## 5 Numerical Studies and Performance Analysis of MPACO and CMPACO  
### Performance Comparisons of MPACO and ACO  
This section provides the performance comparisons between ACO and the proposed algorithm; MPACO under networks with fixed topology. As explained in the previous section, the MPACO algorithm has the advantage of generating an efficient route faster than ACO algorithm in a large network, by simultaneously searching for routes in multiple directions, and updating pheromones based on GPR. In order to verify the validity of the proposed algorithm, numerical experiments and analyses under labyrinth type networks are provided and compared with ACO algorithm.  
Tables 6 and 7 summarize the conditions and the assumptions for the numerical experiments conducted and the parameters used. Table 8 shows the test platform parameters.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Type** & **Detailed Assumptions** \\ \hline Condition No.1 & The same number of ants is used in both ACO and \\  & MPACO tests. \\ \hline Condition No. 2 & The locations for ant colony and food location are the same in each network for both cases. \\ \hline Assumption No. 1 & When ants in different groups meet each other in a \\  & node, the ants stop moving. \\ \hline Assumption No. 2 & When the ants reach the wall/block of the maze, they \\  & go back to find another route. \\ \hline \hline \end{tabular}
\end{table}
Table 6: Experimental conditions and assumptions.  
Figure 6: Estimation of pheromone using a discretized Gaussian distribution.  
Figure 7: Calculation of pheromone in both collision nodes.  
Figure 8 illustrates the tested network, and how the intensity of pheromone is strengthened during the iterations using the proposed MPACO algorithm. As shown in Figure 8a-d, the x and y axes are the coordinates of the node, and the z axis indicates the amount of pheromone. The explained GPR process is applied in every iteration, and the pheromone is deposited around the collision nodes. As a result, it can be seen that the route from each start node to the terminal node is searched effectively.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Value** \\ \hline Evaporation rate (\(\rho\)) & 0.5 \\ Initial pheromone (\(\tau_{ij}\)) & 1 \\ Number of ant (AN) & 12 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameters and their values for experiments.  
Figure 8: Pheromone distribution using the MPACO algorithm. (**a**) Pheromone distribution view in the maze (the 3 rd iteration); (**b**) Pheromone distribution view (the 10 th iteration); (**c**) Pheromone distribution view (the 100 th iteration); (**d**) Final pheromone distribution and the optimal route using MPACO.  
Figure 9 illustrates the change of the cost function value in each iteration using ACO and MPACO, where the x axis indicates the number of iterations, and the y axis the route distance from the start node to the terminal node.  
Figure 9 shows that the proposed MPACO algorithm produces the optimal routes much faster than ACO algorithm. Table 9 summarizes the computation time comparison between both algorithms.  
In order to show the effectiveness of the proposed algorithm, various networks are tested, as shown in Table 10. The tests using various networks provide the effectiveness of the proposed MAPCO framework compared to the existing ACO algorithm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Map** & **Iteration** & \multicolumn{2}{c}{**Rout Search Time**} \\ \cline{3-4}  & & **ACO** & **MPACO** \\ \hline \hline \end{tabular}
\end{table}
Table 10: Test comparison with various networks.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Framework** & **Iteration** & **Computation Time** \\ \hline ACO & 32 & 19.16(s) \\ MPACO & 5 & 4.92(s) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Computation time comparison between ACO and MPACO.  
Figure 9: Route distance comparisons per each iteration between ACO and MPACO.