# 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 c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multicolumn{3}{c}{**GAP**} & \multicolumn{3}{c}{**GAP**} \\  & \multicolumn{3}{c}{**Connected (C)**} & \multicolumn{3}{c}{**Separated (S)**} \\  & **Fitness (\%)** & **Time (s)** & **Time (\%)** & **Fitness (\%)** & **Time (s)** & **Time (\%)** \\ \hline albano & \(7.68\%\) & \(5.04\) & \(-1.66\%\) & \(4.19\%\) & \(-3.54\) & \(1.17\%\) \\ blaz1 & \(-1.04\%\) & \(-109.15\) & \(71.73\%\) & \(-0.94\%\) & \(-99.73\) & \(71.33\%\) \\ blaz2 & \(-1.63\%\) & \(-127.31\) & \(42.19\%\) & \(-1.37\%\) & \(-211.56\) & \(70.57\%\) \\ blaz3 & \(0.13\%\) & \(-105.50\) & \(34.85\%\) & \(-0.19\%\) & \(-99.96\) & \(33.01\%\) \\ dighe1 & \(-0.71\%\) & \(-90.87\) & \(65.81\%\) & \(-0.92\%\) & \(-146.45\) & \(75.22\%\) \\ dighe2 & \(-0.18\%\) & \(-75.42\) & \(76.13\%\) & \(-0.85\%\) & \(-114.69\) & \(74.91\%\) \\ fu & \(-0.48\%\) & \(-131.27\) & \(71.16\%\) & \(-0.57\%\) & \(-99.26\) & \(70.10\%\) \\ inst\_10 pol & \(-1.47\%\) & \(-48.19\) & \(53.42\%\) & \(-0.44\%\) & \(-97.24\) & \(72.01\%\) \\ inst\_16 pol & \(-1.46\%\) & \(-71.59\) & \(54.16\%\) & \(0.37\%\) & \(-106.25\) & \(35.11\%\) \\ inst\_2 pol & \(0.00\%\) & \(-16.21\) & \(67.32\%\) & \(0.00\%\) & \(-17.95\) & \(72.78\%\) \\ inst\_3 pol & \(0.00\%\) & \(-19.10\) & \(70.02\%\) & \(0.00\%\) & \(-22.44\) & \(69.07\%\) \\ inst\_4 pol & \(-0.44\%\) & \(-17.99\) & \(52.54\%\) & \(-0.09\%\) & \(-34.30\) & \(71.23\%\) \\ inst\_5 pol & \(-0.02\%\) & \(-29.67\) & \(70.21\%\) & \(-0.19\%\) & \(-39.40\) & \(68.43\%\) \\ inst\_6 pol & \(-0.70\%\) & \(-26.03\) & \(50.79\%\) & \(-0.33\%\) & \(-53.45\) & \(72.03\%\) \\ inst\_7 pol & \(0.02\%\) & \(-45.77\) & \(72.35\%\) & \(-0.27\%\) & \(-65.27\%\) & \(72.69\%\) \\ inst\_8 pol & \(-0.21\%\) & \(-48.29\) & \(72.03\%\) & \(-0.47\%\) & \(-75.58\) & \(73.66\%\) \\ inst\_9 pol & \(-1.25\%\) & \(-48.36\) & \(58.67\%\) & \(-0.35\%\) & \(-89.28\) & \(73.03\%\) \\ inst\_26 pol & \(10.69\%\) & \(10.21\) & \(-3.34\%\) & \(9.84\%\) & \(-2.97\) & \(0.97\%\) \\ roc1 & \(-0.83\%\) & \(-85.87\) & \(70.38\%\) & \(-0.90\%\) & \(-77.22\) & \(70.00\%\) \\ roc2 & \(-1.67\%\) & \(-208.51\) & \(69.15\%\) & \(-1.24\%\) & \(-190.86\) & \(70.37\%\) \\ roc3 & \(-1.48\%\) & \(-138.98\) & \(45.96\%\) & \(-1.41\%\) & \(-151.76\) & \(50.26\%\) \\ shapes2 & \(-1.74\%\) & \(-198.68\) & \(66.76\%\) & \(-0.95\%\) & \(-177.45\) & \(69.24\%\) \\ shapes4 & \(2.56\%\) & \(4.30\) & \(-1.42\%\) & \(0.04\%\) & \(-50.68\) & \(16.73\%\) \\ spfc & \(-1.19\%\) & \(-167.99\) & \(73.97\%\) & \(-1.05\%\) & \(-150.49\) & \(74.44\%\) \\ trousers & \(6.31\%\) & \(28.88\) & \(-9.28\%\) & \(8.71\%\) & \(-3.86\) & \(1.24\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: GAP comparison BRKGA _x e-_BRKGA to connected (C) and separated (S) instances. tational time) of each strategy. For the two values incorporated in the \"GAP\" column, if the \"fitness\" or \"time (%)\" values are positive, it indicates a percentage gain. This improvement can be either in terms of the average solution quality or the execution speed of ABRKGA compared to \(e\)-ABRKGA or vice versa, depending on the sign. From Table 15, it is possible to verify that \(e\)-ABRKGA obtained the best average in the quality of the results in 48 instances with its set of parameters (positive percentages), tied in 5, and lost in 22 when considering the group of instances (C). For the (S) instances, the e-ABRKGA only lost in the \"trousers\" instance with \(\alpha=\{0.999\}\). Notice that \(e\)-ABRKGA found, on average, a 6.7% improvement, with an execution time 152.1% higher, meaning 125.05 s slower than ABRKGA, considering the \"albano\" instance with \(\alpha=\{0.997\}\). For the group of separate instances (S), the results show similar characteristics to (C), except for the execution time of \(e\)-ABRKGA, which is worse in all instances. In this examination, it is essential to note that in almost all instances \(e\)-ABRKGA presented a superior average fitness.