## 5 Computational Experiment  
### Problem Instance  
The algorithm employed in this research was implemented using Python 3.9 and its performance was evaluated on a computer equipped with an AMD Ryzen 7 5800H with a Radeon Graphics CPU, operating at a clock speed of 3.20 GHz. The computer had eight cores and 8 GB of memory. The test data are sourced from the European Special Interest Group on Cutting and Packing (ESICUP), which operates under the European Operational Research Society (EURO). The data can be accessed at the following website: [[https://www.euro-online.org/websites/esicup/](https://www.euro-online.org/websites/esicup/)]([https://www.euro-online.org/websites/esicup/](https://www.euro-online.org/websites/esicup/)) (accessed on 11 January 2023). The packing instance data utilized in this study are displayed in Table 1, encompassing a range of combinations involving concave polygons and convex polygons. The provided test instances encompass a range of complexities and configurations, thereby mirroring the irregular packing challenges encountered in diverse industries. This extensive assortment of cases facilitates a comprehensive assessment of the performance of our proposed algorithm across various packing scenarios. Concerning the genetic algorithm, the parameters were set to the values presented in Table 2 after conducting the preliminary experiments. Population size: A deliberate decision was made to select an initial population size of 40 individuals in order to strike a balance between ensuring an adequate level of diversity for the optimization process and avoiding the computational burden associated with excessively complex calculations. While an increased population size has the potential to offer a wider range of solutions, it also results in an elevated computational overhead. By conducting a thorough analysis of the trade-off between diversity and computational efficiency, we have empirically determined that a population size of 40 is appropriate. This choice allows for an adequate level of diversity while still keeping the computational complexity at a manageable level. Elite individual ratio: In order to ensure the preservation and promotion of the highest-performing individuals for the next generation, we implemented an elite individual ratio \begin{table}
\begin{tabular}{c c c c} \hline
**Instance** & **Total Number of Pieces** & **Number of Pieces Types** & **Sheet Width** & **Feasible Orientations** \\  & & & **(Degrees)** \\ \hline Albano & 24 & 8 & 4900 & 0 and 180 \\ Blaz1 & 28 & 7 & 15 & 0 and 180 \\ Fu & 12 & 12 & 38 & 0, 90 and 180 \\ Dagli & 30 & 10 & 60 & 0 and 180 \\ Jakobs1 & 25 & 25 & 40 & 0, 90, 180 and 270 \\ Mao & 20 & 9 & 2550 & 0, 90 and 180 \\ Marques & 24 & 8 & 104 & 0, 90, 180 and 270 \\ Shapes0 & 43 & 4 & 40 & 0 \\ Shapes1 & 43 & 4 & 40 & 0 and 180 \\ Shirts & 99 & 8 & 40 & 0 and 180 \\ Swim & 48 & 10 & 5752 & 0 and 180 \\ Trousers & 64 & 17 & 79 & 0 and 180 \\ \hline \end{tabular}
\end{table} 
Table 1: Packing instance data. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Population size & 40 \\ Elite individual ratio & 0.125 \\ Mutation rate & 0.1 \\ Fitness calculation & \(\frac{\theta}{R_{\text{max}}}\) \\ Stopping criteria & Stop after 30 generations \\ \hline \end{tabular}
\end{table} 
Table 2: The GA parameters. of 0.125. This phenomenon facilitated the survival of the most capable individuals across generations, thereby promoting the attainment of optimal solutions. Mutation rate: In relation to genetic mutation, a mutation rate of 0.1 was employed for the specified gene. We conducted a comprehensive series of experiments and found that increasing the mutation rate leads to greater exploration within the solution space, thereby enabling escape from local optima. Conversely, a reduced rate facilitates the exploitation of refined solutions, but it also carries the potential risk of premature convergence.