**1. Introduction**

The key concept at the heart of air traffic managemen<sup>t</sup> (ATM) network operations is air traffic flow and capacity managemen<sup>t</sup> (ATFCM). ATFCM should optimize traffic flows so that airlines can operate safe and efficient flights depending on air traffic control capacity. In Europe, the network manager operations center (NMOC) constantly monitors the balance between the airspace capacity and traffic load. NMOC activities are divided into four –strategic, pre-tactical, tactical and post-operational– phases [1].

The strategic phase is related to capacity prediction at ATC centers by air navigation service providers (ANSPs). ANSPs prepare 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 ATC units and aircraft operators about the ATFCM 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 ATFCM domains and units. Operational processes are measured in order to develop best practices and/or analyze lessons learned after the day of operations.

In this paper, we focus on the pre-tactical phase. This phase has to solve the very important problem of determining how the available air traffic controllers (ATCos) are assigned to each open sector to cover a sectorization structure (established in the strategic phase) for a specified amount of time. This assignment has to comply with a number of strong constraints accounting for the ATCo working conditions.

The sectorization changes throughout the day depend on aircraft traffic. More sectors are opened if the air traffic volume increases. This steps up the demand for ATCos who can only handle a limited amount of traffic.

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. For examples of other timetabling and scheduling problem-solving approaches, see [2,3].

ATCo scheduling software has already been developed within the ATM field [4]. These tools have both strengths and weaknesses [5]. Hardly any of this software has been reported in detail, sometimes because they are in-house tools. Three ATCo scheduling problem codifications were reported alongside three optimization techniques [6]. Another solution [7] is composed of a hybrid technique combining propositional satisfiability problem solving [8] and hill climbing.

A simplified version of the ATCo work shift scheduling problem for Spanish airports was solved by minimizing the number of ATCos required to cover a given airspace sectoring in compliance with Spanish ATCo working conditions [9]. The search process employs regular expressions to check solution feasibility. The solutions that output an optimal number of ATCos are used as the starting point for another optimization process targeting balanced ATCo workloads. This simplified version of the problem analyses straightforward scenarios and accounts for a core with only one type of sector for a 24-h period. Consequently, there are no constraints on ATCo distribution across sectors.

Cores including two sector types (en-route and approach sectors) and accounting for ATCos with different operating credentials were considered in [10]. This proposal focuses on the optimization of only one shift in accordance with a previously specified number of ATCos to cover a specified airspace sector configuration. This proposal adopts a multi-objective approach, accounting for ATCo work and rest periods, positions and workload distribution, the number of control center changes, and the solution structure. It proposes a three-phase problem-solving methodology. In the first phase, a template-based heuristic was used to identify unfeasible solutions. In the second phase, a number of independent simulated annealing (SA) metaheuristic runs were conducted to arrive at feasible solutions using regular expressions to check compliance with ATCo working conditions. In the third phase, simulated annealing was conducted by multiple independent runs to optimize the objective functions of the original feasible solutions again taking into account ATCo working conditions. This optimization process took into account the ordinal information on objective importance using the rank-order centroid function to transform a multiple into a single optimization problem.

In this paper, we consider the same multi-objective problem as [10], albeit using an adaptation of variable neighborhood search (VNS) rather than SA in the three-phase problem-solving methodology. Four representative and complex instances of the problem corresponding to different airspace sectorings provided by the Spanish ATM Research, Development and Innovation Reference Center (CRIDA) are now used to compare the performance of both metaheuristics in the three-phase problem-solving methodology. Moreover, the use of regular expressions to verify the ATCo labor conditions (constraints) is compared against implementation in the code in terms of execution times.

The paper is structured as follows. Section 2 describes the ATCo work-shift scheduling problem. Section 3 describes the proposed problem-solving methodology. Section 3.1 presents a template-based heuristic to identify unfeasible solutions. Then, some notions of VNS and its adaptation to the ATCo work-shift scheduling problem are provided in Section 3.2. Finally, we describe the second and third

phases of the methodology aimed at reaching a feasible and an optimal solution in Sections 3.3 and 3.4, respectively. In Section 4, four real instances are used to illustrate the proposed methodology and to compare the performance of SA against the proposed adaptation of VNS and analyze the use of regular expressions. Finally, some conclusions are provided in Section 6.
