# Non-Cutting Moving Toolpath Optimization with Elitist Non-Dominated Sorting Genetic Algorithm-II
## 4 Simulation Results and Experimental Validations
### Analysis of the Impact of PP on the Distance between Cuts in CNC Processes  
The effectiveness of optimizing PP using the NSGA-II algorithm to reduce the distance between cuts in CNC machining processes is validated with experimental case studies. Throughout our experimentations, consistent parameter settings are maintained for the NSGA-II, as outlined in Table 2.  
NSGA-II parameter values are given in Table 2. \(N\) refers to the number of individual solutions maintained in a population at any given time during the execution of the algorithm. \(G\) indicates the total number of cycles that the algorithm performs to evolve the population from one generation to the next. Each generation represents an iteration of the optimization process. \(n_{c}\) influences the spread of offspring around the parent solutions. It determines how tightly the offspring are clustered around the parents. \(p_{c}\) is the probability that crossover will occur during the reproduction process. \(n_{m}\), similar to the crossover distribution index, controls the spread of mutations applied to an offspring. It affects how mutations are distributed across the chromosome of a solution. \(p_{m}\) defines the probability  
Figure 4: Presentation of the Pareto optimal solution formed by the NSGA-II and MOGA algorithms. of a mutation occurring in an offspring. Mutations introduce random variations into the genetic material, which helps to maintain genetic diversity and potentially discover new and better solutions. Figure 5 illustrates a rectangular workpiece with multiple holes. It can be seen from the figure that the toolpath optimization for holes drilled into a rectangular part is conducted using the NSGA-II algorithm, resulting in a shortened and smoothed path. The algorithm updated the path through shortening the distance to the necessary hole coordinates, tending to draw a smooth path by avoiding the sharp corners as much as possible. Significant improvements are achieved at the end of the optimization. The original toolpath length of 272.4 mm is reduced to 188.7 mm, representing a reduction of approximately 30%. This demonstrates the effectiveness of the NSGA-II in enhancing the efficiency of the machining processes through optimizing the movement and trajectory of the toolpaths. This approach ensures reliable the comparative analysis and robust evaluation of the proposed PP methodology. Simulations are carried out using a computer with Intel i7 13 th Gen Processor 20 Cores, resulting in an execution time of 8.72 s in the Python 3.10.12 simulator. Workpiece 1 and 2 utilize both non-cutting and machining moves while being processed on CNC machines. The machining path is depicted with blue lines, while the non-cutting moves are represented with orange lines. The original PP for Workpiece 1 is shown in Figure 6. It can be seen from the figure that the PP is abundant in non-cutting moves. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Symbol** & **Value** \\ \hline Number of populations & \(N\) & 300 \\ Generation size & \(G\) & 500 \\ Crossover index & \(n_{c}\) & 5 \\ Crossover rate & \(p_{c}\) & 0.9 \\ Mutation index & \(n_{m}\) & 20 \\ Mutation rate & \(p_{m}\) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: NSGA-II parameter values. Figure 5: A rectangular workpiece with multiple holes; (**a**) the original toolpath, (**b**) the optimized toolpath. Figure 7 illustrates the optimized PP for Workpiece 1. It can be seen from the figure that a significant improvement of 29.29% has been achieved in the non-cutting distances. The distance is reduced from 5678.4 mm to 4015.2 mm. This results in 1663.2 mm of travel distance being saved. The experimental results of the non-cutting distance for Workpiece 1 are presented in Table 3.  
Moreover, a new and different workpiece (Workpiece 2) is used to see the effectiveness of the performed approach (Figure 8). It can be seen from Figure 9 that the optimized PP for Workpiece 2 has reduced the non-cutting distances from 4703.6 mm to 3203.2 mm.