristics and dimensionality for the unfolded metaheuristics designed in Experiment 1. (**a**) Cardinality. (**b**) Steps. Figure 6: Frequency of the operator families used for each case category (differentiability and unimodality) and by varying the number of dimensions. From the histogram for all the functions in Figure 6, it is easy to notice that operators from the Genetic Crossover (GC) and Random Search (RS) families were more frequently selected for dealing with two-dimensional problems. This fact is somehow expected due to the'simplicity' of these problems where, in general, a simple stochastic approach would be enough to solve them. Obviously, GC is not such a simple approach, but we believe that its unstructured outcomes with respect to the other operators explain this behaviour. For greater dimensionalities, perturbators from the Central Force (CF) family became more relevant, but it seems to be attenuated when augmenting the number of dimensions. A similar effect is noticed with the GC perturbators and, slightly different, with the RS ones that show an increase in popularity around five and ten dimensions. On the contrary, we found that simple heuristics from Gravitational Search (GS) and Particle Swarm (PS) families seem, in a more notorious way, to be boosted up when dimensionality rises. One may explain such a fact with the'swarm' interaction of the probes representing such operators, making them more suitable for higher-dimensional problems. Regarding the family distribution per dimension, we must remark that this becomes more uniform when increasing the problem dimensionality. Nevertheless, we note that mutation-based operators (from Genetic and Differential Mutation families) are oft-chosen to build heuristic sequences for solving 50-dimensional problems. In general, the shape observed for the overall case, without categories, is preserved when grouping the results as mentioned. To better scrutinise these family distributions, the following lines are dedicated to remark those interesting situations that appear when grouping the results. For non-differentiable problems, operators based on random walks (e.g., RS) seem to be invariant in popularity, with a high-density value, concerning the dimensionality. This effect can be observed in detail at the blocks corresponding to categories 10 and 00. Likewise, the Gravitational Search (GS) family appears to be invariant to dimensionality and exhibits low-density values. This is also noticed in a reduced proportion for the Spiral Dynamic (SD) family. We can notice that GS heuristics start as the less popular ones when dealing with 2D domains for the marginally differentiable problems. Still, they gradually increase in density to a moderate value when the 50D problems are considered. This behaviour is also regarded in the block of marginally multimodal problems. It is crucial to remark that we observe a more uniform density distribution with respect to dimensions and families for this marginal classification. Nevertheless, it is quite obvious that this is mostly inherited from the differentiable and multimodal problems (DU = 10). ### Performance Analysis

Once we have boarded the cardinality, required steps, and nature of the resulting unfolded metaheuristics, which are somehow similar for the two experimental setups, we proceed to study the most crucial feature of them: performance.