**6. Conclusions**

We have proposed a new methodology based on an adaptation of VNS for solving the ATCo work-shift scheduling problem. This problem involves covering a given airspace sectoring with a certain number of ATCos while satisfying a set of ATCo labor conditions according to Spanish regulations. The problem takes into account four objectives: ATCo work and rest periods and position should be as close as possible to fixed values, the solution structure should be similar to the previous template-based solution, the number of control center changes should be minimized, and the ATCo workload distribution should be balanced.

A comparative analysis proves that the constraint verification in search processes is faster with the implementation in the code of such constraints than using regular expressions. Besides, the parallelization of the constraint verification process has also been analyzed in terms of computation times.

Finally, the proposed methodology has been applied to four real complex scenarios selected by Spanish air navigation experts for the purposes of both methodology illustration and performance comparison against two versions of simulated annealing (SA).

Although the SA-derived solutions slightly outperform VNS solutions in terms of the fitness function, air navigation experts regard the quality of both solutions as very similar, where the key factor then is the time it takes to reach that solutions (computation times). VNS clearly outperforms SA in terms of computation times when the instance dimension (number of open sectors and ATCos required) is low or medium, but the improved version of SA is better for high dimensional instances.

Finally, although the CRIDA experts considered than the computation times for the most complex instances, such as instance 4, were good enough for the pre-tactical phase, which takes place one to six days before the day of operations, we propose as a future research line a further analysis to reduce these computation times together with the comparison of the considered metaheuristics with other in the literature.

**Author Contributions:** conceptualization, F.T., A.J.-M. and A.M.; methodology, F.T. and P.L.; software, F.T. and P.L.; validation, F.T., A.J.-M. and A.M.; formal analysis, F.T., A.J.-M., A.M. and P.L.; investigation, F.T., A.J.-M. and A.M.; writing–original draft preparation, F.T., A.J.-M. and A.M.; writing–review and editing, F.T., A.J.-M. and A.M.; visualization, F.T., A.J.-M. and A.M.; supervision, A.J.-M. and A.M.; project administration, A.J.-M. and A.M.; funding acquisition, A.J.-M. and A.M.

**Funding:** This research was funded by Spanish Ministry of Economy and Competitiveness projects gran<sup>t</sup> number MTM2014-56949-C3-2-R and MTM2017-86875-C3-3-R.

**Conflicts of Interest:** The authors declare no conflict of interest.
