# 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 same structure was adopted in Table 8 (BRKGA) and Table 9, except that these two other tables represent execution for the separated group (S). The first point deserving highlight in the analysis of the results is the average computational time of the approaches. Considering the \"average\" column, the average computational time for all executions in Table 6 was 170.70 s. In contrast, in Table 7, it is possible to observe that the _e_-BRKGA required a time 41.30% less than BRKGA, approximately 100.20 s. Considering the group (S), the average execution times of instances (Table 8) is 183.43 s, while the same attribute found in Table 9 is 96.17 s. This fact characterizes a gain of 47.57% of the _e_-BRKGA with respect to the BRKGA. The coefficient of variation (variation column) allows the determination of the precision of the measurements and the variability within samples. Regarding the execution time, we highlight the variation in the instances \"blaz1\", \"dighe2\", and \"inst_9 pol\" from group (C), which were the only ones that showed values higher than 10% (moderate variation). In the instances from Table 7, the entries \"dighe1\", \"inst_2 pol\", \"inst_4 pol\", and \"shapes2\" also fall under a moderate aspect regarding the variation in the samples. The only instance that draws attention to the context of the coefficient of variation is \"inst_6 pol\", with a value of 21.79%. Even so, there still needs to be a sizable relative variability in the data about the mean. Tables 8 and 9 present the execution time variation, presenting all values less than 10%. In summary, we can consider the execution times based on their respective averages due to the low value of the coefficient of variation attribute. We also highlight that 76% of the instances in group (C) and 80% in group (S) have GAPs (see Table 10) more significant than 50% concerning the difference in execution times for _e_-BRKGA. Even in the column of the best and worst solutions found, when the developed Eulerian heuristic is applied to the initial population, it presents gains in execution time when applying _e_-BRKGA to LCPP. In the group of instances (C), only four instances had a longer execution time for e-BRKGA. However, for the set (S), no _e_-BRKGA execution took more computational time than BRKGA. All evolution graphs (best individual fitness x number of generation) are available in Data Availability Statement. \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 & 108.17 & 304.68 & 113.10 & 303.76 & 110.97 & 304.17 & 1.46 & 0.54 & 1.32 & 0.18 \\ blaz1 & 154.71 & 146.10 & 155.86 & 148.94 & 155.09 & 152.18 & 0.37 & 17.03 & 0.24 & 11.19 \\ blaz2 & 269.94 & 302.04 & 271.95 & 301.69 & 270.81 & 301.74 & 0.75 & 0.35 & 0.28 & 0.12 \\ blaz3 & 423.96 & 303.11 & 428.01 & 302.96 & 425.39 & 302.75 & 1.42 & 0.48 & 0.33 & 0.16 \\ dighe1 & 70.57 & 158.89 & 70.88 & 140.12 & 70.72 & 138.07 & 0.10 & 8.24 & 0.14 & 5.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: BRKGA applied to the connected set of instances (C).