# Rank-Based Ant System with Originality Reinforcement and Pheromone Smoothing
## 4 Experimental Results
### Analysis of the Proposed ACO Algorithm
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Instance** & **Algorithm** & **Avg.** & **Std.** & **Best** & **PD\({}_{\text{arg}}\) (\%)** & **PD\({}_{\text{best}}\) (\%)** & **Time (s)** \\ \hline \multirow{4}{*}{brazil58} & AS\({}^{ps}_{Rank}\) & 25,628 & 126 & 25,400 & 0.92 & 0.02 & 14 \\  & AS\({}^{ps}_{Rank}\) & **25,480** & 118 & **25,395** & **0.33** & **0** & 14 \\  & AS\({}^{ps}_{ORank}\) & 25,677 & 124 & 25,400 & 1.11 & 0.02 & 13 \\  & AS\({}^{ps}_{ORank}\) & **25,487** & 121 & **25,395** & **0.36** & **0** & 15 \\ \hline \multirow{4}{*}{kroA100} & AS\({}^{ps}_{Rank}\) & 21,683 & 252 & 21,306 & 1.89 & 0.11 & 45 \\  & AS\({}^{ps}_{Rank}\) & **21,357** & 100 & **21,282** & **0.35** & **0** & 45 \\  & AS\({}^{ps}_{ORank}\) & 21,591 & 186 & 21,331 & 1.45 & 0.23 & 43 \\  & AS\({}^{ps}_{ORank}\) & 21,378 & 120 & **21,282** & 0.45 & **0** & 45 \\ \hline \multirow{4}{*}{ch130} & AS\({}^{ps}_{Rank}\) & 6235 & 39 & 6169 & 2.04 & 0.97 & 120 \\  & AS\({}^{ps}_{Rank}\) & 6193 & 47 & 6141 & 1.36 & 0.51 & 120 \\  & AS\({}^{ps}_{ORank}\) & 6241 & 48 & 6154 & 2.14 & 0.72 & 115 \\  & AS\({}^{ps}_{ORank}\) & **6170** & 33 & **6136** & **0.98** & **0.43** & 120 \\ \hline \multirow{4}{*}{tsp225} & AS\({}^{ps}_{Rank}\) & 4026 & 31 & 3989 & 2.80 & 1.86 & 422 \\  & AS\({}^{ps}_{ORank}\) & 3949 & 23 & **3916** & 0.84 & **0** & 422 \\  & AS\({}^{ps}_{ORank}\) & 4034 & 33 & 3978 & 3.02 & 1.58 & 407 \\  & AS\({}^{ps}_{ORank}\) & **3942** & 17 & **3916** & **0.65** & **0** & 424 \\ \hline \multirow{4}{*}{gr48} & AS\({}^{ps}_{Rank}\) & 5117 & 40 & 5066 & 1.40 & 0.40 & 9 \\  & AS\({}^{ps}_{ORank}\) & 5104 & 35 & 5054 & 1.15 & 0.16 & 9 \\  & AS\({}^{ps}_{ORank}\) & 5135 & 39 & 5074 & 1.76 & 0.55 & 9 \\  & AS\({}^{ps}_{ORank}\) & **5091** & 28 & **5049** & **0.88** & **0.06** & 11 \\ \hline \multirow{4}{*}{pr76} & AS\({}^{ps}_{Rank}\) & 111,609 & 1064 & 109,392 & 3.19 & 1.14 & 60 \\  & AS\({}^{ps}_{Rank}\) & 110,357 & 1187 & 108,238 & 2.03 & 0.07 & 61 \\  & AS\({}^{ps}_{ORank}\) & 111,507 & 810 & 110,100 & 3.10 & 1.79 & 60 \\  & AS\({}^{ps}_{ORank}\) & **109,922** & 900 & **108,159** & **1.63** & **0** & 64 \\ \hline \multirow{4}{*}{gr202} & AS\({}^{ps}_{Rank}\) & 41,803 & 435 & 40,960 & 4.09 & 1.99 & 324 \\  & AS\({}^{ps}_{Rank}\) & 41,095 & 255 & 40,609 & 2.33 & 1.12 & 325 \\  & AS\({}^{ps}_{ORank}\) & 41,602 & 340 & 41,022 & 3.59 & 2.15 & 317 \\  & AS\({}^{ps}_{ORank}\) & **41,056** & 228 & **40,554** & **2.23** & **0.98** & 328 \\ \hline \multirow{4}{*}{ftv35} & AS\({}^{ps}_{Rank}\) & 1497 & 9 & **1473** & 1.65 & **0** & 5 \\  & AS\({}^{ps}_{Rank}\) & 1488 & 10 & **1473** & 0.98 & **0** & 5 \\  & AS\({}^{ps}_{ORank}\) & 1497 & 11 & **1473** & 1.64 & **0** & 5 \\  & AS\({}^{ps}_{ORank}\) & **1483** & 11 & **1473** & **0.71** & **0** & 5 \\ \hline \multirow{4}{*}{ftv64} & AS\({}^{ps}_{Rank}\) & 1867 & 21 & 1848 & 1.50 & 0.49 & 17 \\  & AS\({}^{ps}_{ORank}\) & **1859** & 8 & 1848 & **1.07** & 0.49 & 17 \\  & AS\({}^{ps}_{ORank}\) & 1862 & 14 & **1839** & 1.23 & **0** & 16 \\  & AS\({}^{ps}_{ORank}\) & **1858** & 9 & **1839** & **1.05** & **0** & 17 \\ \hline \multirow{4}{*}{ftv70} & AS\({}^{ps}_{Rank}\) & 2010 & 44 & 1957 & 3.10 & 0.36 & 21 \\  & AS\({}^{ps}_{Rank}\) & 1999 & 35 & 1957 & 2.49 & 0.36 & 21 \\ \cline{1-1}  & AS\({}^{ps}_{ORank}\) & **1989** & 30 & **1950** & **1.98** & **0** & 20 \\ \cline{1-1}  & AS\({}^{ps}_{ORank}\) & **1989** & 37 & 1989 & **1.99** & 0.20 & 21 \\ \hline \end{tabular}
\end{table} 
Table 3: Results for symmetric and asymmetric TSP instances obtained by Rank-based ACO variants. The best results for each problem instance are highlighted in bold.