# 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
On the other hand, the remaining components are problem-agnostic and can be reused in a general-purpose framework. Using an Eulerian path heuristic, we also consider a specialized initial population and new individual mutation. Algorithm 3 provides the pseudocode for our adaptation for ABRKGA to tackle LCPP. The procedure's input parameters are the required cutting edges (\(n\)) and the specific ABRKGA parameter \(\gamma\). \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Suggested Value Range** \\ \hline \(p\) & \(\approx[100,1000]\) \\ \(p_{e}\) & \(max[3,k_{e}*p]\) where \(k_{e}\in[0.10,0.25]\) \\ \(p_{m}\) & \(max[1,k_{m}*p]\) where \(k_{m}\in[0.05,0.20]\) \\ \(\rho_{e}\) & \(0.65\leq\rho_{e}\leq 0.80\) \\ \(max_{gen}\) & \([50,500]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter definitions and recommended value ranges. \begin{table}
\begin{tabular}{c c c} \hline
**Search Property** & \(max_{gen}\) & \(\gamma\) \\ \hline \(p1\) & 271 & 0.999 \\ \(p2\) & 740 & 0.998 \\ \(p3\) & 2017 & 0.997 \\ \hline \end{tabular}
\end{table} 
Table 2: Available options for parameter \(\gamma\). ```
1 procedure\(ABRKGA(n,\gamma,V_{c},V_{m})\)
2 ifinput layout has a Eulerian circuitthen
3 apply an offset (use \(V_{m}\)) to the point closest to the origin individual = path back to origin
4 compute fitness of individual returnindividual
5
6 end if
7
8 else
9 Apply Eulerian path heuristic at input data /* see Section 4.3 */
10 Create a population \(POP\), consisting of p vectors of \(2n+2\) random keys while(\(max_{gen}\) not reached)do
11 Compute the fitness (use \(V_{m}\) and \(V_{c}\)) of each newly generated solution within \(Pop\). 12 Update \((p,\alpha,k_{c},k_{m})\) /* see Equations (7) and (10)-(12) */
13 Divide \(Pop\), into two distinct sets: \(RCL\) and \(non-RCL\). 14 Initialize the population of the next generation. \(POP^{+}=RCL\) Apply perturbations to the similar solutions within \(POP^{+}\) /* use the intensity parameter \(\beta\). */
15 Create mutants \(POP_{m}\), for the next generation. \(POP^{+}=POP^{+}\cup POP_{m}\) /* crossover process lines */
16 for(\(i=1\) to \(p-|RCL|-|POP_{m}|\))do
17 Select parent \(a\) at random from \(RCL\)
18 Select parent \(b\) at random from \(non-RCL\)
19 Set \(\rho_{e}\) with random-key \(b[2n+1]\) /* see Equation (14) */
20 for(\(j=1\) to \(2n+2\))do
21 if\(random()<\rho_{e}\)then
22\(c[j]=a[j]\)
23 end if
24 else
25\(c[j]=b[j]\)
26 end if
27\(POP^{+}=POP^{+}\cup\{c\}\)
28 end if
29 Update population \(POP=POP^{+}\)
30 end for
31\(return\) individual with \(argmin\{fitness(x)|x\in Pop\}\)
32 end for
```

**Algorithm 3**Pseudocode of ABRKGA to tackle LCPP

## 5 Computational Results

The computational experiments were performed on a machine with a max turbo clock speed of 4.90 GHz, 11 th Gen Intel(R) Core(TM) i7-11700 processor, 16 GB RAM, and running the Windows 10 operational system. All algorithms were implemented using the Python language version 3.7. The input data consisted of layout instances extracted using the research developed by [PERSON] et al. [PERSON] et al. (2017), resulting in SVG files. The subsequent sections provide details about the benchmark instances (Section 5.1), the results obtained with the execution of the LCPP model (Section 5.2), a comparison between BRKGA x \(e\)-BRKGA (Section 5.3), ABRKGA without and with an improved initial population with the Eulerian heuristic (Section 5.4), called here \(e\)-ABRKGA, and the \(e\)-BRKGA x \(e\)-ABRKGA in Section (5.5). Please refer to the Data Availability Statement for complete information on the data used and constructed in this paper. ### Instances

We utilized a set of 50 problem instances to evaluate the presented approaches. These instances were categorized based on the possible presence of space between the pieces in the input layout, categorized as connected (C) instances (see Figures A1 and A2 in Appendix A.1) or separated (S) instances (see Figures A3-A5 in Appendix A.2).