## 2 The Relaxed Flexible Flow Shop Problem Model with Sequence-Dependent Setup Times (RFFSP) Methods
### Simulated Annealing Algorithm
The nomenclature, defined in this work, to name the benchmarks is the following:

\[\text{RFFS\_SDST\_rxmxk}\text{ }\text{ }\text{x}\text{ }\text{r}_{\text{b}}\]

where RFFSP_SDST means relaxed flexible flow shop problem with sequence dependent setup times, \(n\) is the number of jobs, m is the number of stages, k is the number of parallel machines on each stage, r is the ratio of setup time, dependent on the sequence related to the processing time, and b is the problem number which can be from 1 to 5. Five benchmarks are generated to create sets of five instances, to be solved with the HGAG. Table 2 shows these instances. Readers can access the data of the instances that are resolved in this work in [34]. ### HGAG Sensibility Analysis

The HGAG has different parameters that must be tuned to find the appropriate values to get the best algorithm performance. The sensibility analysis is carried out with an automatically distributed tuning methodology. Three metaheuristics are used in the HGAG: one cooperative genetic algorithm, an ant colony system algorithm (ACS), and a simulated annealing algorithm (SA). For each algorithm, its input parameters are tuned. First, the GA parameters to be tuned are (1) the population size P, which is directly proportional to the number of grid processors, (2) the selection operator S, (3) the crossover-rate C, and (4) the number of generations G. Next, the ACS parameters to be tuned are: (1) the number of ants h, (2) the importance coefficient \(\alpha\), (3) the importance coefficient \(\beta\), (4) the evaporation coefficient \(\gamma\) for the local transition, (5) the evaporation coefficient \(\delta\) for a global transition, (6) the ratio between exploration/exploitation \(q\), and (7) the stop criterion \(p\). Finally, the SA parameters to be tuned are: (1) the initial temperature \(T_{o}\), (2) the Markov chain length \(m\), (3) the cooling rate \(\mu\), and (4) the final temperature \(T_{f}\). The ACS and SA algorithms are independent, and they are tuned simultaneously. The CGA algorithm is tuned posteriorly. The sensibility analysis is realized in the Cuexcomate cluster. It is carried out for every set of different size instances, as shown in Table 2.

As shown in Table 2, the only parameter modified to create every instance set is the number of jobs \(n\); the other parameters are kept constant. The analysis of sensibility \begin{table}
\begin{tabular}{c c c c} \hline
**INSTANCES SET** & **NAME** & **Description** & **SIZE** \\ \hline
1. & RFFS\_SDST\_20 \(\times\) 4 \(\times\) 4 \(\times\) 25 & \(n=\) 20, \(m=\) 4, \(k=\) 4, \(ST_{jk}=\) 25\% & **Small** \\
2. & RFFS\_SDST\_40 \(\times\) 4 \(\times\) 4 \(\times\) 25 & \(n=\) 40, \(m=\) 4, \(k=\) 4, \(ST_{jk}=\) 25\% & **Medium** \\
3. & RFFS\_SDST\_60 \(\times\) 4 \(\times\) 4 \(\times\) 25 & \(n=\) 80, \(m=\) 4, \(k=\) 4, \(ST_{jk}=\) 25\% & **Large** \\
4. & RFFS\_SDST\_140 \(\times\) 4 \(\times\) 4 \(\times\) 25 & \(n=\) 140, \(m=\) 4, \(k=\) 4, \(ST_{jk}=\) 25\% & **Large** \\ \hline \end{tabular}
\end{table} 
Table 2: Instances for tuning the HGAG. is realized by tuning the following parameters in the following order: first, the ACS parameters \(h\), \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(p\), \(q\), then the SA parameters \(T_{o}\), \(m\), \(\mu\), \(T_{f}\), and finally, the HAGG parameters S, G, C, P. The following steps are applied to tune the algorithms:

1. The design, codification, and tuning of the ACS and SA algorithms are carried out. 2. An analysis of previous works described in the existing literature to find the values commonly used for the ACS parameters is conducted. It is found that the \(\alpha\) and \(\beta\) values fluctuate between 0 and 5. The values for the pheromone evaporation factors, \(\gamma\), \(\delta\), and the importance factor \(p\), are in 0 and 1. The h value has a maximum of 200, and some researchers set it to with the same value as the number of jobs to the problem. The q parameter takes values higher than 2000, depending on the available computing resources.