## 3 Methodology  
We set up the four studied hyper-heuristic controlling five LLH (GDE3, IBEA, NSGAII, SPEA2, and mIBEA) to solve the eighteen continuous real-world optimization problems presented in Table 1. Different configurations were employed for these problems. In particular, the number of generations and population size were slightly different for P17-P18. This was due to the high computational effort demanded in these applications, which takes almost three months for experiments using the same setup considered for the problems P01-P16.  
Table 2 presents all these parameter used in our experiments. We set up HHLA and HHRL parameters according to [3], MOABHH according to [32] and HHCF according to [6] for P01-P16. For P17 and P18, we set MOABH \(g=1\), \(\beta=0.5\) and HHLA and HHRL \(g=1\). For these two problems, parameters were defined empirically.  
All the original hyper-heuristics were designed to work with populational genetic algorithms. They have specific procedures to manipulate the current population of solutions. In this comparison, we add two other MOEAs (mIBEA and GDE3), which are modeled the same as the three others. For this reason, it does not demand deep changes in the compared hyper-heuristics, changes that would define new hyper-heuristics. Thus, algorithms such as MOEA/D [76] and SMPSO [55] cant be considered without proposing four new algorithms.  
For genetic algorithms, we followed [3; 6]. For GDE3, we followed [32]. We analyzed the performance comparing both hyper-heuristic results with the five single meta-heuristic results. All MOEAs used are implemented by jMetal 5.7 [71]. Each experiment is repeated 30 times.  
\begin{table}
\begin{tabular}{c l c c c} \hline \hline
**ID** & **Problem Name** & **Obj.** & **Vars.** & **Const.** & **Source** \\ \hline P01 & Water & 5 & 3 & 7 & [33] \\ P02 & Machining & 4 & 3 & 3 & [35] \\ P03 & CarSidelImpact & 3 & 7 & 0 & [34] \\ P04 & CrashWorthiness & 3 & 5 & 0 & [19] \\ P05 & FourBarTruss & 2 & 4 & 0 & [39] \\ P06 & Golinski & 2 & 7 & 11 & [40] \\ P07 & Quagliaella & 2 & 16 & 0 & [41] \\ P08 & Poloni & 2 & 2 & 0 & [42] \\ P09 & Vibrating Platform Design & 2 & 5 & 0 & [38; 72] \\ P10 & Optical Filter & 2 & 11 & 0 & [38; 73] \\ P11 & Welded Beam Design & 2 & 4 & 0 & [38; 74] \\ P12 & Disk Brake Design & 2 & 4 & 0 & [38; 75] \\ P13 & Heat Exchanger & 2 & 16 & 0 & [38] \\ P14 & Hydro Dynamics & 2 & 6 & 0 & [38] \\ P15 & Area Under Curve & 2 & 10 & 0 & [38] \\ P16 & Kernel Ridge Regression & 2 & 5 & 0 & [38] \\ P17 & Facility Placement & 2 & 20 & 0 & [38] \\ P18 & Neural Network Controller & 2 & 24 & 0 & [38] \\ \hline \hline \end{tabular}
\end{table}
Table 1: A brief description real-world multi-objective problems containing the number of objectives, variables constraints, and source.  
We employed Hypervolume and IGD+ averages obtained from the 30 executions. First, for each problem, we joined all results obtained by all algorithms, found the nadir point, necessary for Hypervolume calculation, and took the non-dominated set in order to generate the _known Pareto Front_ (\(\mathit{PF}_{\mathit{known}}\)), necessary for IGD+ calculation. Then, we calculated the quality indicator averages and compared them using Kruskal-Wallis as the statistical test with a confidence level of 95%. In order to perform this, we first identify which algorithm has the best average according to the quality indicator. Thus, all the other algorithms are compared to the best, generating a set of \(p\)-values. We define an algorithm tied statistically with the best when a given \(p\)-value is superior to the significance level, which in this case is 0.05.