# Ant Colony Optimization with Warm-Up
## 4 Computational Experiments
### Results Obtained on Generic TSP Instances  
The results obtained on the generic TSP instances are reported in Tables 2-4. In particular, results concerning the cost of the best solution found are reported in Table 2, results concerning the number of iterations needed to find the best solution are reported in Table 3, and computational times are in Table 4. For sake of clarity, the results obtained by the proposed ACOWU are written in bold when it outperformed the classic ACO without warm-up, and highlighted in gray when it outperformed all the other comparison algorithms.  
As visible in Table 2 and represented in Figure 1, the proposed warm-up allows the ACOWU to outperform all the other algorithms in almost all the experiments. The Dai algorithm also performs better than the classic ACO, but it rarely reach equals the results of the proposed one. The Bellaachia algorithm provides reasonably good results, although it struggle to reach even the ACO. It is reasonable to believe the authors of Bellaachia algorithm focused more on the reduction of iterations needed to converge to a good solution than on the quality of the solution itself.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**G**} & \multirow{2}{*}{**N**} & \multicolumn{2}{c}{**ACO**} & \multicolumn{2}{c}{**ACOWU**} & \multicolumn{2}{c}{**Dai**} & \multicolumn{2}{c}{**Bellaachia**} \\ \cline{3-10}  & & & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** \\ \hline
0 & 20 & 4453 & 162 & 4471 & 159 & 4496 & 130 & 5154 & 463 \\
0 & 30 & 6079 & 513 & **5539** & 118 & 5497 & 230 & 5994 & 134  
Figure 1: Comparison in terms of cost on generic TSP instances.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**G**} & \multirow{2}{*}{**N**} & \multicolumn{3}{c}{**ACO**} & \multicolumn{3}{c}{**ACOOWU**} & \multicolumn{3}{c}{**Dai**} & \multicolumn{3}{c}{**Bellaichia**} \\ \cline{3-10}  & & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** \\ \hline  
*566** & 357 & 815 & 956 & 1178 & 1168 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results obtained on generic TSP instances in terms of solutions explored before finding best one.  
Tables 3 and 4, Figures 2 and 3 show the comparison in terms of computational time and solutions explored by the algorithms before finding the best one. In this sense, the Dai and Bellaachia algorithms are the best ones, although, the difference with the proposed ACOWU is not that big--i.e., 100-300 milliseconds, which translate into a few milliseconds. Moreover, even the ACOWU is able to outperform all the others in some experiments. The experiments in which the ACOWU needs less iterations (and consequently, computational time) to find the best solutions are also the most complicated instances where the number of nodes to visit is higher (i.e., 40-60 nodes).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**G**} & \multirow{2}{*}{**N**} & \multicolumn{2}{c}{**ACO**} & \multicolumn{2}{c}{**ACOWU**} & \multicolumn{2}{c}{**Dai**} & \multicolumn{2}{c}{**Bellaachia**} \\ \cline{3-10}  & & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** & **Avg.** & **St.Dev.** \\ \hline
0 & 20 & 0.639 & 0.302 & 0.794 & 0.074 & 0.617 & 0.211 & 0.504 & 0.264 \\
0 & 30 & 1.034 & 0.69 & 1.19 & 0.519 & 1.1