## 5 Numerical Results  
Once the calibration phase has been completed, an extensive comparison campaign has been performed in order to assess the performances of the proposed GARS against the latest alternative metaheuristic procedures proposed by the relevant literature in the field of FSDGS. To this end, the hybrid Particle Swarm Optimization (PSO) algorithm developed by [PERSON], [PERSON] and [PERSON] [27] and the hybridizing metaheuristic procedure combining genetic algorithm and simulated annealing (GSA) devised by [PERSON] and [PERSON] [28] have been considered. The same experimental benchmark employed for the calibration phase has been adopted. This time, two distinct replicates have been randomly generated for each combination of experimental factors. Therefore, a total amount of 54 + 162 + 54 = 270 separate test cases have been created. Each instance has been solved by the three optimization procedures, running with a stopping criterion of \(N\times M\) seconds of CPU time. On the whole, a number of 270 \(\times\) 3= 810 experimental runs have been executed. In order to perform a fair comparison among the procedures considered, the Relative Percentage Deviation from the best heuristic solution has been computed for each algorithm and for each experimental instance as follows:  
\[RPD=100\times\frac{ALG_{sol}-Best_{sol}}{Best_{sol}} g=1,2,...,G \tag{5}\]  
where \(ALG_{sol}\) is the solution provided by a given algorithm with reference to a certain test problem and \(Best_{sol}\) is the best heuristic solution available for the same problem, _i.e._, the lowest total flow time value among those provided by the three aforementioned algorithms. The average random percentage deviations \(RPD\)s obtained by GARS, PSO, and GSA are reported in Tables 7-9. Each table refers to a given sub-benchmark of the proposed experimental design, at varying number of machines. Boldfaced values put in evidence the best average \(RPD\) the combination experimental factors changes. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Notation** & **Values** \\ \hline Population size & \(N_{pop}\) & (30, 50, **70**) \\ Crossover probability & \(p_{cr}\) & (0.5, 0.7, **0.9**) \\ Mutation probability & \(p_{m}\) & (0.05, **0.1**, 0.2) \\ Size of the sub-population to be modified through BRS procedure & \(N_{best}\) & ( 0.3 - \(N_{pop}\), 0.5 - \(N_{pop}\), \(N_{pop}\) ) \\ \hline \end{tabular}
\end{table} 
Table 6: Genetic Algorithm with Random Sampling GARS experimental calibration. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Level of Factors**} & \multicolumn{3}{c}{**Average _RPD_**} \\ \hline
**Number of groups** & **Number of jobs in a group** & **Setup times of machine \(M_{i}\)** & **GARS** & **PSO** & **GSA** \\ \hline
1 & 1 & 1 & **0.000** & **0  
The obtained results confirm the effectiveness of GARS in solving the FSDGS problem at hand. With reference to two-machine problems (See Table 7), the proposed algorithm achieves the best average _RPD_ 25 times out of 27. Its performance is strengthen by the lowest grand average _RPD_ value equal to 0.103, against PSO and GSA that gain 0.846 and 2.947, respectively. The same trend in terms of grand average _RPD_ values may be observed for problems with three-machines (See Table 8), according to which GARS assures the best average _RPD_ 72 times out of 81, while PSO and GSA gather with 16 and 13 wins, respectively. As far as six-machine problems are concerned, GARS still reports the best performances, providing the lowest average _RPD_ 24 times out of 27 and a grand average _RPD_ significantly lower than PSO and GSA. In order to infer some statistical conclusion over the difference observed among the tested algorithms, an ANOVA analysis has been performed by means of Stat-Ease(r) Design-Expert(r) 7.0.0 version commercial tool, calculating Least Significant Difference (LSD) intervals at 95% confidence level for the _RPD_s connected to each optimization procedure. The corresponding chart (see Figure 6)