# Rank-Based Ant System with Originality Reinforcement and Pheromone Smoothing
## 4 Experimental Results
### Analysis of the Proposed ACO Algorithm  
With the purpose of analysing the benefits of the extended Rank-based Ant System with originality reinforcement and pheromone smoothing, denoted onwards as AS\({}_{ORank}^{ps}\)  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{10}{c}{**Percentage (\%)**} \\ \cline{2-11}  & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** & **90** & **100** \\ \hline brazil58 & 22 & 7 & 6 & 5 & 3 & 2 & 1 & 1 & 1 & 0 \\ \hline kroA100 & 17 & 12 & 9 & 7 & 4 & 3 & 3 & 2 & 2 & 1 \\ \hline ch130 & 18 & 11 & 8 & 6 & 5 & 4 & 3 & 3 & 3 & 0 \\ \hline tsp225 & 21 & 18 & 14 & 12 & 10 & 7 & 4 & 1 & 0 & 0 \\ \hline gr48 & 14 & 7 & 4 & 2 & 2 & 2 & 2 & 1 & 0 & 0 \\ \hline pr76 & 14 & 9 & 6 & 5 & 5 & 4 & 4 & 2 & 2 & 1 \\ \hline gr202 & 16 & 14 & 9 & 7 & 5 & 4 & 3 & 2 & 1 & 0 \\ \hline frv35 & 18 & 14 & 10 & 7 & 4 & 4 & 4 & 3 & 3 & 3 \\ \hline frv64 & 17 & 13 & 9 & 8 & 7 & 7 & 5 & 3 & 1 & 0 \\ \hline frv70 & 21 & 15 & 5 & 4 & 2 & 2 & 1 & 1 & 0 & 0 \\ \hline ESC78 & 9 & 6 & 5 & 5 & 3 & 2 & 1 & 0 & 0 & 0 \\ \hline fr70.1 & 18 & 13 & 11 & 7 & 5 & 2 & 2 & 1 & 0 & 0 \\ \hline p43.1 & 15 & 12 & 8 & 6 & 6 & 5 & 3 & 2 & 1 & 0 \\ \hline p43.4 & 25 & 20 & 18 & 15 & 13 & 10 & 6 & 5 & 5 & 3 \\ \hlinewith respect to the original AS\({}_{Rank}\), this section compares their performance over several TSP and SOP instances. Furthermore, in order to analyse how each of the two new proposed strategies contributes to the performance of the algorithm, we also analyse the performance of the algorithm when only one of the two strategies is considered--either with the originality reinforcement (denoted as AS\({}_{ORank}\)), or with the pheromone smoothing strategy (denoted as AS\({}_{Rank}^{ps}\)). The comparative results obtained by these four variants of Rank-based Ant System are shown in Table 3 for the symmetric and asymmetric TSP instances, and in Table 4 for the SOP instances. Both tables are organized in a similar way: the different instances are shown in the first column; for each instance, columns 2-8 show: the algorithm used for execution (second column), the mean length of the best-ant tours and its standard deviation over 30 independent runs of the algorithm (third and fourth columns, respectively), the fitness of the best solution found by the 30 runs (fifth column), the average percentage deviation of the solutions from the optimal solution given by PD\({}_{avg}=(Avg-Opt)/Opt\cdot 100\) (sixth column) and the percentage deviation of the length of the best solution of the 30 runs from the optimal solution, given by PD\({}_{best}=(Best-Opt)/Opt\cdot 100\) (seventh column) and computed according to [30], and lastly, the mean computational time over the independent runs of the algorithms in seconds (eighth column). Furthermore, for each problem instance, the best (lowest) average length tour (_Avg._), best-found tour (_Best_), average (PD\({}_{avg}\)) and best (PD\({}_{best}\)) percentage deviation are highlighted in bold. The comparative results show that the proposed Rank-based extension AS\({}_{ORank}^{ps}\) clearly outperforms the original AS\({}_{Rank}\), obtaining lower mean path lengths in all TSP, ATSP and SOP instances. For example, in the case of the symmetric TSP instance _ch130_, AS\({}_{Rank}\) obtained a mean path length of 6235, which corresponds to an average percentage of deviation from the optimal solution of PD\({}_{avg}=2.04\%\), while AS\({}_{ORank}^{ps}\) obtained a mean path length of 6170 and PD\({}_{avg}=0.98\%\). Moreover, for the same problem instance, the fitness of the best solution found during the 30 runs by AS\({}_{Rank}\) was 6169, which corresponds to a percentage of deviation from the optimal solution of PD\({}_{best}=0.97\%\), while AS\({}_{ORank}^{ps}\) obtained a path length of 6136 and PD\({}_{best}=0.43\%\). Additionally, no significant differences are observed between the computational times.