# 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 results from the second round of test analyses are strongly supported by the data presented in Table 15. The strategy of initializing the population with the Eulerian heuristic for LCPP had a direct impact, now negative, assuming the relation solution quality versus computational execution speed. Particularly noteworthy is the observation that in the datasets of groups (C) and group (S), the difference in fitness values when the average ABRKGA result is superior is relatively small, being less than 3%. Simultaneously, the ABRKGA approach demonstrated execution times up to 500% shorter than the \(e\)-ABRKGA method. These results show that the ABRKGA approach outperforms the \(e\)-ABRKGA approach for the LCPP problem, particularly when considering computational execution times. Therefore, for practical applications where efficiency is a significant factor, the ABRKGA strategy is the preferred choice. ### Comparison of Results for \(e\)-Brkga vs. \(e\)-ABRKGA

This section shows the third comparison between strategies incorporating the Eulerian heuristic in Sections 5.3 and 5.4. The \(e\)-BRKGA is compared using the parameters \(p=1000\); \(p_{e}=0.30\); \(p_{m}=0.15\) and the \(e\)-ABRKGA with the parameters \(\alpha=\{0.997,0.998,0.999\}\). In this third analysis, the first point that deserves attention in analyzing the results is the average time of the approaches. Considering the \"average\" column, the average time of all executions in Table 7 was 100.20 s. On the other hand, in Table 12, it is possible to observe that the \(e\)-ABRKGA required a time 174.63% greater than the \(e\)-BRKGA, approximately 275.18 s. Considering group (S), the average execution time of the instances (Table 9) is 96.17 s, while the same attribute found in Table 14 is 179.04 s. This fact characterizes a gain of 86.17% for \(e\)-BRKGA with respect to \(e\)-ABRKGA. To consider the quality of the solutions as the second point of analysis of the results, Tables 7, 9, 12, 14 and 16 reveal the best and average solutions of each approach and the \"GAP\" column that presents the comparison between the potentials (solution quality and computational time) of each strategy. For the two values incorporated in the \"GAP\" column, if the \"fitness\" or \"time (%)\" values are positive, this indicates a percentage gain. This improvement can be either in terms of the average solution quality or the execution speed of \(e\)-BRKGA compared to \(e\)-ABRKGA or vice versa, depending on the sign. From Table 16, it is possible to verify that \(e\)-ABRKGA obtained the best average in the quality of the results in 41 instances with its set of parameters (positive percentages), tied in 6, and lost in 28 when considering the group of instances (C). For the (S) instances, the \(e\)-ABRKGA lost in 30, tied in 7, and won in 38 instances. Note that \(e\)-ABRKGA found, in the best result, an improvement of 4.65%, with an execution time 847.78% higher, that is, 2882.43 s slower than \(e\)-BRKGA, considering instance \"albano\" with \(\alpha=\{0.999\}\) for group (C). For group (S), the result was similar, an improvement of 5.15%, with an execution time 512.91% higher, that is, 1570.11 s slower than \(e\)-BRKGA, considering the instance \"albano\" with \(\alpha=\{0.997\}\). The findings of this third round of test analyses are strongly substantiated by the data presented in Table 14. Using the Eulerian heuristic in initializing the population for LCPP impacted the two aspects under consideration: solution quality and computational execution speed. It is worth noting that in the group (C) and group (S) datasets, the disparity in fitness values when the average \(e\)-BRKGA result is superior is relatively minor, generally below 3%. Moreover, the \(e\)-BRKGA approach exhibited significantly shorter computational execution times, often up to 1000% faster. These results consistently point to the superiority of the \(e\)-BRKGA approach over \(e\)-ABRKGA when considering the LCPP problem and its computational execution times.