# Metaheuristics for the Minimum Time Cut Path Problem with Different Cutting and Sliding Speeds
## 4 Results and Discussion
### GA and BRKGA Hyper-Parameter Configuration  
GA and BRKGA hyper-parameter configurations used in the experiments were selected after preliminary tests: population of 10,000, 5000, and 1000 individuals; crossover of 70%, 75%, and 80%; and mutation of 10%, 15%, and 20%. For the elite group in BRKGA, values of 30%, 20%, and 10%; mutated population of 10%, 15%, and 20%; \(p_{a}\) of 70%. The stopping criteria were met when there was no improvement in the best solution found for 100 generations or when the execution time exceeded 300 s. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multirow{2}{*}{**Vertices**} & \multicolumn{6}{c}{**Edges**} & \multirow{2}{*}{**Items**} \\ \cline{3-3} \cline{5-8}  & & \multicolumn{2}{c}{**Original**} & & \multicolumn{2}{c}{**Adapted**} & \\ \cline{2-8}  & **(C)** & **(S)** & **(C)** & **(S)** & **(C)** & **(S)** & **(C)** & **(S)** \\ \hline albano & 156 & 164 & 164 & 164 & 173 & 164 & 24 & 24 \\ blaz1 & 39 & 44 & 44 & 44 & 46 & 44 & 7 & 7 \\ blaz2 & 70 & 80 & 88 & 80 & 89 & 80 & 14 & 13 \\ blaz3 & 97 & 132 & 132 & 132 & 130 & 132 & 21 & 21 \\ dighe1 & 20 & 54 & 46 & 54 & 38 & 54 & 15 & 15 \\ dighe2 & 20 & 46 & 38 & 46 & 30 & 46 & 10 & 10 \\ fu & 37 & 43 & 43 & 43 & 51 & 43 & 12 & 12 \\ inst\_01\_10 pol & 20 & 40 & 39 & 40 & 29 & 40 & 10 & 10 \\ inst\_01\_16 pol & 27 & 128 & 64 & 128 & 42 & 128 & 16 & 32 \\ inst\_01\_2 pol & 7 & 8 & 8 & 8 & 8 & 8 & 2 & 2 \\ inst\_01\_3 pol & 8 & 12 & 12 & 12 & 10 & 12 & 3 & 3 \\ inst\_01\_4 pol & 10 & 16 & 16 & 16 & 13 & 16 & 4 & 4 \\ inst\_01\_5 pol & 12 & 20 & 19 & 20 & 16 & 20 & 5 & 5 \\ inst\_01\_6 pol & 13 & 24 & 23 & 24 & 18 & 24 & 6 & 6 \\ inst\_01\_7 pol & 15 & 28 & 27 & 28 & 21 & 28 & 7 & 7 \\ inst\_01\_8 pol & 16 & 32 & 31 & 32 & 23 & 32 & 8 & 8 \\ inst\_01\_9 pol & 18 & 36 & 35 & 36 & 26 & 36 & 9 & 9 \\ inst\_01\_26 pol & 210 & 264 & 264 & 264 & 237 & 264 & 66 & 66 \\ rec01 & 33 & 36 & 36 & 36 & 40 & 36 & 7 & 7 \\ rec02 & 62 & 72 & 72 & 72 & 81 & 72 & 14 & 14 \\ rco3 & 82 & 108 & 108 & 108 & 116 & 108 & 21 & 21 \\ shapes2 & 68 & 70 & 70 & 70 & 78 & 70 & 8 & 8 \\ shapes4 & 127 & 140 & 140 & 140 & 147 & 140 & 16 & 16 \\ spfc\_instance & 55 & 55 & 55 & 55 & 63 & 55 & 11 & 11 \\ trousers & 350 & 388 & 388 & 388 & 424 & 388 & 64 & 64 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of instances used in the computational experiments. Tables A1 (see Appendix B.1) and A2 (see Appendix B.2) summarize the computational results. For every connect and separate instance, the results include the objective function value (FO), the elapsed time for the best execution, the average objective values, and the elapsed computational time. It is possible to observe that the BRKGA is more suitable than the standard GA for addressing the tested instances. Every hyper-parameter setting was tested for both GA and BRKGA, and each type of layout, connected (C) and separated (S). Figure 11 presents the number of times that a BRKGA hyper-parameter setting achieved the best solution found. For connected layouts (C), the best GA configuration addressing _relation-1_ (instances/executions) was obtained using the following hyper-parameters after 49 times: \(P=10{,}000\), \(txCross=0.8\), and \(txMut=0.1\). In the separated instances group (S), the best solution was obtained after 20 executions, and the configuration was the following: \(P=5000\), \(txCross=0.75\), and \(txMut=0.2\). Figure 12 presents the minimum time required to reach the best value in each execution. It is possible to observe that, for each instance, the objective function value was above the average best. A consideration of the results exposes that it is not suitable to define the best configuration for _relation-1_. While it achieves the best result in all executions for some instances, it led to the below-average value for the remaining. Figure 11: Analysis of the proposed configurations for BRKGA (_relation-1_). Figure 10: Input and output representation.