# Rank-Based Ant System with Originality Reinforcement and Pheromone Smoothing
## 5 Conclusions and Future Work
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Instance** & **Algorithm** & **Avg.** & **Std.** & **Best** & **PD\({}_{avg}\)(\%)** & **PD\({}_{best}\)(\%)** & **Time (s)** \\ \hline \multirow{4}{*}{ESC78} & AS & 20,631 & 227 & 19,950 & 13.17 & 9.43 & 182 \\  & AS\({}_{Rank}\) & 18,609 & 147 & 18,415 & 2.08 & 1.01 & 184 \\  & MMAS & 18,464 & 28 & 18,405 & 1.29 & 0.96 & 187 \\  & ACS & 18,477 & 97 & **18,290** & 1.35 & **0.33** & 196 \\  & AS\({}_{Rank}^{P^{\text{S}}}\) & **18,460** & 69 & 18,300 & **1.26** & 0.38 & 192 \\ \hline \multirow{4}{*}{ft70.1} & AS & 44,110 & 469 & 43,221 & 12.20 & 9.94 & 81 \\  & AS\({}_{Rank}\) & 41,403 & 419 & 40,646 & 5.32 & 3.39 & 81 \\  & MMAS & 40,903 & 441 & 40,192 & 4.05 & 2.24 & 81 \\  & ACS & 42,562 & 715 & 41,014 & 8.27 & 4.33 & 87 \\  & AS\({}_{Rank}^{P^{\text{S}}}\) & **40,768** & 341 & **40,092** & **3.70** & **1.98** & 81 \\ \hline \multirow{4}{*}{p43.1} & AS & 28,776 & 64 & 28,615 & 2.26 & 1.69 & 21 \\  & AS\({}_{Rank}\) & 28,333 & 107 & **28,220** & 0.69 & **0.28** & 21 \\  & MMAS & 28,258 & 57 & **28,220** & 0.42 & **0.28** & 21 \\  & ACS & 28,461 & 88 & 28,245 & 1.40 & 0.37 & 24 \\  & AS\({}_{Rank}^{P^{\text{S}}}\) & **28,236** & 34 & **28,220** & **0.34** & **0.28** & 22 \\ \hline \multirow{4}{*}{p43.4} & AS & 84,218 & 115 & 83,950 & 1.46 & 1.14 & 68 \\  & AS\({}_{Rank}\) & 83,693 & 119 & 83,415 & 0.83 & 0.49 & 67 \\ \cline{1-1}  & MMAS & 83,514 & 130 & 83,360 & 0.61 & 0.43 & 67 \\ \cline{1-1}  & ACS & 83,601 & 146 & 83,270 & 0.72 & 0.32 & 69 \\ \cline{1-1}  & AS\({}_{Rank}^{P^{\text{S}}}\) & **83,416** & 113 & **83,265** & **0.49** & **0.31** & 68 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results for SOP instances obtained by ACO algorithms. The best results for each problem instance are highlighted in bold. procedures to the ACO algorithms can improve their performance for certain problems significantly [17]. In line with this, we plan to combine our method with different local search strategies and analyse their performance for the TSP, SOP and other combinatorial optimization problems. We also plan to compare the performance of our method with other ACO implementations after they are all paired with those local search procedures, under the premise that the superior behaviour of our method with respect to best ACO performers without any local search should remain when the local search is added to all methods in a similar fashion. Other interesting extensions are the comparison of this new ACO algorithm with other powerful metaheuristics, such as genetic algorithms, particle swarm optimization or differential evolution, as well as analysing whether the performance of our method can be improved by its hybridization with some of those metaheuristics. These ideas are all part of our plans for future work in the field. Conceptualization, S.P.-C. and A.G.; methodology, software, and validation, S.P.-C., A.G. and A.I. ; formal analysis, S.P.-C. and A.G.; investigation, S.P.-C. and A.I. ; resources, S.P.-C.; data curation, S.P.-C. and A.I. ; writing--original draft preparation, S.P.-C.; writing--review and editing, S.P.-C., A.G. and A.I. ; supervision, project administration and funding acquisition, A.G. and A.I. All authors have read and agreed to the published version of the manuscript. This research work was funded by the European project PDE-GIR of the European Union's Horizon 2020 research & innovation program (Marie Sklodowska-Curie action, grant agreement No 778035), and by the Spanish government project #PID2021-127073 OB-I00 of the MCIN/AEI/10.13039/501100011033/FEDER, EU \"Una manera de hacer Europa\". The data used for the analysis belong to the TSPLIB benchmark library [38], which is accessible at [[http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/](http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/)]([http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/](http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/)) (accessed on 24 October 2022). The authors declare no conflict of interest.