## 5 Results and Discussion
### Discussion on PSO Results  
The simulation was simulated in C ++ and applied on a computer using the C ++ language on the computer having the Windows 10 operating system with the specifications of 8 Gigabyte ram and 2.3 GHz Core (M) 2 Duo CPU processor. In order to measure the execution of the proposed approaches, WELL-based PSO (WE-PSO), Torus-based PSO (TO-PSO), and Knuth-based PSO (KN-PSO), a group of fifteen non-linear benchmark test functions were utilized to do the comparison of WE-PSO, TOPSO, and KN-PSO with standard PSO, SO-PSO, and HPSO. These functions are normally applied to investigate the performance of any technique. Hence, in our study, we used them to examine the optimization outcomes of the quasi-random-based approach of WE-PSO, TO-PSO, KN-PSO, SO-PSO, and H-PSO. The list of those functions is available in Table 3. In Table 1, D(Dimensions) shows the dimensionality of the problem, S (Search Space) represents the interval of the variables, I(Iterations), Pop(Population size) and in Table 2, \(f_{\min}\) denotes the common global optimum minimum value. The parameters for the simulation use c1 = c2 = 1.45, inertia weight w is used in the interval [0.9, 0.4], and swarm size is 50. For simulation, the function dimensions are \(D\) = 10, 20, and 30 and a maximum number of epochs is 3000. All techniques were applied to similar parameters for comparatively effective results. In order to check the performance of each technique, all algorithms were tested for 30 runs. The purpose of this study continues to observe whereby the unique characteristics of experimental results rely on dimensions of these standard benchmark functions. The objective of this study is to find the most suitable initializing approach for the PSO and during the first experiment, the proposed WE-PSO, TO-PSO, and KN-PSO with other approaches SO-PSO, H-PSO, and standard PSO was investigated. The objective of the second simulation is to find the nature of the dimension regarding standard function optimization. Lastly, the simulation results of WE-PSO, TO-PSO, and KN-PSO were compared with standard PSO, SO-PSO, and H-PSO. In the rest of the paper, simulation results are discussed in detail. Figures 7-22 contain the graphical representation of the comparisons of the proposed WE-PSO, TO-PSO, and KN-PSO with standard PSO, H-PSO, and SO-PSO. In the \(x\)-axis, dimensions of the problem 10, 20, and 30 are presented while the \(y\)-axis represents the mean best against each dimension of the problem. We can see that the majority of the figures contain a better convergence curve for KN-PSO on functions F1, F2, F3, F4, F5, F6, F17, F8, F9, F10, F11, F12, F13, F14, and F15 over WE-PSO, TO-PSO, H-PSO, SO-PSO, and standard PSO on all dimensions comprehensively. The other proposed approach TO-PSO provides better results over WE-PSO on functions F1, F2, F4, F6, F17, F8, F9, F10, F14, F15, and F16 and beats on all functions for PSO, SO-PSO, and TO-PSO. In this simulation, PSO is initialized with the WELL sequence (WE-PSO), [PERSON] sequence (TO-PSO), and [PERSON] sequence (KN-PSO) instead of uniform distribution. The variant proposed WE-PSO, TO-PSO, and KN-PSO is compared with other initialized approaches, [PERSON] sequence (SO-PSO), [PERSON] sequence (H-PSO), and standard PSO. The experimental results give superior results in higher dimensions for KN-PSO on other SO-PSO, H-PSO, PSO, and proposed approach TO-PSO and WE-PSO. Figure 21: Convergence curve on F15. Figure 22: Convergence curve on F16. The core objective of this simulation setup is to find the superiority of results depending upon the dimension of the functions that are to be optimized. In experiments, three dimensions for benchmark functions \(D=10\), \(D=20\), and \(D=30\) were used.