## 2 The Relaxed Flexible Flow Shop Problem Model with Sequence-Dependent Setup Times (RFFSP) Methods
### Simulated Annealing Algorithm
\begin{table}
\begin{tabular}{c 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. ### HGAG Algorithm Convergence

The HGAG convergence in the Cuexcomate cluster is evaluated using a medium-size instance: RFFS_SDST_60\(\times\) 4 \(\times\) 4 \(\times\) 25. Figure 8 shows the algorithm's convergence curve as a function of the number of generations and the objective function, using a population size of 60 and processing one individual by cluster core. The results show constant improvement in the objective function value until generation 30, where convergence is reached. ### Grid Efficacy

Figure 9 shows the experimental studies results considering different population sizes, which evolve with 30 generations. This Figure shows the average of 30 tests for each variation in population size, until reaching a size of 70 individuals: 60 in the Cuexcomate cluster, and 10 in the Texcal cluster. It is observed that as the size of the population increases, the quality of the solution improves. The grid impacts the solutions' quality by using a more significant number of cooperative processes distributed throughout it. \begin{table}
\begin{tabular}{c c c c c c} \hline
**INSTANCE** & **G** & **C** & **P** & **ACS** & **SA** \\ \hline RFFS\_SDST\_60X4X4X25 & 30 & 0.5 & 240 & ACS & SA \\ \hline RFFS\_SDST\_140X4X4X25 & 5 & 0.5 & 120 & ACS & SA \\ \hline \end{tabular}
\end{table} 
Table 6: HGAG tuning results. Figure 8: HGAG convergence for the RFFS_SDST_SDST_60\(\times\) 4 \(\times\) 4 \(\times\) 25 instance. ### Process cooperation in HGAG

To perform an analysis on the cooperation of the processes in the grid, the following elements are used: 5 RFFS_SDST_ [20, 40, 60, 80, 140] test instances, six groups of cores (5, 10, 15, 30, 60, 120) and eight population sizes (5, 10, 15, 30, 60, 120, 240, 480). 30 tests are carried out by each of the following combinations of population sizes and number of cores: (5,5), (10,10), (20,20), (30,30), (60,60), (120,120), (240,120), (480,120). Each group of cores is distinguished by being closer together in an attempt to avoid bottlenecks in the grid. Two populations show core overload (240 and 480) since they must be distributed over 120 cores. ### Process cooperation in small instances

Table 7 shows the results for the RFFS_SDST_20 \(\times\) 4 \(\times\) 4 \(\times\) 25 instance. In this table, P is the number of processes, N is the number of cores, Np is the number of cores, Cmax is the mean of the 30 tests, t is the average time of the tests, and best the best makespan obtained. It is observed that the best result is obtained when an overload of processes is done and the population increases twice, from 120 to 240 individuals. This behavior is due to the algorithm conducting more exploration and exploitation of the solution space. However, the algorithm's execution time practically increases more than double (since the processes' communications increase) when compared to running 120 processes. In the other tests presented in Table 7, the number of individuals remains at 120. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**RFFS_SDST_20 \(\times\) 4 \(\times\) 4 \(\times\) 25**} & \\ \hline
**P** & **N** & **Np** & **Cmax** & **t s** & **Best** \\ \hline
5 & 5 & 24 & 481.43 & 361 & 478 \\ \hline
10 & 10 & 12 & 479.67 & 363 & 472 \\ \hline
15 & 15 & 8 & 479.17 & 364 & 476 \\ \hline
30 & 30 & 4 & 477.70 & 364 & 473 \\ \hline
60 & 60 & 2 & 476.23 & 366 & 473 \\ \hline
120 & 120 & 1 & 475.10 & 415 & 471 \\ \hline
240 & 120 & 1 & 473.80 & 1087 & 469 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: RFFS_SDST_20 \(\times\) 4 \(\times\) 4 \(\times\) 25 results.