# 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
\begin{table}
\begin{tabular}{c r r r r r r r r r r} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Instances**}} & \multicolumn{2}{c}{**Best**} & \multicolumn{2}{c}{**Worst**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Std. Deviation**} & \multicolumn{2}{c}{**Variation**} \\  & \multicolumn{1}{c}{**Fitness**} & \multicolumn{1}{c}{**Exec.**} & \multicolumn{1}{c}{**Fitness**} & \multicolumn{1}{c}{**Exec.**} & \multicolumn{1}{c}{**Fitness**} & \multicolumn{1}{c}{**Exec.**} & \multicolumn{1}{c}{**Fitness**} & \multicolumn{1}{c}{**Exec.**} & \multicolumn{1}{c}{**Fitness**} & \multicolumn{1}{c}{**Exec.**} \\ \hline albano & 113.09 & 303.06 & 116.14 & 303.15 & 113.74 & 303.53 & 0.95 & 0.64 & 0.84 & 0.21 \\ blaz1 & 162.23 & 145.88 & 163.41 & 142.45 & 162.82 & 139.82 & 0.32 & 7.05 & 0.20 & 5.04 \\ blaz2 & 292.01 & 301.86 & 293.67 & 300.93 & 292.88 & 299.80 & 0.60 & 4.46 & 0.21 & 1.49 \\ blaz3 & 493.35 & 302.32 & 499.90 & 302.43 & 496.15 & 302.84 & 1.77 & 0.34 & 0.36 & 0.11 \\ dighe1 & 96.18 & 186.45 & 96.66 & 198.26 & 96.40 & 194.69 & 0.16 & 12.22 & 0.17 & 6.28 \\ dighe2 & 78.66 & 137.75 & 79.16 & 147.81 & 78.96 & 153.10 & 0.18 & 13.34 & 0.22 & 8.71 \\ fu & 28.00 & 137.50 & 28.20 & 140.58 & 28.10 & 141.59 & 0.06 & 4.76 & 0.21 & 3.36 \\ inst\_10 pol & 193.09 & 134.11 & 194.14 & 135.72 & 193.48 & 135.04 & 0.29 & 6.15 & 0.15 & 4.55 \\ inst\_16 pol & 172.74 & 303.31 & 174.69 & 303.12 & 173.59 & 302.65 & 0.63 & 0.65 & 0.36 & 0.21 \\ inst\_2 pol & 36.05 & 24.20 & 36.05 & 24.20 & 36.05 & 24.66 & 0.00 & 1.31 & 0.00 & 5.29 \\ inst\_3 pol & 48.10 & 32.16 & 48.10 & 32.16 & 48.10 & 32.49 & 0.00 & 1.00 & 0.00 & 3.08 \\ inst\_4 pol & 72.15 & 48.82 & 72.15 & 48.82 & 72.15 & 48.15 & 0.00 & 3.31 & 0.00 & 6.87 \\ inst\_5 pol & 90.15 & 53.17 & 90.20 & 60.56 & 90.16 & 57.58 & 0.02 & 4.41 & 0.03 & 7.67 \\ inst\_6 pol & 114.23 & 73.97 & 114.55 & 77.93 & 114.43 & 74.20 & 0.10 & 4.67 & 0.09 & 6.29 \\ inst\_7 pol & 138.47 & 104.92 & 138.91 & 80.33 & 138.70 & 89.79 & 0.16 & 7.09 & 0.11 & 7.90 \\ inst\_8 pol & 156.68 & 101.20 & 157.18 & 102.50 & 156.92 & 102.61 & 0.18 & 6.62 & 0.11 & 6.45 \\ inst\_9 pol & 187.10 & 117.63 & 187.65 & 130.63 & 187.34 & 122.25 & 0.21 & 5.22 & 0.11 & 4.27 \\ inst\_26 pol & 218.68 & 308.02 & 219.84 & 304.34 & 219.16 & 306.23 & 0.33 & 1.40 & 0.15 & 0.46 \\ rool & 162.78 & 109.34 & 163.88 & 116.15 & 163.25 & 110.32 & 0.30 & 6.24 & 0.19 & 5.66 \\ roo2 & 317.76 & 264.95 & 319.63 & 262.81 & 318.71 & 271.21 & 0.60 & 6.22 & 0.19 & 2.29 \\ roo3 & 480.66 & 301.41 & 485.71 & 302.02 & 481.99 & 301.92 & 1.38 & 0.42 & 0.29 & 0.14 \\ shapes2 & 227.83 & 237.60 & 229.71 & 266.72 & 228.87 & 256.27 & 0.54 & 13.71 & 0.24 & 5.35 \\ shapes4 & 453.30 & 302.06 & 455.87 & 303.49 & 454.13 & 302.92 & 0.84 & 0.51 & 0.18 & 0.17 \\ spfc & 147.77 & 195.91 & 148.81 & 196.75 & 148.21 & 202.18 & 0.30 & 17.37 & 0.21 & 8.59 \\ trousers & 343.11 & 312.32 & 347.59 & 306.85 & 346.43 & 309.98 & 1.33 & 2.3 & 0.38 & 0.72 \\ \hline Exec. time (avg) & & 181.60 & & 183.63 & & 183.43 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: BRKGA applied to the separated set of instances (S). ment with a 3.34% longer execution time. In \"trousers\", the average fitness gain was 6.31% in 28.88 s (\(-\)9.28%) more for the algorithm to stop. On the other hand, in some instances, _e_-BRKGA improved compared to BRKGA, both in the average fitness and execution time. The entries \"blaz3\" and \"inst_7 pol\" demonstrated slight gains in fitness, 0.13% and 0.02%, respectively. However, in terms of execution time, the improvement reaches 34.85% (105.50 s) for \"blaz3\" and 72.35% (45.77 s) for \"inst_7 pol. For the group of separated instances (S), the results exhibit similar characteristics of (C), except for the execution time of _e_-BRKGA, which is better in all instances. In this examination, it is crucial to note that out of the five instances where _e_-BRKGA showed a superior average fitness: \"albano\", \"inst_16 pol\", \"inst_26 pol\", \"shapes4\", and \"trousers\".