## 2 Materials and Methods
### Methodology
#### 2.4.5 Sensitivity Analysis and Parameter Tuning of GA  
To validate the robustness of the GA and select efficient parameter configurations, a sensitivity analysis was performed on key evolutionary parameters, population size, number of generations, crossover probability (\(P_{c}\)),and the weights \(w_{1}\) and \(w_{2}\) used in the composite fitness function. The aim was to assess how these values affect layout quality, station interference, and execution time.  
This analysis consisted of multiple algorithms runs, where each parameter was varied systematically while holding the others constant. The impact on performance metric, such as convergence time, number of collisions, and total area were recorded for each setting.  
Crossover Probability (\(P_{c}\)): This parameter controls how frequently the recombination operator is applied to the selected parents. Since crossover is the only evolutionary operator used, a high \(P_{c}\) (between 0.9 and 1.0) is recommended to promote broad exploration and prevent premature convergence. Slightly lower values (e.g., 0.8 or 0.85) may be considered if population diversity declines.  
Population Size and Number of Generations: These parameters were empirically tuned based on algorithm performance under different test cases. Larger populations improve genetic diversity but increase computation time. Similarly, more generations enable deeper exploration but at higher computational cost. Tests showed that using between 30 and 100 individuals and 50 to 150 generations provided a good balance between solution quality and processing efficiency.  
Weights w\({}_{1}\) and w\({}_{2}\) in the Composite Function: Each candidate solution was evaluated using a weighted fitness function (9) that combines two objectives: minimizing the total layout area (\(f_{area}\)) and minimizing physical interferences between stations (\(f_{interference}\)). The following weight combinations were tested: \(w_{1}=1\), \(w_{2}=100\); \(w_{1}=1\), \(w_{2}=500\); \(w_{1}=1\), \(w_{2}=1000\).  
The final selection was based on solutions that avoided interference while maintaining compact layouts. Interference reduction was prioritized due to its critical impact on the physical feasibility of the RC.  
Table 1 presents a representative subset of the results obtained, specifically the top 10 scenarios evaluated during the sensitivity analysis. Each row shows the parameter combinations that delivered the best performance in terms of layout compactness, interference elimination, and execution time. This selection highlights the most efficient configurations and provides empirical support for the final parameter choices used in the GA implementation.  
The sensitivity analysis enabled the identification of efficient configurations of the GA in terms of both solution quality and execution time. The most effective parameter combination was achieved with a population size of 60,100 generations, a crossover probability \(P_{c}=0.95\), and weights \(w_{1}=1\) and \(w_{2}=500\). This configuration produced a layout with an occupied area of 81 square units, no interferences between stations, and an execution time of only 15.12 min.  
This result shows that moderate parameterization can be sufficient to reach optimal solutions (in terms of population size and number of generations), provided that adequate selective pressure is maintained through crossover and that geometric feasibility is prioritized via proper weighting in the objective function.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Population** & **Generations** & **P\({}_{\mathbf{c}}\)** & **w\({}_{\mathbf{1}}\)** & **w\({}_{\mathbf{2}}\)** & **Area (u\({}^{2}\)) 1** & **Interferences** & **Execution Time (min)** \\ \hline
60 & 100 & 0.95 & 1 & 500 & 81 & 0 & 15.12 \\
60 & 100 & 0.95 & 1 & 500 & 95 & 0 & 17.25 \\
60 & 150 & 0.95 & 1 & 1000 & 71 & 1 & 24.92 \\
30 & 50 & 0.90 & 1 & 100 & 85 & 1 & 23.04 \\
100 & 100 & 1.00 & 1 &