**1. Introduction**

Air transport is growing exponentially, from the 3.8 billion air travelers in 2016 to 7.2 billion passengers expected to travel in 2035, according to the International Air Transport Association [1]. Therefore, air transport and the resulting shortage of laborers in the civil aviation industry has become a serious problem. Air traffic controller (ATC) requires long hours of training, with more than one year needed to train ATCs.

The role of the network manager (NM) is to establish a balance between air traffic demand and airspace/airport capacity in Europe. However, currently, this role is merely moderation between aircraft operators and capacity providers, since the NM has limited instruments to influence either capacity or demand side planning decisions [2,3]. The European Commission also recognizes that the lack of the NM's clear executive powers in practice means that the NM tends to decide by consensus, which often results in weak compromises [4]. The European Commission however stresses that an optimization of the network performance requires an extended operation scope of actions by the NM, a view also shared by Ryanair [5].

Although the NM initiates planning several months before the day of operations [2,3], most of demand-capacity imbalance situations are still resolved on the day of operations by means of demand managemen<sup>t</sup> actions, predominantly by delaying flights. For instance, the total en-route air traffic flow managemen<sup>t</sup> delay was 8.7 million minutes in Europe in 2016, corresponding to a traffic of more than 10 million flights [3,6].

More than 55% of total en-route delay is attributed to lack of capacity and staffing reasons, while approximately half of that delay occurred during peak summer months—June, July and August [3,6]. The Performance Review Commission notes that the capacity requirements are frequently not met by some area control centres, but also that maximum capacity is not delivered at the times when it is needed [3,6].

One of the underlying causes for capacity/demand mismatch is seasonal traffic variability. If traffic is highly variable and there is limited flexibility to adjust the capacity provision according to actual demand, the result may be poor service quality or an under utilization of resources [6]. If addressed proactively, traffic variability can be mitigated or resolved to a certain degree by utilizing previous experience, roster staffing levels to suit and to make more operations staff available by reducing ancillary tasks performed by ATCs during the peak period [6]. Meanwhile delay costs occur when there is no sufficient capacity provision for aircraft operators.

In Reference [7], air traffic control scenarios are classified using decision trees. The authors conclude that decision trees and classification rules perform well in prediction, stability and interpretability.

The activities of the network manager operations center (NMOC) are divided into four phases [8]—strategic, pre-tactical, tactical and post-operational. The strategic phase is related to capacity predictions at air traffic control centers by air navigation service provides, preparing a routing scheme with the help of NMOC seven days ahead of operations.

The pre-tactical phase is related to the definition of the initial network plan. The NMOC publishes the agreed plan for the day of operations, informing air traffic control units and aircraft operations about the air traffic flow and capacity managemen<sup>t</sup> measures affecting European airspace from one to six days ahead of operations.

The tactical phase updates the plan for the day of operations according to real-time traffic demand where the NMOC monitors the situation and continuously optimizes capacity. Delays are minimized by providing aircraft affected by changes with alternative solutions on the day of operations.

The post-operational phase is related to operational process improvement by comparing planned and measured outcomes covering all air traffic flow and capacity managemen<sup>t</sup> domains and units. Operational processes are measured in order to develop best practices and/or analyze lessons learned after the days of operations.

In this paper we focus on the tactical phase, that is, the day of operation. A specific sectorization has been established depending on the aircraft traffic for the considered day and the ATCs have been assigned to the open sectors. We assume that an incident arises, which involves a sectorization change and possibly less available ATCs, and that the available ATCs must be reassigned to the open sectors from that moment until the end of the shift taking into account the work done by each of them from the beginning of the shift to the instant of the incident.

The problem under consideration is similar but different to the shift and break assignment problems, which are referred to in the literature with different names, such as shift design [9,10], shift scheduling [11–14], break scheduling [15–17] and both process shift design and break assignment with a large number of breaks [16].

Such problems have been extensively investigated in Operations Research and have also recently been tackled with Artificial Intelligence techniques. Also, this is a timetabling and scheduling problem. Timetabling and scheduling problems are combinatorial problems, which, on the grounds of size and complexity, cannot be solved by exact methods within a reasonable computation time. For examples of other timetabling and scheduling problem-solving approaches, see References [18,19].

A general mathematical model and specific models for personnel scheduling problems are presented in Reference [20]. Complexity issues regarding personnel scheduling problems are addressed by identifying polynomial solvable and NP-hard special cases. More recently, a general mathematical model and specific models for personnel scheduling problems were proposed in Reference [21] enabling the implementation of various heuristic algorithms and their application to a wide range of problems.

Reference [22] conducted a literature review related to shiftwork managemen<sup>t</sup> within air traffic management, addressing shiftwork impact on health and safety, productivity and efficiency and discussing issues concerning working time organization in accordance with EC Directive 93/104.

A study on shiftwork practices in both ATM and other fields, such as medicine, the police force and the airline industry, is presented in Reference [23]. It concludes that, although there are a range of software tools, in many cases involving ATM, they are costly and not completely suited to the needs. The strengths and weaknesses of automated scheduling tools have already been outlined [24].

Reference [25] used propositional satisfiability (SAT, [26]), MaxSAT, the pseudo-Boolean, satisfiability modulo theory, constraint satisfaction and integer linear problem solvers to address a number of different month- or year-long scheduling requirements. According to the results of applying three different optimization techniques jointly with the above problem solvers, SAT approaches appear to come out on top. Then Stojadinovic [27] combined SAT problem solving with the hill climbing method. The hill climbing method is applied to a feasible solution output by the SAT solver to solve more ATC requirements. The SAT solver is then applied again to further improve the solution. This cycle is iterated until the resulting solution is optimal.

A preliminary approach for modeling many of the features of the ATC scheduling problem was proposed in Reference [28]. The model divides time into 30-min slots, and an ATC should not work for more than 2 h, followed by at least a 30-min break.

A simplified version of the ATC work shift scheduling problem for Spanish airports was solved by minimizing the number of ATCs required to cover a given airspace sectoring in compliance with Spanish ATC working conditions [29] in the pre-tactical phase. This problem was mathematically modeled as a mixed integer problem. A simple sectorization for a whole day with 40 available ATCs involved 751,200 variables and more than 161,669 constraints. This makes it hard to reach good solutions in a reasonable time, leading to the use of a metaheuristic.

The search process employs regular expressions to check solution feasibility. Both search processes use regular expressions (regex) [30]. A regex is a sequence of characters that define a search pattern, used in search algorithms to find strings. The strings represent solutions. The patterns in our approach represent breaches of ATC working conditions.Thanks to the high testing speed and modularity of regex, the optimization model is simpler to implement and maintainable.

Both search processes use regular expressions (Regex) [30]. A Regex is a sequence of characters that define a search pattern, used in search algorithms to find strings. The strings represent solutions. The patterns in our approach represent breaches of ATC working conditions.Thanks to the high testing speed and modularity of regex, the optimization model is simpler to implement and maintainable.

A further optimization process is applied to the resulting optimal number of ATCs to balance ATC workloads. There are no constraints on ATC distribution across sectors in this simplified workshift scheduling problem, which accounts for a 24-h period in a core with just one sector.

The proposal reported in Reference [31] addresses cores with two (en-route and approach) sector types and ATCs with different credentials. Adopting a multi-objective approach, one shift is optimized for a sectorization specified during the pre-tactical phase according to a set number of ATCs considering ATC work and break periods, ATC positions and workload, control center changes and solution structure. The methodology is divided into three optimization phases using a rank-order centroid function to convert a multiple into a single optimization problem taking account ordinal information on objectives (i.e., objectives ranked by importance). First, a template-based method identified unfeasible solutions. Second, independent simulated annealing (SA) output feasible solutions applying regex to check for compliance with ATC working conditions. Third, further independent SA runs optimized the objective functions of these solutions again checking for ATC working condition compliance.

Tello et al. [32] replaced SA in the third phase of the above methodology with an adaptation of variable neighborhood search. They compared the performance of the two metaheuristics on four different sectorizations supplied by the Spanish ATM Research, Development and Innovation Reference Center (CRIDA). They also compared the run times between regex use and implementation in code to verify ATC working conditions (constraints).

Unlike the problems considered previously [31,32], in this paper we focus on the tactical rather than the pre-tactical phase, at the time an incident arises. As said before, a specific sectorization has been established depending on the aircraft traffic for the considered day and the ATCs have been assigned to the open sectors. We assume that an incident happens. Incidents can be of different types: ATCs who cannot continue their working hours because they are unwell or change the sectorization due to a significant increase in air traffic due to diverted air traffic from another airport that has closed for weather reasons.

The incident usually involves a sectorization change. From the moment the incidents occur until the end of the shift the ATCs must be reassigned to the new open sectors taking into account the work done by each of them from the beginning of the shift to the time of the incident and ATC working conditions.

It is important to note that response time is critical in the tactical phase for this variant of the problem. Thus, the use of metaheuristics is even more justified than in the pre-tactical phase.

To solve the problem we propose a methodology that utilizes the metaheuristics simulated annealing and variable neighborhood search, which consists of two phases. In the first phase, it derives a solution, which does not have to be feasible and even need a greater number of ATCs than available ones. This solution in the second phase, applying the metaheuristic, is transformed into a solution in which it need as many ATCs as the number of available ATCs and try to achieve feasibility.

The paper consists of three more sections. Section 2 shows the proposed methodology for a real time adaptation of the ATC work shift scheduling problem to deal with incidents in airports control centers. The adaptation of SA and VNS to the ATC work shift scheduling problem is explained in Section 2 Phase 1 and Phase 2, respectively. Section 3 illustrates the methodology in a real incident from the Barcelona control center. SA and VNS performance has been tested on a set of representative instances in Section 4. Finally, some conclusions are provided inn Section 5.

#### **2. Problem Description**

Airspace sectorization is the partitioning of the airspace into a given number of sectors. Sectors that are open at any one time should cover total airspace capacity. Additionally, sectors can be clustered together to constitute a *core*. Nevertheless, a sector may be part of more than one core. An air traffic control center may be in charge of managing one or more cores. Cores are managed separately, although ATCs can be assigned to two or more cores, if sectors they manage belong to more than one core. Sectors are divided into two types depending on their distance from the airport: approach (5 to 10 nautical miles from the airport) and en-route sectors (more than 10 nautical miles from the airport).

Each sector is assigned to a team of air traffic controllers, each of which can handle a limited amount of traffic. Sectors are operated by ATCs with different roles: planner ATCs and executive ATCs. Planner ATCs foresee possible conflicts between aircraft that they report to executive ATCs who instruct pilots on how to avoid loss of minimum separation.

The sector configuration depends on the volume of air traffic. More smaller sectors are opened as the volumen of air traffic grows, increasing the demand for ATCs and vice versa. Therefore, sector configuration and number of ATCs changes as sectors are dynamically divided and merged according to air traffic variations.

Figure 1 shows an example of an airspace sectorization for the Barcelona eastern route. Each time slot has an associated configuration. The number stands for the number of open sectors, and the letter symbolizes the sector configuration. For example, sectorizations 5A and 5B have the same number of sectors with a different spatial distribution.

**Figure 1.** Barcelona eastern route airspace sectoring.

Besides, in Spain, working conditions are compiled and published in the Official State gazette, Royal Decree 1001/2010, and Law 9/2010. ATC working conditions are as follows:


More details about the problem description are available in Reference [31] .
