# Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem
## 5 Experimental Results
### Exploratory Ability of the Algorithm  
We observed the behavior of the algorithm in exploration and development through the iterative curve and the diversity curve. We took the S51D1, S76D4, and kelly01 datasets as examples to draw the convergence curve of the algorithm, as shown in Figure 6.  
As shown in Figure 6, the initial solution found by the IACA is of high quality. In Figure 6a,c, the initial solution of the IACA is superior to other algorithms. In Figure 6b, the initial solution of the IACA is similar to the ACGCT and better than the remaining algorithms. This indicates that the proposed initial pheromone distribution method based on a greedy strategy in this paper is effective. With the guidance of pheromones, ants can find higher quality initial solutions. From Figure 6, it can also be observed that the IACA achieves a better balance between convergence speed and global search ability. Compared to other algorithms, the IACA can converge to better solutions. This indicates that the proposed adaptive threshold selection can better adapt to different stages of the ant colony search process, thereby achieving a better balance between convergence speed and global search ability of the algorithm. In addition, as the number of iterations increases, the positive feedback mechanism of the pheromone leads to the accumulation of a large amount of pheromone on the current optimal path, causing these algorithms to gradually become trapped in local optima. Due to the use of the pheromone backtracking mechanism and the adaptive pheromone volatile factor during the pheromone handling stage, the IACA exhibits a stronger ability to escape from local optima. Even in the middle and late stages of the algorithm, the IACA can still jump out of the local optimum. In Figure 6a, the IACA jumps out of the local optimal value when the algorithm iterates approximately 150 times, and in Figure 6b, the IACA jumps out of the local optimal value when the algorithm iterates approximately 160 times. In Figure 6c, the IACA can still find a better solution after almost 360 iterations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **TACA/s** & **ASGCT/s** & **ACGCT/s** & **MMGCT/s** & **IACA/s** \\ \hline kelly01 & 1157.84 & 1104.85 & 993.33 & 989.58 & 994.36 \\ kelly02 & 2039.38 & 1943.89 & 1733.78 & 1757.55 & 1747.29 \\ kelly03 & 3251.28 & 3100.41 & 2860.84 & 2912.49 & 2904.04 \\ kelly04 & 4111.77 & 4008.17 & 3978.55 & 4004.51 & 3923.41 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Average running time of algorithms on large-scale datasets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **ASGCT/\%** & **ACGCT/\%** & **MMGCT/\%** & **IACA/\%** \\ \hline kelly01 & 4.58 & 14.21 & 14.53 & 14.12 \\ kelly02 & 4.68 & 14.98 & 13.82 & 14.32 \\ kelly03 & 4.64 & 12.01 & 10.42 & 10.68 \\ kelly04 & 2.52 & 3.24 & 2.61 & 4.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Reduction rate compared to the running time of the TACA. From Figure 6, it can also be observed that during the mid-phase of the algorithm, the ants do not find better solutions for an extended period. This indicates that further improvements can be made to enhance the algorithm's performance during the mid-phase, such as improving the pheromone update strategy. We took the datasets of S51D1, S76D1, and S101D1 as examples to draw the solution diversity curves of the four improved ant colony algorithms, as shown in Figure 7.  
The diversity of solutions refers to the number of different solutions found by ants in each iteration. The higher the diversity of solutions, the more likely the ants are to find a better solution. Since the 2-opt strategy can improve the diversity of solutions, the IACA without the 2-opt strategy is used here to only observe the situation of ants when exploring solutions. As shown in Figure 7, with the progress of iteration, the diversity of solutions gradually decreases and then begins to fluctuate within a certain range.