*4.1. Illustrative Instances*

**Instance 1. Morning shift in Barcelona control center (10 different open sectors)**. Figure 9 shows the airspace sectoring of the Barcelona control center. It consists of a morning shift covering from 5:20 to 13:00 with 10 different open sectors:


**Figure 9.** Sectorization of instance 1.

The problem posed by this instance is the percentage resting time constraint, where all ATCos must rest for 25% of the shift. If 16 ATCos cover 6 sectors (two templates of 3 sectors with 8 ATCos, which is optimized), the ATCos must work exactly 75% of the time. This implies that the work distribution slack is zero, and the working time for all ATCos is exactly equal, whereas all other constraints are met.

**Instance 2. Morning shift in Barcelona control center (11 open sectors).** Figure 10 shows the airspace sectoring of the Madrid control center. It consists of a morning shift, which covers from 5:20 to 13:00 with 11 open sectors: 3 open sectors from 5:20 to 8:40, 4 open sectors from 5:40 to 6:40, 6 open sectors from 6:40 to 7:20, 5 open sectors from 7:20 to 10:00, 5 open sectors from 10:00 to 11:00, and from 11:00 to 13:00. All the sectors and ATCos involved belong to the *eastern route* core. There are 15 available ATCos, whom are CON accredited.

**Figure 10.** Sectorization of instance 2.

The problem posed by this instance is sector opening and closing. We must also cover 6 sectors opened for 40 min with only 15 controllers. Note that an additional ATCo would be necessary if these sectors were to remain open for longer.

**Instance 3. Morning shift in Barcelona control center (9 open sectors).** Figure 11 shows the airspace sectoring of the Barcelona control center for a morning shift, covering from 6:20 to 14:00 with 9 sectors open: 2 open sectors from 6:20 to 8:20, 4 sectors open from 8:20 to 10:20, 5 sectors open from 10:20 to 12:20, and 4, from 12:20 to 14:00. All the sectors and ATCos involved belong to the *eastern route* core. There are 14 available ATCos, all whom hold CON accreditation.

**Figure 11.** Sectorization of instance 3.

It is a relatively simple example, but there are quite a few changes of sectors that complicate the fulfillment of all constraints.

**Instance 4. Afternoon shift in Barcelona control center (6 open sectors).** Figure 12 shows the airspace sectoring of the Barcelona control center for an afternoon shift, covering from 14:00 to 21:20. There are 6 open sectors in the Barcelona *western route* core: 4 sectors open from 14:00 to 19:20, and 2 sectors open from 19:20 to 21:20; and 9 sectors open in the Barcelona *eastern route* core: 5 sectors open from 14:00 to 19:20, 4 sectors open from 19:20 to 21:00, and 2 sectors open from 21:00 to 21:20. The number of available ATCos is 28, whose accreditation type is CON, 14 belonging to the eastern and 14 to the western core.

**Figure 12.** Sectorization of instance 4.

The main problem posed by this instance is its size. A large number of ATCos are needed to cover all sectors. In addition, there is a progressive closure of sectors, where ATCos are highly unlikely to comply with the minimum consecutive work constraint, among others.

Figures 13–15 illustrate an initial solution of this last instance, the corresponding feasible solution and the optimal solution reached using VNS in Phases 2 and 3, respectively. Looking at Figure 13, we find that optimized templates are used where three ATCos cover a sector for 96 time slots and the number of necessary ATCos (43) is greater than the number that are actually available ATCos (28). Figure 14 shows the feasible solution derived from Phase 2 using VNS. Now the number of ATCos matches 28, and all labor conditions are met. Finally, Figure 15 shows the optimal solution derived in Phase 3 using VNS. If we compare the initial feasible and the optimal solutions, it is clear that the structure of the optimal solution is like the previous template-based solution, where work and rest periods are more concentrated.

**Figure 13.** An initial solution in instance 4.

**Figure 14.** A feasible solution in instance 4.

**Figure 15.** Optimal solution in instance 4.

Table 1 shows three out of the four original objectives, optimizing the ATCo work and rest periods and positions, the number of control center changes (rest periods), and the distribution of the ATCo workload.


**Table 1.** Objective functions in the initial and optimal solution in Instance 4.

The first column (sub-object. 1) shows the percentage of cases with a difference less than or equal to 10, 15 and 25 min, respectively, with respect to the goal of 45 min. The second column (sub-object. 2) shows the percentage of cases with a difference less than or equal to 15, 20 and 25 min, respectively, with respect to the goal of 90 min. The third column (sub-object. 1) shows the ATCo percentages whose differences are lower than or equal to 5%, 10% and 15%, respectively. The fourth column shows the number of rest periods, and, finally, the fifth column (ATCo workloads) lists the standard deviation, and the minimum and the maximum value of the ATCo workloads.

As expected, the optimal solution outperforms the initial one for all the objectives under consideration.
