# Evolutionary Algorithms for Optimization Sequence of Cut in the Laser Cutting Path Problem
## 4 Overview of Application of ABRKGA to LCPP
### An Eulerian Heuristic to Generate Improved Individuals for LCPP
The conclusion of this initial round of test analyses is well supported in Table 10. The strategy of initializing the population with the Eulerian heuristic for LCPP directly influenced the two aspects discussed here: solution quality and computational execution speed. Notably, in both the group (C) and group (S) datasets, the difference in fitness values when BRKGA averages better is relatively small, less than 2%. This information leads to the preference of the _e_-BRKGA approach over BRKGA for the LCPP problem, considering the computational execution times. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multicolumn{2}{c}{**Best**} & \multicolumn{2}{c}{**Worst**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Std. Deviation**} & \multicolumn{2}{c}{**Variation**} \\  & **Fitness** & **Exec.** & **Fitness** & **Exec.** & **Fitness** & **Exec.** & **Fitness** & **Exec.** & **Fitness** & **Exec.** \\ \hline albano & 107.88 & 301.77 & 109.96 & 301.59 & 108.97 & 299.99 & 0.56 & 4.34 & 0.51 & 1.45 \\ blaz1 & 163.59 & 39.04 & 166.29 & 37.48 & 164.34 & 40.09 & 0.83 & 2.27 & 0.51 & 5.66 \\ blaz2 & 295.40 & 89.47 & 298.28 & 82.66 & 296.89 & 88.25 & 0.94 & 3.86 & 0.32 & 4.38 \\ blaz3 & 495.29 & 194.61 & 500.55 & 180.00 & 497.09 & 202.88 & 1.32 & 12.68 & 0.27 & 6.25 \\ dignel & 97.09 & 48.74 & 97.57 & 49.69 & 97.28 & 48.24 & 0.18 & 2.73 & 0.19 & 5.66 \\ dighe2 & 79.37 & 40.46 & 80.09 & 39.89 & 79.63 & 38.40 & 0.20 & 1.73 & 0.25 & 4.50 \\ fu & 28.16 & 41.89 & 28.35 & 39.42 & 28.25 & 42.33 & 0.07 & 3.76 & 0.24 & 8.88 \\ inst\_10 pol & 193.90 & 37.91 & 195.08 & 37.28 & 194.33 & 37.80 & 0.42 & 2.43 & 0.21 & 6.42 \\ inst\_16 pol & 172.20 & 220.74 & 173.85 & 189.20 & 172.95 & 196.40 & 0.57 & 12.97 & 0.33 & 6.60 \\ inst\_2 pol & 36.05 & 5.96 & 36.05 & 5.96 & 36.05 & 6.71 & 0.00 & 0.49 & 0.00 & 7.23 \\ inst\_3 pol & 48.10 & 10.19 & 48.10 & 10.19 & 48.10 & 10.05 & 0.00 & 0.36 & 0.00 & 3.63 \\ inst\_4 pol & 72.15 & 14.61 & 72.40 & 12.80 & 72.21 & 13.85 & 0.09 & 0.94 & 0.13 & 6.82 \\ inst\_5 pol & 90.15 & 18.59 & 90.73 & 18.70 & 90.34 & 18.18 & 0.19 & 1.53 & 0.21 & 8.42 \\ inst\_6 pol & 114.60 & 19.37 & 115.23 & 20.15 & 114.80 & 20.75 & 0.23 & 1.26 & 0.20 & 6.07 \\ inst\_7 pol & 138.66 & 23.88 & 139.60 & 22.55 & 139.08 & 24.52 & 0.33 & 1.37 & 0.24 & 5.58 \\ inst\_8 pol & 157.20 & 27.25 & 158.25 & 24.11 & 157.65 & 27.03 & 0.32 & 1.63 & 0.20 & 6.03 \\ inst\_9 pol & 187.03 & 30.68 & 188.83 & 32.33 & 188.00 & 32.97 & 0.47 & 1.70 & 0.25 & 5.17 \\ inst\_26 pol & 196.50 & 304.46 & 201.26 & 303.18 & 197.61 & 303.26 & 1.44 & 0.65 & 0.73 & 0.21 \\ rcol & 164.06 & 31.18 & 165.53 & 36.59 & 164.71 & 33.10 & 0.47 & 2.02 & 0.28 & 6.09 \\ rco2 & 321.88 & 83.18 & 323.96 & 77.48 & 322.66 & 80.36 & 0.85 & 3.10 & 0.26 & 3.86 \\ rco3 & 486.86 & 153.02 & 491.18 & 159.72 & 488.79 & 150.16 & 1.49 & 7.87 & 0.31 & 5.24 \\ shapes2 & 230.52 & 79.09 & 231.73 & 76.11 & 231.04 & 78.82 & 0.46 & 3.62 & 0.20 & 4.59 \\ shapes4 & 452.26 & 243.26 & 455.89 & 231.39 & 453.94 & 252.24 & 1.12 & 16.11 & 0.25 & 6.39 \\ spfc & 149.41 & 55.16 & 150.18 & 55.40 & 149.76 & 51.69 & 0.26 & 2.80 & 0.17 & 5.42 \\ trousers & 308.86 & 306.82 & 334.58 & 305.88 & 316.26 & 306.12 & 8.61 & 0.79 & 2.72 & 0.26 \\ \hline Exec. time (avg) & \multicolumn{2}{c}{96.85} & \multicolumn{2}{c}{93.99} & \multicolumn{2}{c}{96.17} \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _e_-BRKGA applied to the separated set of instances (S). ### Comparison of Results for ABRKGA vs. e-ABRKGA

This section presents the second comparison between the ABRKGA strategies within this paper, explicitly pitting them against the same strategy that incorporates the Eulerian heuristic in the mechanism for creating the initial population. Through this comparison, we aimed to evaluate the performance of both ABRKGA variants and the Eulerian heuristic-based procedure. The set of three parameters for \(\alpha=\{0.997,0.998,0.999\}\) were applied for both approaches.