## 2 The Relaxed Flexible Flow Shop Problem Model with Sequence-Dependent Setup Times (RFFSP) Methods
### Simulated Annealing Algorithm
In some works, the \(\alpha\) parameter value is generally set as one, and only the importance factor \(\beta\) is varied [35]. 3. An analysis of previous works presented in the existing literature to find the established values for the SA parameters is also carried out. It is found that the values defined for the four parameters of the algorithm, \(T_{o}\), \(m\), \(\mu\), \(T_{f}\), are called low, medium, and high. 4. Ranges are established for the HAGG parameters. The ranges of values found in the literature [36] consider two selection operators: roulette and tournament. For the crossover operator, the researchers commonly use PMX (Partially Mapped Crossover), OP (One Point order crossover), TP (Two Point order crossover), and OX (Order Crossover), among others. The crossover rate is defined in the range [0.1, 0.5]. The population size is fixed in the range between 20 and 50. Table 3 shows the definition of values for tuning the ACS, SA, and HAGG algorithms. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Values** & **Increments (Units)** \\ \hline \hline \multicolumn{3}{c}{ACS parameters} \\ \hline \(h\) & 1–20 & 1 \\ \hline \(\alpha\) & 1 & - \\ \hline \(\beta\) & 0.1–0.9; 1–10 & 0.1 \\ \hline \(\gamma\) & 0.1–0.9 & 0.1 \\ \hline \(\delta\) & 0.1–0.9 & 0.1 \\ \hline \(q\) & 0.1–0.9 & 0.1 \\ \hline \(p\) & 500–2000 & 250 \\ \hline \multicolumn{3}{c}{SA parameters} \\ \hline \(T_{o}\) & 1–110 & 10 \\ \cline{2-3}  & 120–500 & 20 \\ \hline \multirow{3}{*}{\(m\)} & 1–9 & 1 \\ \cline{2-3}  & 10–30 & 2 \\ \cline{2-3}  & 30–50 & 4 \\ \cline{2-3}  & 50–75 & 5 \\ \hline \multirow{3}{*}{\(\mu\)} & 0.800–0.982 & 0.14 \\ \cline{2-3}  & 0.984–0.990 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values for tuning ACS, SA, and HAGG. ### Sensitivity analysis results

The tuning process for the instances of five sizes (Table 1) is carried out in the Cuexcomate cluster, and Tables 4-6 show the SA, ACS, and HGAG algorithm results tuning, respectively. Whereas 30 tests are conducted for each set of instances for the first two algorithms, 30 tests are carried out with the instance of 60 jobs for the HGAG. Considering one of the worst cases, where the number of generations G is 30, and a population size \(P\) is 60, perfect distribution of processes is observed since the number of cores of the Cuexcomate cluster is equal to the size of the population. By distributing the processes on each core, one process is executed per core, so there is no overhead. It is recommended that the HGAG execution be one process per core to avoid overload on the grid. The main cycle of the HGAG algorithm is given by the number of generations G, within which each process applies the tuned mutation hybrid operator using the ACS and SA algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline
**INSTANCE** & \(T_{0}\) & \(m\) & \(\mu\) & \(T_{f}\) \\ \hline RFFS\_SDST\_20x4x4x25 & 450 & 78 & 0.989 & 0.0015 \\ \hline RFFS\_SDST\_40x4x4x25 & 425 & 56 & 0.890 & 0.0001 \\ \hline RFFS\_SDST\_60x4x4x25 & 395 & 71 & 0.989 & 0.0001 \\ \hline RFFS\_SDST\_80x4x4x25 & 160 & 65 & 0.990 & 0.0001 \\ \hline RFFS\_SDST\_140x4x4x25 & 40 & 70 & 0.990 & 0.0001 \\ \hline \end{tabular}
\end{table} 
Table 4: SA tuning results. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Values** & **Increments (Units)** \\ \hline  & 0.8–0.2 & \(-\)0.2 \\ \cline{2-3} \(T_{f}\) & 0.08–0.02 & \(-\)0.02 \\ \cline{2-3} \(T_{f}\) & 0.008–0.002 & \(-\)0.002 \\ \cline{2-3} \(0.0001\) & - \\ \cline{2-3} \(1\) & - \\ \hline \multicolumn{3}{c}{HGAG parameters} \\ \hline \(S\) & Roulette & - \\ \hline \(G\) & 10–30 & 2 \\ \hline \(C\) & 0.10–1 & 0.10 \\ \hline \(P\) & 20–70 & 5 \\ \hline \end{tabular}
\end{table} 
Table 3: _Cont._