# 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 significance of 3600 s was considered reasonable by our team. We aim to consider the optimal results obtained and check with the BRKGA, ABRKGA, and variations. Tables 4 and 5 present results for the MIP flow model in the set of instances connected and separated. The column instance notes the name of the instance; column solution reports the best solution obtained by the model; column time is the elapsed time in seconds if the model reached the time limit of 1 h, this is represented by LIMIT; column gap registers the gap as presented by the solver at the end of execution; and column nodes reports the number of nodes in the branch and bound tree. We filter instances containing a maximum of 40 nodes to carry out tests with the model. The results of Tables 4 and 5 show that the MIP model can only solve a small number of the instances (four in both cases), and the instances that were solved have a small size (number of vertices plus the number of edges). One can verify that the models reach the time limit for most instances. Furthermore, for some instances, it was trapped in the root node of the branch and bound tree, and for the instance blaz2 of the separated benchmark, the model could not even be produced due to memory issues. All these remarks lead us to conclude that, although this model presents a theoretical contribution to LCPP, it still needs to be a practical method, and heuristic methods are needed to tackle large instances. However, we intend to use the results presented by the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance (Genrected)** & **Solution** & **Time** & **GAP** & **Nodes** \\ \hline inst\_2 pol & 33.13 & 0.02 & 0.0 & 0 \\ inst\_3 pol & 39.25 & 1.54 & 0.0 & 469 \\ inst\_4 pol & 57.38 & 24.61 & 0.0 & 25,463 \\ inst\_5 pol & 69.63 & 2595.58 & 0.0 & 1,393,088 \\ inst\_6 pol & 102.88 & LIMIT & 1.0 & 587,126 \\ inst\_7 pol & 109.38 & LIMIT & 1.0 & 314,325 \\ inst\_8 pol & 113.61 & LIMIT & 1.0 & 189,994 \\ inst\_9 pol & 181.54 & LIMIT & 1.0 & 114,572 \\ inst\_10 pol & 157.59 & LIMIT & 1.0 & 70,963 \\ dighe1.txt & 75.64 & LIMIT & 1.0 & 25,198 \\ dighe2.txt & 84.56 & LIMIT & 1.0 & 54,868 \\ rocl.txt & 345.63 & LIMIT & 1.0 & 8798 \\ blaz1.txt & 449.59 & LIMIT & 1.0 & 2118 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for the MIP flow model on the connected set of instances. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance (Separated)** & **Solution** & **Time** & **GAP** & **Nodes** \\ \hline inst\_2 pol & 36.05 & 0.21 & 0.0 & 152 \\ inst\_3 pol & 48.10 & 6.55 & 0.0 & 1618 \\ inst\_4 pol & 72.15 & 138.18 & 0.0 & 48,868 \\ inst\_5 pol & 90.15 & 1039.98 & 0.0 & 95,880 \\ inst\_6 pol & 184.66 & LIMIT & 1.0 & 52,374 \\ inst\_7 pol & 244.31 & LIMIT & 1.0 & 12,773 \\ inst\_8 pol & 260.18 & LIMIT & 1.0 & 21,074 \\ inst\_9 pol & 300.32 & LIMIT & 1.0 & 11,166 \\ inst\_10 pol & 395.04 & LIMIT & 1.0 & 3874 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results for the MIP flow model on the separated set of instances. model as an initial guide for the solutions of the computational experiments proposed in this research. ### Comparison of Results for BRKGA vs. e-BRKGA

This section shows the first comparison between some strategies discussed in the context of this paper. Thus, the results of BRKGA, as presented by [9], are confronted with the same strategy. However, we incorporate the Eulerian heuristic into the mechanism for creating the initial population. We will refer to this approach as _e_-BRKGA to simplify the terms. The same parameters (\(p=1000;p_{e}=0.30;p_{m}=0.15\)) were applied to both approaches, and the stopping rule was approximately 300 s of time execution or 100 generations without improvement on the best solution found. Table 6 (BRKGA) and Table 7 (_e_-BRKGA) present the best, worst, average, standard deviation, and coefficient of variation for the ten computational runs of each instance from the connected group (C).