## 4 Methodology
2:Best heuristic sequence \(\vec{h}_{*}\)
3:\(\vec{h}\leftarrow\textsc{ChooseRandomly}(H_{PF},\,P_{H_{PF}})\)\(\triangleright\)\(P_{H_{PF}}(h)\) is the probability distribution of \(H_{PF}\)
4:for\(r=\{1,\ldots,N_{r}\}\)do\(\triangleright\) Repeat the subsequent evaluations \(N_{r}\) times
5:\(X_{*}\
i\vec{x}_{r,*}\leftarrow\textsc{EvaluateMH}(\vec{h})\)\(\triangleright\) Evaluate and record the candidate sequence
6:endfor
7:\(\mathrm{perf}(\vec{h})\leftarrow\textsc{UpdateUpdate}(\vec{h}_{*})\)\(\triangleright\) Update the best global position using the \(t\)-th selector
8:endprocedure
```  
**Pseudocode 2** Hyper-heuristic based on Simulated Annealing (SAHH)  
```
1:Domain \(\mathfrak{X}\), objective function \(f\), heuristic collections \(H_{PF},H_{SV}\subset\mathfrak{H}\) (cf. Table 2), initialiser \(h_{i}\), performance \(Q(\vec{h}|\mathfrak{X},f)\), and population size \(N\). Additional parameters: \(\Theta_{0}\), \(\Theta_{\min}\), and \(\delta\). 2:Best heuristic sequence \(\vec{h}_{*}\)
3:\(\vec{h}\leftarrow\textsc{ChooseRandomly}(H_{PF},\,P_{H_{PF}})\)\(\triangleright\)\(P_{H_{PF}}(h)\) is the probability distribution of \(H_{PF}\)
4:for\(r=\{1,\ldots,N_{r}\}\)do\(\triangleright\) Repeat the subsequent evaluations \(N_{r}\) times
5:\(X_{*}\
i\vec{x}_{r,*}\leftarrow\textsc{EvaluateMH}(\vec{h})\)\(\triangleright\) Evaluate and record the candidate sequence
6:endfor
7:\(\mathrm{perf}(\vec{h})\leftarrow\textsc{Update}(\vec{h}_{*})\)\(\triangleright\) Update the best global position using the \(t\)-th selector
8:endprocedure
```  
**Pseudocode 3** Hyper-heuristic based on Simulated Annealing (SAHH)  
```
1:Domain \(\mathfrak{X}\), objective function \(f\), heuristic collections \(H_{PF},H_{SV}\subset\mathfrak{H}\) (cf. Table 2), initialiser \(h_{i}\), performance \(Q(\vec{h}|\mathfrak{X},f)\), and population size \(N\). Additional parameters: \(\Theta_{0}\), \(\Theta_{\min}\), and \(\delta\). 2:Best heuristic sequence \(\vec{h}_{*}\)
4:\(\vec{h}\leftarrow\textsc{ChooseRandomly}(H_{PF},\,P_{H_{PF}})\)\(\triangleright\)\(P_{H_{PF}}(h)\) is the probability distribution of \(H_{PF}\)
5:for\(r=\{1,\ldots,N_{r}\}\)do\(\triangleright\) Repeat the subsequent evaluations \(N_{r}\) times
6:\(X_{*}\
i\vec{x}_{r,*}\leftarrow\textsc{EvaluateMH}(\vec{h})\)\(\triangleright\) Evaluate and record the candidate sequence
7:endfor
8:\(\mathrm{perf}(\vec{h})\leftarrow\textsc{Update}(\vec{h})\)\(\triangleright\) Estimate the performance metric
9:\(\vec{h}_{*}\leftarrow\vec{h}\)\(\triangleright\) Initialise the best heuristic sequence
10:\(s\gets 0\), and \(\Theta\leftarrow\Theta_{0}\)\(\triangleright\) Initialise the step counter and temperature
11:while\((\Theta>\Theta_{\min})\) and \((s\leq s_{\max})\)do
12:\(a\leftarrow\textsc{ChooseAction}(H_{SV},a\,,\sigma)\)\(\triangleright\) Choose an action at random from the action set \(a\in H_{SV}\)
13:\(\vec{h}_{c}\gets a\{\vec{h}\}\)\(\triangleright\) Obtain a neighbour sequence \(\vec{h}_{c}\) by applying \(a\) to \(\vec{h}\)
14:for\(r=\{1,\ldots,N_{r}\}\)do\(\triangleright\) Repeat the subsequent evaluations \(N_{r}\) times
15:\(X_{*}\
i\vec{x}_{r,*}\leftarrow\textsc{EvaluateSequence}(\vec{h}_{c})\)\(\triangleright\) Evaluate and record the candidate sequence
16:endfor
17:\(\mathrm{perf}(\vec{h}_{c})\gets Q(\vec{h}_{c}|\mathfrak{X},f)\) via Equation (5)\(\triangleright\) Estimate the performance metric
18:if\(\mathcal{U}(0,1)\leq\exp(-(\mathrm{perf}(\vec{h}_{c})-\mathrm{perf}(\vec{h}_{c}))/\Theta)\)then\(\triangleright\) Apply Metropolis selection
19:\(\vec{h},\mathrm{perf}(\vec{h})\leftarrow\vec{h}_{c},\mathrm{perf}(\vec{h}_{c})\)
20:endif
21:if\(\mathrm{perf}(\vec{h}_{c})<\mathrm{perf}(\vec{h}_{*})\)then\(\triangleright\) Apply Greedy selection
22:\(\vec{h}_{*

## 5 Results and Discussion

After carrying out the experiments described in the previous section, we study all the results using different approaches to validate our proposal. These approaches are organised as we now mention.