## 4 Methodology
Subsequently, we analyse the results by carrying out multiple comparisons using the pairwise (one-sided) Wilcoxon signed-rank test. Thus, we compute the true statistical significance for multiple pairwise comparisons such as \(p=1-\prod_{i}(1-p_{i})\) by following the procedure and conditions given in [45; 46]. We employed the performance value reached by the metaheuristics, for each problem and dimension via (5), as the measure of one run of the algorithm. Thus, we set the unfolded metaheuristic generated by SAHH as the control parameter and the basic metaheuristics as the other parameters. We look for the statistical significance of uMHs outperforming at least one of the basic MHs in a given problem. In doing so, we justify the fact that we do not require prior knowledge of which problems a particular MH is an excellent choice. Keeping this in mind, we state the null and alternative hypotheses as:

**Hypothesis 0**. : _(Null) The unfolded metaheuristic, generated by the SAHH approach, performs equal to or worse than at least one basic metaheuristic with a significance level 0.05._

**Hypothesis 1**. : _(Alternative) The unfolded metaheuristic, generated by the SAHH approach, outperforms at least one basic metaheuristic with a significance level of 0.05._

Figure 9 shows the \(p\)-values from this [PERSON]'s analysis, and also grouped by category and dimensionality. At a general glance, we notice that it is possible (\(p<0.05\)) to reject the null hypothesis \(\mathrm{H}_{0}\) for several scenarios in both experimental setups. Particularly, for those liaised with the multimodal domains, numerous basic metaheuristics are not suitable for dealing with them. It is also worth mentioning that the number of dimensions seems to increment the \(p\)-values determined for these scenarios. Nevertheless, it does not mean that SAHH generates poor-performing unfolded metaheuristics. We attribute this tendency to fail to reject the null hypothesis because these problems become harder to solve with any technique. It is also valuable to comment that Experiment 2 represents a better configuration than Experiment 1 for most problems with a dimensionality beyond five. On the opposite side, we can find those problem domains with a unimodal landscape where it

Figure 8: Average Success Rate achieved for the unfolded metaheuristic tailored using the proposed methodology compared against the basic metaheuristics. (**a**) Experiment 1. (**b**) Experiment 2.

is hard to find a method incapable of solving them. Likewise, increasing the dimensionality has a similar effect as the one we just commented. However, we only fail to reject H\({}_{0}\) when regarding the nondifferentiable and unimodal problems in 2D and Experiment 2. This result is quite understandable because this setup prefers to generate longer heuristic sequences from longer HH searching procedures. It may also be considered a drawback of using a memoryless metaheuristic approach to deal with a hyper-heuristic problem. Naturally, if the practitioner decides to apply our approach for low-dimensional problems, we highly recommended a configuration similar to Experiment 1. Otherwise, Experiment 2 would suffice as a good starting point for dealing with more than twenty dimensions. ### Time Complexity Analysis

All the terms mentioned below follow the nomenclature described in Sections 2 and 3. After analysing the nature of the resulting heterogeneous heuristic sequences, we are interested in studying the computing cost of our proposal. Keep in mind that once a uMH is designed for any problem, its implementation or further tuning for a relatively similar problem requires a low computing burden; it could be even lower than that required for a standard metaheuristic. Thus, we only focus our attention on the hyper-heuristic process for designing these uMHs. In that sense, SAHH has a time complexity similar to that of a regular metaheuristic, given by

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