## 4 Experimental Calibration and Test Cases  
Before comparing the proposed GARS against alternative optimization procedures emerging from the recent literature in the field of the FSDGS problems, a comprehensive calibration phase has been fulfilled in order to properly set all the relevant parameters of the developed metaheuristic. To this end, the experimental benchmark proposed by [PERSON], [PERSON], and [PERSON] [26] has been employed. Therefore, three-distinct sub-benchmarks, entailing problems with 2, 3, and 6 machines, respectively, have been considered. Within each sub-benchmark, test cases have been generated combining three factors, namely number of groups, number of jobs within groups and setup times of groups on each machine, in a full-factorial experimental design, as shown in Tables 3-5. On the whole, a total of \(27+81+27=135\) separate instances, describing a consistent dataset for the presented problem have been generated. All instances have been created by extracting job processing times according to a uniform distribution in the range [1,20]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Factor** & **Level** & **Value** \\ \hline Number of groups & 1 & U [1,5] \\  & 2 & U [6,10] \\  & 3 & U [11,16] \\ \hline Number of jobs in a group & 1 & U [2,4] \\  & 2 & U [5,7] \\  & 3 & U [8,10] \\ \hline  & 1 & \(M_{1}\rightarrow\) U [1,50] & \(M_{2}\rightarrow\) U [17,67] \\ Setup times of machine \(M_{i}\) & 2 & \(M_{1}\rightarrow\) U [1,50] & \(M_{2}\rightarrow\) U [1,50] \\  & 3 & \(M_{1}\rightarrow\) U [17,67] & \(M_{2}\rightarrow\) U [1,50] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sub-benchmark of instances for the 2-machine FDSGS problem. Table 6 illustrates the tested parameters, and denotes in bold the best combination of values, chosen after an ANOVA analysis [36] performed by means of Stat-Ease(r) Design-Expert(r) 7.0.0 commercial tool. As it can be noticed, four experimental factors have been tested at three different levels. Therefore, 34 = 81 distinct configurations of GARS procedure have been considered, gaining a total of 135 \(\times\) 81 = 10,935 experimental runs. GARS procedure has been coded in MATLAB(r) language and executed on a 2 GB RAM virtual machine embedded on a workstation powered by two quad-core 2.39 GHz processors. The response variable chosen for the calibration phase was the Relative Percentage Deviation (_RPD_) from the best heuristic solution available, calculated according to the following formula: \begin{table}
\begin{tabular}{c c c} \hline
**Factor** & **Level** & **Value** \\ \hline Number of groups & 2 & U [1,5] \\  & 3 & U [11,16] \\ \hline  & 1 & U [2,4] \\ Number of jobs in a group & 2 & U [5,7] \\  & 3 & U [8,10] \\ \hline  & 1 & \(M_{1}\rightarrow\) U [1,50] \\  & 2 & \(M_{1}\rightarrow\) U [17,67] \\  & 3 & \(M_{1}\rightarrow\) U [45,95] \\  & 4 & \(M_{1}\rightarrow\) U [1,50] \\ Setup times of machine Mi & 5 & \(M_{1}\rightarrow\) U [1,50] \\  & 6 & \(M_{1}\rightarrow\) U [17,67] \\  & 7 & \(M_{1}\rightarrow\) U [17,67] \\  & 8 & \(M_{1}\rightarrow\) U [45,95] \\  & 9 & \(M_{1}\rightarrow\) U [45,95] \\ \hline \end{tabular}
\end{table} 
Table 4: Sub-benchmark of instances for the 3-machine FDSGS problem. \begin{table}
\begin{tabular}{c c c c} \hline
**Factor** & **Level** & **Value** \\ \hline Number of groups & 1 & U [1,5] \\  & 2 & U [6,10] \\  & 3 & U [11,16] \\ \hline  & 1 & & U [2,4] \\ Number of jobs in a group & 2 & U [5,7] \\  & 3 & U [8,10] \\ \hline  & 1 & \(M_{1}\rightarrow\) U & \(M_{2}\rightarrow\) U & \(M_{3}\rightarrow\) U \\  & 1 & [1,50] \\ Setup times of machine \(M_{i}\) & 2 & \(M_{1}\rightarrow\) U & \(M_{2}\rightarrow\) U \\  & 2 & \(M_{1}\rightarrow\) U & \(M_{3}\rightarrow\) U \\  & 1,50] & [1,50] \\  & 3 & \(M_{1}\rightarrow\) U & \(M_{2}\rightarrow\) U \\  & 3 & \(M_{1}\rightarrow\) U & \(M_{3}\rightarrow\) U \\  & 300,350] & [170,220] & [92,142] \\ \hline \end{tabular}
\end{table} 
Table 5: Sub-benchmark of instances for the 6-machine FDSGS problem.