A VNS metaheuristic for solving the aircraft conflict ...

Journal of Global Optimization manuscript No. (will be inserted by the editor)

A VNS metaheuristic for solving the aircraft conflict detection and resolution problem by performing turn changes Antonio Alonso-Ayuso · Laureano F. Escudero · F. Javier Mart´ın-Campo · Nenad Mladenovi´ c

Received: date / Accepted: date

Abstract The aircraft Conflict Detection and Resolution problem (CDR) in air traffic management consists of finding a new configuration for a set of aircraft such that conflict situations between them are avoided. A conflict situation arises if two or more aircraft violate the safety distances that they must maintain in flight. In this paper we propose a VNS approach for solving the CDR by turn changes. This metaheuristic compares favorably with previous best known methods for solving the Mixed Integer Nonlinear Programming model proposed elsewhere. It is worth pointing out the astonishingly short time in which the first feasible solution is obtained. This is crucial for this specific problem, where a response must be provided almost in real time if it is to be useful in a real-life problem. A comparative study between the performance of the new approach, a state-of-the-art MINLP solver and our SILO approach proposed elsewhere is reported, using a testbed of instances with up to 25 aircraft. Keywords Aircraft collision detection and resolution problem · Air traffic management · Variable Neighborhood Search A. Alonso-Ayuso Departamento de Estad´ıstica e Investigaci´ on Operativa, Universidad Rey Juan Carlos E-mail: [email protected] L.F. Escudero Departamento de Estad´ıstica e Investigaci´ on Operativa, Universidad Rey Juan Carlos E-mail: [email protected] F.J. Mart´ın-Campo Departamento de Estad´ıstica e Investigaci´ on Operativa II, Universidad Complutense de Madrid E-mail: [email protected] N. Mladenovi´ c Mathematical Institute, Serbian Academy of Sciences and Arts E-mail: [email protected]

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Antonio Alonso-Ayuso et al.

1 Introduction Due to increasing demand in air traffic operations, systems to automatize the maneuvers of aircraft so as to avoid conflict situations are required. The Conflict Detection and Resolution (CDR) problem is one of the most widely studied problems in air traffic management including fields such as Operations Research and engineering. Once the flight configuration for a set of aircraft is known, the aim of the problem is to draw up a new configuration that will avoid any conflict situation. A conflict situation is an event in which two or more aircraft breach the minimum safety distance that they must maintain in flight. Those distances are 5 nautical miles (nm) horizontally and 1000 feet in the vertical plane. The immediate consequence is that every aircraft is at the center of a cylinder with a radius of 2.5 nm and a height of 1000 feet. To avoid conflict situations an aircraft can change its velocity, heading angle or altitude. Kuchar and Yang (2000) [24] present a survey with the most important approaches to the CDR problem presented up to year 2000. However, there have since been many more papers that have contributed to this problem. The most important of them are reviewed in Martn-Campo (2012) [26]. Pallottino et al. (2002) [33] introduce a seminal approach for modeling the velocity change (VC) maneuvers based on geometric constructions. The VC is based on Mixed Integer Linear Programming (MILP for short) and solves the problem by making velocity changes. In Dell’Ollmo and Lulli (2003) [16], a MILP model is presented and solved even for large-scale problems by using both exact and heuristic methodologies consisting of choosing way points. Christodoulou and Costoulakis (2004) [15], based on [33], propose a Mixed Integer NonLinear Programming approach (MINLP for short) to solve the CDR problem by making changes in velocity and heading angle. In Alonso-Ayuso et al. (2011) [1] a tight MILP model is presented, referred to as Velocity and Altitude Changes (VAC), where both maneuvers are performed to avoid conflict situations and the optimal solution is obtained in a very short computing time by using a state-of-the-art solver. Alonso-Ayuso et al. (2012) [2] extend the VC model by including continuity in the velocity changes, since the original considers that all changes happen instantaneously. Recently, Rey et al. (2012) [35] have presented a MILP model that aims to avoid conflict situations including uncertainty due, for instance, to weather conditions. Also, Cafieri and Durand (2013) [11] propose a MINLP model based on velocity regulation that considers different time instants in which the velocity changes are made. In [4], Alonso-Ayuso et al. (2013) present an approximation for coordinating different Air Traffic Controllers Officers (ATCO) in different air sectors. Two MILP models, the first of which solves the problem using altitude changes and the second a combination of altitude and velocity changes, are proposed in Alonso-Ayuso et al. (2013) [6]. Notice that horizontal maneuvers are in general preferred by passengers because they are more comfortable than vertical ones, though in financial terms, the latter are cheaper if they entail climbing.

On solving the aircraft conflict detection and resolution problem

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Additionally, velocity changes are not very efficient; see Frazzoli et al. (2001) [20], Jardin (2003) [23] and Peyronne et al. (2012) [34], among others. This paper tackles the aircraft collision and resolution problem by means of angle changes in aircraft directions. Therefore, the aircraft velocity is assumed to be constant, as in many other papers, see Pallottino et al. (2002) [33], Christodoulou and Costoulakis (2004) [15], Treleaven (2007) [37], Gao et al. (2012) [21] and Cafieri and Durand (2013) [11], among others. Notice that the ERASMUS (En Route Air Traffic Soft Management Ultimate System) project (see Bonini et al. (2009) [9]) suggests the bounds on variations of -6% and 3% of the current velocity, so this assumption is not very restrictive in real-life situations. This paper presents an algorithmic approach for solving the nonconvex MINLP model introduced in Alonso-Ayuso et al. (2013) [3] where turn change maneuvers are considered based on the geometric construction introduced in [33]. Besides presenting the CDR model by Turn Changes (TC), [3] introduces a Quadratic Programming (QP) model to force each aircraft to return to the initial flight configuration. The problem solving is based on both, an exact method and a heuristic one. The exact scheme needs a high computing time, making this methodology impractical for application in real life, where the answer is needed in few seconds. A heuristic, based on Sequential Integer Linear Optimization (SILO for short) provides good solutions in a short time for realistic medium-size instances. In order to make this approach usable in real life, the computing effort to obtain a feasible solution must be reduced. See Alonso-Ayuso et al. (2013) [5] for the presentation of the preliminary guidelines to this work. The approach presented in this paper is based on the Variable Neighborhood Search (VNS for short) metaheuristic framework. The basic idea of VNS consists of changing neighborhood structures in search of a better solution, in both the perturbation (diversification) and the local search (intensification) phases. Good solutions are usually obtained in short computing time by using this methodology in a variety of problems (see Mladenović and Hansen (1997) [30], Hansen et al. (2008) [22], Brimbergh et al. (2010) [10] and Mladenović et al. (2012) [32]). Other metaheuristic methods have been used to solve the problem as Ant Colony optimization (see Durand and Alliot (2009) [18] and Meng and Qi (2012) [28] for solving the problem by performing angle change maneuvers). Genetic Algorithms (see Medioni et al. (1996) [27] by also performing turn changes and Vivona et al. (2006) [38] for prescribed maneuvers). Particle Swarm optimization (see Gao et al. (2012) [21] where again the turn change maneuver is used) and Neural Networks (see Durand et al. (2000) [19], Christodoulou and Kontogeourgou (2008) [14] and Cetek (2009) [13] for performing velocity changes), among others. The rest of the paper is organized as follows. Section 2 introduces the geometric construction that allows conflict situations to be detected and solved. Section 3 presents with detail our VNS based proposed approach. Section 4 presents an extensive computational experience comparing the results of the

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VNS approach and those obtained by using the state-of-the-art MINLP solver Minotaur and the inexact SILO approach introduced in [3]. Finally, Section 5 concludes and outlines directions for future research.

2 The conflict detection and resolution problem As indicated above, many papers tackling the CDR problem have been published. The TC model, which is the first step of the algorithmic approach in [3], is based on the geometric construction introduced in [33]. Once the aircraft have avoided the conflict situation by performing an angle change, they must return to their initial flight configuration. To that end, a set of QP models are also presented in [3] and the solution is obtained in a very short computing time. In order to present carefully our approach in detail we must introduce the geometric construction for detecting and solving conflict situations in the VC model introduced in [33]. It is depicted in Fig. 1. Let us assume a set of aircraft F = {1, . . . , n} in a given air sector. Let us also denote vi and vj the velocity vectors of aircraft i and j, respectively. The main idea of the model is based on the constructionof the relative velocity vector vi − vj , such y −y

r +r /2

j , respectively, that let ωij and αij denote arctan xii −xjj and arcsin idij /2 where xi , yi are the abscissa and ordinate (i.e., the current configuration of the position of aircraft i), ri is the safety radius of aircraft i and dij is the distance between aircraft i and j. Depending on the tangent of angles lij = ωij + αij and gij = ωij − αij and the tangent of the relative velocity vector, a conflict situation can be detected. Although the notation in the geometric construction is based on vectors, it can be decomposed into the two components, abscissa and ordinate, in the mathematical model. Therefore, there is no conflict between aircraft i and j if one of the following expressions is satisfied,

(vi ) sin(mi + µi ) − (vj ) sin(mj + µj ) > tan(lij ) (vi ) cos(mi + µi ) − (vj ) cos(mj + µj ) (vi ) sin(mi + µi ) − (vj ) sin(mj + µj ) 6 tan(gij ), (vi ) cos(mi + µi ) − (vj ) cos(mj + µj )

(1a) (1b)

where mi is the current direction of motion and µi is the angle variation (optimization variable), such that now the angles of motion mi +µi and mj +µj avoid the conflict between the aircraft i and j. Notice that the left-hand side expressions in (1) can involve a null denominator. These cases are referred to as pathological situations, see [1]. They produce unstable solutions since a specific conflict between two aircraft may be wrongly solved due to the null denominator, even forcing the aircraft to crash in the worst case. These situations are detected in the preprocessing phase when |xi − xj | < ri + rj . To tackle these situations, the parameters mi ,


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v i − vj

rj dij vj mj

shadow

αij gij

ωij

lij

vi mi

(a) Geometric construction for conflict detection

(b) Angles for the conflict detection

Fig. 1 Geometric Construction for the VC problem

mj , tan(lij ) and tan(gij ) can be replaced with the ones turned π2 rad, since after this turn |xi − xj | > ri + rj , and, then, there is not a pathological situation anymore. Notice that this configuration is only considered in the constraints for the pair of aircraft i and j but it does not mean that the aircraft take a heading angle of π2 rad. The parameter pcij takes the value 1 in the model if the preprocessing phase detects a pathological situation between aircraft i and j and 0 otherwise; see below. The TC model solves the CDR problem by making instantaneous turn changes, i.e., all the configurations must change at the same time to avoid conflict situations. This does not mean that each time a new aircraft enters a sector, every other aircraft must change its angle of motion, since it may even happen that no conflict situation is detected. We propose in the paper that the model should be solved whenever an aircraft enters the sector and, if a conflict is detected, then the model optimization solves it. Additionally, we consider constant velocity as do many other papers in the relevant literature.

3 Variable Neighborhood Search approach VNS has been used for solving continuous global optimization problems in the relevant literature in two different ways: with and without neighborhoods induced by using an ℓp norm. A natural approach in applying VNS to solve global continuous nonconvex optimization problems is to induce neighborhood structures denoted as Nk (x) from the ℓp metric (see Mladenović et al. (2003) [31], Audet et al. (2008) [7] and Bierlaire et al. (2010) [8]). A VNS-based approach for solving the generally constrained global optimization problem is suggested in Mladenović et al. (2008) [29], where the problem is first transformed into a sequence of box constrained problems within the well-known point method. In our paper we will also use this type of transformation. If ℓp norm is not chosen, the two different neighborhoods, N1 (x) and N2 (x), are used in the VNS-based approach suggested in Toksari and G¨ uner (2007)

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[36]. In N1 (x), r random directions from the current point x are generated and a one-dimensional search along each direction is performed, where r is an input parameter. The best point (out of r) is selected as a new starting solution for the next iteration, provided that it is better than the current one. Otherwise, the search continues within the next neighborhood N2 (x). The new point in N2 (x) is obtained as follows: The current solution is moved to each (new) = µi + ∆i or µi (i = 1, . . . , n) by value ∆i , taken at random, i.e., µi (new) = µi − ∆i . Points obtained by the plus or minus sign for each variable µi (new) to the right gives a better define neighborhood N2 (x). If a change of µi (new) solution than for µ , the + sign is chosen; otherwise – is chosen. In our diversification move we use a similar neighborhood. More details are given below. The computational experiments reported below for the VNS approach (presented in Subsections 3.1 and 3.2) have provided very good results (in terms of solution quality and computing time) by performing an iterative discretization of the incumbent aircraft’s direction of motion. Good results from this type of metaheuristics have also been obtained in other contexts as well, see [7,8,12, 29,36], among others.

3.1 Penalty function reformulation As it was shown in Section 2, the feasibility condition for the MINLP model is (1). In order to reformulate the problem in the form of a penalty cost function, the infeasibility of the problem must be expressed so that it can be penalized and a feasible solution is quickly obtained. Taking expression (1) as support, the infeasibility condition can be expressed as follows,

tan(gij ) 6

vi sin(mi + µi ) − vj sin(mj + µj ) 6 tan(lij ). vi cos(mi + µi ) − vj cos(mj + µj )

Notice that this expression is the infeasibility condition for a pair of aircraft i and j. A pathological case can also occur in which the above expression takes a value close to the asymptotic part of the tangent function. If this occurs, the problem becomes unstable. In order to check for this case, a parameter cpij for a pair of aircraft i and j, is set to 1 if | tan(lij )| > 1.5 or | tan(lij )| > 1.5 and 0, otherwise. If there is a pathological case, the above expression must be replaced with the following one,

− cot(gij ) 6

vi sin(mi + µi + π/2) − vj sin(mj + µj + π/2) 6 − cot(lij ), vi cos(mi + µi + π/2) − vj cos(mj + µj + π/2)

where the angles involved in the geometric construction are turned by π/2 rad.


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So, a general expression is needed in which all pairs of aircraft are taken into account. This could be as follows,  X  max 0, min{tan(lij ) − tij , tij − tan(gij )} if cpij = 0   i tr then break; end until t > tmax ;

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/* Next neighborhood */;

(maximum size allowed for the neighborhood structures), tr (computing time allowed since the last improvement is performed in order to execute the restarting procedure), and tmax (computing time limit); see Subsection 4.1 for the values used in our computational experimentation. Notice that the restarting scheme is used in order to start from the beginning once the procedure is blocked. The VNS algorithm is observed to be blocked very frequently and too much time is spent without improving the incumbent solution.

4 Computational experiment This section is organized as follows. The parameter estimations for the VNS approach are presented in Subsection 4.1. For doing so, an extensive computational experiment is performed by using the three testbeds that have been used in the experiment presented in [3]. Testbed 1 is presented in Subsection 4.2, where an illustrative instance is tested in order to assess the validation of the VNS approach. Subsection 4.3 reports the main results of a broad computational experiment for Testbed 2 using the state-of-the-art nonconvex MINLP solver Minotaur [25], the SILO approach presented in [3], and the VNS approach introduced in this paper. We experimented with three pilot cases involving n = 5, 7 and 10 aircraft. Finally, the results for Testbed 3 which includes realistic-size cases involving n = 12, 15, 20 and 25 aircraft are reported in Subsection 4.4 where the SILO and VNS approaches are compared. Note: Given the unaffordable computing time required by Minotaur in experimenting with Testbed 2, it is not used for Testbed 3. Remember that the CDR problem requires a solution in almost real-time if it is to be useful in real-life applications.


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Table 1 Results for the 25 instances with 25 aircraft by using high and low penalty parameter values and reformulation Study Reform High Low

z∗ 3.0315 3.0309 3.0291

z ¯ 3.3567 3.3007 3.3079

Sz 0.2944 0.2398 0.2350

t∗ 14.70 13.27 15.38

¯ t 12.18 14.34 12.61

St 7.30 6.91 7.56

¯ tf 0.67 0.93 4.90

Stf 0.63 1.14 5.38

The simulations for each instance in testbeds 2 and 3 are obtained by uniformly generating positions, velocities and angles of motion (forcing each pair of aircraft to be more than 5 nm. from each other) within an squared air sector. The experiments are performed by considering 25 random instances for each of the 7 pilot cases under consideration (3 cases for Testbed 2 and 4 cases for Testbed 3). In total, 175 simulations are performed. The following HW/SW platform is used for the computational experiment: 4xIntel Core i5-2430M, 2.40 GHz, 4 GB RAM and Linux Xubuntu 11.10 OS.

4.1 Preliminary results Our VNS approach strongly depends on some user-defined parameters: M (infeasibility penalty), kmax (maximum size for neighborhood structures), ang (angle discretization value), tr (time when the VNS approach must be restarted if there is no more improvement) and tmax (time limit). This subsection is devoted to present our approach for computing them. 4.1.1 Infeasibility penalties in the VNS global objective function Three alternative ways of estimating the infeasibility penalties have been tested. Alternative 1 consists of a high penalty parameter value (M = 100n, 100 times the number of aircraft under consideration); alternative 2 consists of a low penalty (M = n, i.e., number of aircraft); and, alternative 3 consists of a mixture of the other two in such a way that the algorithm starts with the high penalty (M = 100n) in order to get a feasible solution in a short computing time, and once that solution is obtained, the problem is reformulated using the low penalty (M = n). The procedure continues until a new feasible improvement is found, and alternates back to the high penalty and so on. The results reported in Table 1 involve 25 aircraft. They are the average of the results of the 25 instances that are solved for each alternative. The headings of the table are as follows: Reform, High and Low, alternatives 3, 2 and 1, respectively; z ∗ is the best solution found by using VNS; z¯ and Sz are the average and standard deviations of the solution values, respectively; t∗ is the computing time (in seconds) to obtain the best solution; t¯ and St are the average and standard deviations in computing time, respectively; and t¯f and Stf are the average and standard deviations in the computing time needed to obtain the first feasible solution, respectively.

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Antonio Alonso-Ayuso et al. b b

b

(a) Initial situation

b

b

b

b

b

b b

b b

(b) Resolution

Fig. 2 Illustrative 6 aircraft instance for testing the algorithmic approaches

Table 1 reports similar results for the three alternatives in terms of solution quality. However, observe that when the Low penalty alternative is used the VNS approach takes too long to obtain the first feasible solution, which is crucial for our problem. Although the results of the alternatives Reform and High are similar, we have decided to use the High alternative for our computational experiments since the results are slightly better. 4.1.2 Estimation of other parameters in the VNS approach π We set kmax at 45◦ since the bounds for angle variation are −π 4 and 4 in the MINLP model [3]. Notice that k = 1, . . . , kmax denotes the angle variation in the shaking procedure. On the other hand, tmax is set at the number of aircraft (in seconds). We found this to be a reasonable computing time limit for the requirements of our problem. Parameter tr is set at tmax /10 and, finally, the angle parameter ang for the first local search is set at 1◦ .

4.2 Illustrative case – Testbed 1 Fig. 2(a) depicts the classical hard instance but not a realistic one for testing the validity of the VNS approach: see [17,21] among others. As an illustration, 6 aircraft are situated on a circumference, all flying towards the center, at the same height and the same velocity. As shown in Fig. 2(b), the resolution of the MINLP model [3] forces all aircraft to change their heading angle by -0.2527 rads. (notice that another solution would be the opposite, i.e. 0.2527 rads.). Table 2 reports the comparative results. The additional headings are as follows: Inst. is a configuration that denotes the number of aircraft that are considered in the instance; Minotaur is the exact MINLP engine of choice [25]; SILO is the metaheuristic algorithm introduced in [3]; z is the solution value obtained by each approach; t is the computing time (sec.) needed to obtain


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Table 2 TC model comparison results for the circumference case (Testbed 1) Minotaur Inst I2 I3 I4 I5 I6 I7 I8 I9

z 0.2507 0.4345 0.7108 1.0715 1.5161 2.0457 2.6620 3.3673

t 0.02 0.03 0.11 1.23 9.32 143.66 567.92 4332.67

SILO z 0.2659 0.4416 0.7140 1.0944 1.5772 2.1341 2.7123 3.4338

t 0.05 0.11 0.08 0.28 0.29 1.25 4.45 15.50

VNS z 0.2507 0.4345 0.7108 1.0715 1.5161 2.0462 2.6620 3.3697

t 0.21 1.86 0.10 1.83 0.98 1.44 1.45 4.77

GAP (Minotaur) SILO VNS 6.06 0.00 1.63 0.00 0.45 0.00 2.14 0.00 4.03 0.00 4.32 0.02 1.89 0.00 1.97 0.07

Table 3 TC model comparison average results (Testbed 2)

Case C05 C07 C10

Best known z 0.1670 0.2254 0.3822

Minotaur z 0.1690 0.2255 0.3832

t 0.16 1.67 342.27

SILO z 0.1743 0.2317 0.4029

t 0.09 0.19 0.51

VNS z 0.1671 0.2255 0.3824

t 1.56 2.68 4.78

GAP (Best known) Minot SILO VNS 0.16 6.52 0.00 0.00 5.74 0.05 0.15 6.80 0.08

z; GAP is the relative difference in the solution values obtained by the SILO and VNS approaches with respect to the Minotaur solution value, denoted by M zM . It is computed as z−z zM · 100%. Notice that the VNS approach is repeated 10 times in total (from the beginning and considering restarting) and the best solution is reported. We used the Minotaur engine, as representative of exact MINLP engines, since it provides the best results in the experiments reported in [3]. We can observe in Table 2 that Minotaur never gives solution values z which are worse than the values given by the other approaches. The SILO approach has a very small GAP. VNS appears to obtain the same solution as Minotaur in all but two instances: their GAPs are 0.07% and 0.02%. Notice that the computing time required by Minotaur for instances with more than 7 aircraft is unaffordable for this type of problem. Finally, the VNS approach requires a very short computing time, much less than the time required by SILO for 8 and 9 aircraft. 4.3 Small size instances – Testbed 2 Table 3 reports the results for the pilot cases with 5, 7 and 10 aircraft (which are not as difficult as the circumference case) as an average of 25 instances generated for each case by data variations. The additional headings are: Best known, the best solution value obtained on average, since Minotaur does not always provide the smallest solution value; the Gap of each of the three approaches from the best known solution value, denoted as zb , computed as z−zb zb · 100%. The table shows that for the instances with 5 and 7 aircraft, Minotaur obtains good solutions in less than 2 seconds. However, it needs an unaffordable

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Table 4 TC model dimensions and computational experiment for the SILO and VNS approaches (Testbed 3) SILO Case C12 C15 C20 C25

ml 606 960 1730 2725

n0−1 516 735 1180 1725

z 0.6990 1.2033 2.1118 3.1776

VNS t 0.71 2.18 4.11 256.94

z 0.6877 1.1543 2.0480 3.0309

t 6.76 8.23 12.46 13.27

GAP (Best known) SILO VNS 3.06 1.31 5.62 0.67 3.34 2.04 6.90 2.53

computing time for 10 aircraft. The SILO approach obtains a solution value that on average is between 5.74% and 6.80 % worse than the best known one (see its GAP). However, it requires less than 1 second of computing time. On the other hand, the VNS approach obtains an average solution value that is always smaller than the one provided by SILO. The GAP values for Minotaur and VNS are almost zero. Given the GAP of VNS (compared to that of SILO) and the average time required by each of the two approaches, there is no doubt that for problems of these sizes, VNS is the approach of choice. Notice that the average of the best-known solution value reported for the three cases is smaller than the one obtained by the VNS scheme since sometimes VNS does not reach the best known solution, while Minotaur does. It is worth pointing out that the computing time to obtain the first feasible solution using VNS is very small. 4.4 Real-size instances – Testbed 3. The real-size instances considered here have 12, 15, 20 and 25 aircraft. The Minotaur engine is not used for experimenting with Testbed 3 due to the long computing time that it required for Testbed 2. Therefore, only the SILO and VNS approaches are considered. Table 4 reports the average results for the four cases. The new headings are ml , n0−1 and nc , representing the number of constraints, 0–1 variables and continuous variables, respectively, for the SILO approach. It can be clearly observed that VNS improves on the solution value obtained by the SILO approach, and in particular VNS improves on the computing time, which remains very small, for the instances involving 25 aircraft. 5 Conclusions and future research A basic VNS heuristic is presented to tackle the aircraft conflict detection and resolution problem (CDR), where only a change of aircraft direction is considered to avoid conflict. To that end, the problem is reformulated as an unconstrained optimization problem using a penalty approach. We use a large value for the penalty coefficient in order to accelerate the obtaining of a feasible solution, which is crucial in an operational context. The first improvement local


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search uses a change of direction by ±ang ◦ (a parameter) for each aircraft, while a shaking operator consists of moving several aircraft at once by the angle k · Rand(0, 1), k = 1, . . . , kmax . An extensive computational experiment is performed based on the same testbeds used in [3]. In that paper only the SILO approach was compared with the MINLP Minotaur engine. SILO was the approach of choice, given the unaffordable computing time of the MINLP engine for realistic-sized instances and the good results provided by SILO for small-to-medium sized instances. Its solution quality and computing time are still comparable with the results of the VNS approach presented in this work. However, VNS outperforms the SILO approach for larger instances. As a direction for future research, we plan to continue improving the VNS approach, mainly in the local search scheme, so that velocity changes can be considered as a new maneuver in combination with turn change. Acknowledgements The authors are grateful for the help of Sven Leyffer of the Mathematics and Computer Science Division, National Laboratory, Chicago, USA, for making available to us his nonconvex MINLP Minotaur engine. This research has been partially supported by the projects MTM2012-36163-C06-06 and 174010 funded by the Ministerio de Econom´ıa y Competitividad, Spain, and by the Serbian Ministry of Sciences, respectively.

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