## 6 Conclusions and Future Work  
A thorough and comprehensive performance analysis on the efficiency of three GVNS algorithms has been presented in this work. The main difference lies on the perturbation strategy used. Our comparative performance analysis involves problems that are modelled as asymmetric and Symmetric TSP instances that are resolved using GVNS. The well-known TSP benchmarks from the TSPLIB are used for extensive experimental testing. We believe that the experimental results are quite conclusive, as they confirm that for asymmetric TSP instances GVNS_1 outperforms the other two methods and GVNS_2 consistently provides better solutions in all cases compared to GVNS_3. Simultaneously, the perturbation strategy does not seem to critically affect the solutions of Symmetric TSP instances.  
It is also worth emphasizing that even though the improvement stage of the GVNS schemes has been given limited attention, the present paper shows that the tested approaches to the solution, can be quite promising. This is justified by the solutions produced, which are significantly better than other metaheuristic approaches in recent literature.  
The investigation of alternative neighborhood structures and neighborhood change movements in VND under the GVNS framework could be a possible direction for future work. In the same vein, one might potentially study modifications or specific combinations with more than one perturbation strategy during the perturbation phase in an effort to determine whether optimal solutions can be achieved even closer, especially on large asymmetric benchmarks.  
\begin{table}
\begin{tabular}{l l} \hline GA & Genetic Algorithm \\ SA & Simulated Annealing \\ TS & Tabu Search \\ ACO & Ant Colony Optimization \\ TPO & Tree Physiology Optimization \\ \hline \end{tabular}
\end{table}
Table 20: Abbreviations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Instance** & **OV** & **GVNS\_1** & **GVNS\_2** & **Average** & **Average Time** \\ \hline eil51.tsp & 426 & 426 & 426 & 428.98 & 454.1 \\ berlin52.tsp & 7542 & 7542 & 7542 & 7544.36 & 7946.4 \\ st70.tsp & 675 & 676 & 675 & 677.11 & 700.72 \\ kroA100.tsp & 21282 & 21282 & 21312 & 21695.79 & 22726.2 \\ ch130.tsp & 6110 & 6137 & 6208 & 6153.72 & 6610.8 \\ rat195.tsp & 2323 & 2349 & 2448 & 2453.81 & 1382.34 \\ rd400.tsp & 15281 & 15684 & 16272 & 16250.21 & 1953.49 \\ pcb1173.tsp & 56892 & 61479 & 63387 & 63435.95 & 9531.54 \\ pcb442.tsp & 50778 & 52381 & 54108 & 50800.24 & 2183.27 \\ \hline \end{tabular}
\end{table}
Table 19: Comparisons between GVNS_1 and GVNS_2 with BI for 2 mins with the recent VNS work of [PERSON] et al.  
gathering all the necessary material. [PERSON] assisted [PERSON] in designing the methods used in the main part. [PERSON] was responsible for supervising the construction of this work. [PERSON] was responsible for the interlinking between the theoretic model and the actual application. [PERSON] contributed to the appropriate typing of the formal definitions and the math used in the paper.  
This research received no external funding.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} TSP & Traveling Salesman Problem \\ VNS & Variable Neighborhood Search \\ GVNS & General Variable Neighborhood Search \\ GA & Genetic Algorithm \\ SA & Simulated Annealing \\ TS & Tabu Search \\ ACO & Ant Colony Optimization \\ TPO & Tree Physiology Optimization \\ \end{tabular}