The flow management problem: recent computational algorithms

Control Engineering Practice 6 (1998) 727—733

The flow management problem: recent computational algorithms Giovanni Andreatta!,*, Lorenzo Brunetta", Guglielmo Guastalla# !Dipartimento di Matematica pura ed applicata, Universita% degli Studi di Padova, Via Belzoni 7, 35131 Padova, Italy "Dipartimento di Elettronica ed Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy #Operations Research Center, Massachusetts Institute of Technology, 77 Mass Ave, E40-130, Cambridge, MA 02139, USA Received 11 December 1997

Abstract The air traffic flow management problem, together with various policies to address it, is described in this paper. A survey of optimization algorithms for the ground-holding (and ‘free flight’) policies is provided. An exact algorithm, based on the integration of a heuristic algorithm with an integer linear programming model is presented next. This approach provides exact solutions in a much shorter computational time than previous algorithms proposed in the literature. Computational results for large-size instances with over 20 000 flights based on the OAG data for a full day in the USA air traffic network are reported. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Air traffic control; models; combinatorial mathematics; heuristics; integer programming

1. Introduction An air traffic network is composed of airports, airways and sectors (subsets of the airspace). Each of these elements has its own limited capacity. The airport capacity, measured in terms of movements (landings and takeoffs) per hour, is a quantity that can be estimated with good accuracy. It is determined by the airport characteristics (location, number of runways, topology, etc.), safety requirements and weather conditions. The sector capacity, on the other hand, is defined as the number of aircraft that can simultaneously be controlled by the air traffic controllers of a sector in a given time interval (usually an hour). In recent decades, the increasing volume of air traffic has led to heavier use of air traffic networks, while their capacities have not grown accordingly. When air traffic demand exceeds capacity, it produces congestion. Congestion leads to delays in departures and queues before landing, causing inconvenience to passengers and big losses to air companies. It can also potentially affect airspace safety. Delays have a cost: for American air companies, the annual cost due to air traffic congestion was estimated to be over 2 billion dollars in 1990, and 2 billion and a half in 1991 (Vranas et al., 1994).

*Corresponding author. E-mail: [email protected].

In Europe, in 1993, there occurred 106 000 hours of delays and a 19% non punctuality, with a cost of 5 billion dollars. In some of the most important European airports, e.g., London—Heathrow, the saturation point, in terms of capacity, is reached during most of the operating periods. In the European airspace, many sectors are often congested. Furthermore, this situation is not likely to improve in the future: the International Air Transport Association (IATA) has forecast for the year 2000 that 16 European airports will not be able to meet their demand for movements, unless their capacity is substantially increased. For the same year, the number of congested sectors is foreseen to be four times the current one. Building new airports or new runways would certainly increase the network capacity, but it is a policy that is not always easy to implement, and it produces effects only in the long term. For more information please refer to Odoni (1987) and to Adams et al. (1996) for the USA case, and to Matos and Ormerod (1995), Matos et al. (1996) and Vranas (1996) for the European case. In the short term, the best that can be achieved by the system is to try to limit the size and the impact of the delays produced by congestion, or, in other words, to manage the air traffic flows in order to avoid the demand exceeding the available capacity. This activity is known as air traffic flow management.

0967-0661/98/$ — See front matter ( 1998 Elsevier Science Ltd. All rights reserved PII S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 0 7 8 - 1

728

G. Andreatta et al. / Control Engineering Practice 6 (1998) 727—733

The measures adopted to reduce congestion range from changing the routes of flights (re-routing) to controlling aircraft speeds, and from monitoring the rate of some traffic flows (metering) to imposing a delay prior to some flight departures (ground holding). This paper first outlines the difference in air traffic management between United States and Europe; then the ground-holding policy is presented, and a survey of the optimization models proposed in the literature is provided. The ‘free flight’ concept and its similarities with ground holding are presented next. Finally, an exact and efficient approach that solves large real-world instances is discussed in detail. Although one could question the practical usefulness of an exact approach, the following point should be stressed: in order to assess the performance of a heuristic, exact results are essential. The fast, exact approach presented in this paper will make it possible to solve large-scale instances of the problem at optimality, thus providing useful benchmarks for assessing the quality of various heuristics. The key idea in this exact approach is first to obtain a feasible solution to the problem by using a heuristic, then to transform this feasible solution into a basic feasible solution, and finally, starting from this basic feasible solution, to apply Cplex to find the optimal solution. This approach provides exact solutions in a much shorter computational time than the previous algorithms proposed in the literature: the computational experience is based on large, realistic instances from the literature; see Andreatta et al. (1996) for more details. In the last section, computational results for instances based on real OAG data for the USA air traffic network are reported.

2. Air traffic flow management: the situation in the United States and in Europe In the United States, the air traffic flow management is in the hands of a central unit in Washington DC, the Air Traffic Control System Command Center (ATCSCC). Congestion problems are daily occurrences at airports in the USA. In Europe, a continent with many countries and a correspondingly divided airspace, the problem of controlling and managing air traffic flows is more complex. The majority of flights take between one to two hours, but have to cross many sectors (in different countries). Congestion is not concentrated mainly at airports, as in the USA, but it is also present in many sectors. The main effort required to control and manage air traffic in Europe is to coordinate the operations. The Central Flow Management Unit (CFMU) at Eurocontrol was created for these reasons, and it will manage air traffic for the 32 nations of the European Civil Aviation Conference. Computer scientists, engineers and flow-managers have

built a system that allows a constant comparison between demand and capacity. When needed, some flows of the air traffic are changed in either space (re-routing) or time (slot allocation). The hardware and software system that manages the air traffic flows comprises four main components: TACT (demand/capacity comparison), IFPS (flight-plan processing), CASA (Computer Aided Slot Allocation), and ARCHIVE. A new software version is released every six months, and flow controllers participate to the building of this system by collaborating with the technical and research sections. The timing and procedures of the activities vary tremendously between the USA and Europe: in the United States the planning is done few hours before flight departures by the ATCSCC, while in Europe it begins six months before the actual flights, and requires input not only from air traffic controllers, but also from representatives of different countries and air companies. The terminology itself changes: in the USA the planning done before departure is defined as strategic, while it is tactical after departure. In Europe there is a strategic planning from six months to few days ahead of the flight, pretactical planning a couple of days ahead, and tactical planning the day of departure. Research into air traffic flow management began in the late eighties and has mainly focused on optimization models for ground-holding. The great majority of models have considered the USA situation, where congestion may only arise at destination airports. Some of the most recent models also consider congestion on the sectors of the route, and they allow re-routings (Bertsimas and Stock, 1994), in particular those thought appropriate for the European case (Matos et al., 1996; Mugis, 1995). Furthermore, an optimization model is being integrated in SPORT, a pre-tactical decision-support system for air traffic flow management (ATFM) (Mugis, 1995). The air traffic management problem with reference to the USA situation, its characteristics and the tools to approach it have been defined by Odoni (1987); current developments and future scenarios are presented in Adams et al. (1996). A similar effort has not yet been produced for the European situation, but the first papers are appearing (Vranas, 1996; Matos and Ormerod, 1995; Matos et al., 1996; Mugis, 1995; Delahaye et al., 1994). An attempt to draw up the needs and priorities for improving the modelling tools in the near future is given in a recent study at MIT (Odoni et al., 1997), where the strengths and weaknesses of 27 of the most popular and widely used ATM and airport models has been considered. The main conclusion of that survey is that the state of the art differs considerably between the categories of models analyzed. While models for estimating the capacity, and the corresponding delays, in a single airport are able to give satisfactory and useful answers in a reasonable amount of time, the models for conflict resolution, as well as those for ground-holding or ‘free

G. Andreatta et al. / Control Engineering Practice 6 (1998) 727—733

flight’ policies (see Sections 3 and 4), are not yet flexible, powerful or fast enough.

3. Ground-holding policies A ground-holding (GH) policy imposes on selected aircraft a ground-holding prior to their departure, so that congestion during peak periods of time may be smoothed away. The usefulness of these policies stems from the following facts. First, air delays are much costlier than ground delays. Second, the capacity of an airport is affected by weather conditions. In the USA airports a reduction of capacity is fairly common. Third, if pilots were free to depart at will the situation could get completely out of control (i.e., too many flights in a certain part of the air traffic network) and the air traffic controllers would not be able to provide any instructions, with serious safety risks. Cost and safety are sufficient to justify the study of methods for managing air traffic in unstable weather conditions. These policies rely on the fact that costs are lower when delays are imposed on the ground rather than on the air. The ground-holding problem (GHP) consists of determining the amount of delay to be imposed on the ground on each flight, in order to minimize the overall cost of delays (in the ground and in the air) in the network (Andreatta et al., 1993). Since a large number of planes fly along ‘itineraries’, with many consecutive legs on the same day (i.e., the same aircraft performs several consecutive flights), the network effects might be serious: when an arrival flight is delayed, the successive flights could be automatically delayed. Furthermore, in ‘‘hub’’ airports in which passengers commute from a plane to another, the effect of delaying one flight might affect more than one successive flight. The importance of ground-holding policies is well recognized: they have already been adopted, for example, in the FAA control center ATCSCC. However, decisions made in the ATCSCC are based on the opinion of ‘expert’ traffic controllers rather than on optimization models. Optimization models for the ground-holding problem in a network of airports were introduced in Vranas et al. (1994). Heuristic algorithms have been proposed only recently by Richetta for a single airport (Richetta 1995) and by Andreatta, Brunetta and Guastalla for the ground-holding problem in a USA network (Andreatta et al., 1997), while Delahaye et al. (1994) and Mugis (1995) dealt with the European network. In Mugis (1995) four different heuristic approaches are tried (based on local search, simulated annealing, tabu search, e-approximate schemes), but no performance guarantee is discussed. This problem is common to many papers on the ATFM topic; in fact, it is very hard to obtain real, or even

729

‘realistic’ GHP instances, to evaluate the quality of one procedure over the others; therefore, the above-cited models have been tested on a very small set of instances. It should be pointed out that the above-mentioned models have been studied for the case in which each flight has at most one predecessor and at most one successor. The so-called ‘banking’ phenomenon, in which a flight may have more than one successor or predecessor has been studied only in Navazio and Romanin-Jacur (1995).

4. Free flight Many airlines in the USA have been complaining about the GH policies, and have been pushing toward the new concept of ‘free flight’. They are asking the FAA to provide them only with an arrival timeslot, leaving them the ‘freedom’ of selecting, for each flight, its departure time, route and speed, as long as they are able to arrive at the assigned timeslot. More rigorously, free flight is ‘a safe and efficient flight operating capability under instrument flight rule in which the operators have the freedom to select their path and speed in real time. Air traffic restrictions are only imposed to ensure separation, to preclude exceeding airport capacity, to prevent unauthorized flight through special use airspace, and to ensure safety of flight. Restrictions are limited in extent and duration to correct the identified problem’ (see the FAA web page, at the URL http: //www.faa.gov/ asd/ff – ov.htm). For more information about the Free Flight the interested reader may see, for instance, the special issue of Air ¹raffic Control Quarterly edited by Margaret Jenny (Jenny, 1997). The algorithms presented in this paper can easily fit into the ‘free flight’ concept. In this case, the ‘delays’ suggested by the algorithms have to be interpreted as time differences between the arrival time slot imposed by the FAA, or other analogous Authority, and the time slot originally scheduled (leaving up to the Aircarriers to decide how much delay should be absorbed on the ground, how much through speed reduction or through rerouting, etc.). Under a ground-holding policy all planned delays are assumed to be incurred on the ground prior to departure.

5. Notation The important parameters are airport capacities, together with the daily timetable of flights (planned or ‘desired’ arrival times). Since capacity is expressed per time unit, time is discretized, i.e, the day is divided into a set of time intervals of the same length (for example, 15 minutes or 5 minutes).

730

G. Andreatta et al. / Control Engineering Practice 6 (1998) 727—733

The feature that distinguishes the GHP for a network of airports from the GHP for a single airport is the need to consider ‘connected’ flights, i.e., a couple of flights performed in succession by the same aircraft. For every flight, there exists a turnaround time, i.e., the minimum amount of time that the aircraft must spend on the ground before performing another flight. Turnaround time is needed for loading/unloading passengers, cleaning the aircraft, refueling, etc. With respect to the time between the scheduled arrival and the successive departure of an aircraft (i.e., the arrival time of a flight and the departure time of the one connected to it), the complement of the turnaround time is the slack time. If the delay for the arriving flight is less than or equal to the slack time there is no consequence for the connected flight. Otherwise, there is not enough time to complete the turnaround operations before the scheduled departure time; the departing flight ‘inherits’ a delay from the arriving one. The following notation will be used in the following pages:

Ormerod (1995), Matos et al. (1996) and in Vranas (1996): these papers address the ‘slot allocation problem’ and Vranas shows that it is equivalent to the multi-airport ground-holding problem when congestion may arise only at the destination airport. As has already been said, the ground-holding problem (GHP) consists of determining the amount of delay to be imposed on each flight on the ground, in order to minimize the overall cost of delays (on the ground and in the air) in the network. Then the problem is to minimize the objective function defined by + c * , subject to the f|F f f following sets of constraints:

"set of time intervals; "generic time interval (t3¹); "set of airports; "generic airport (z3Z); "set of flights; "generic flight ( f3F); "set of pairs of connected flights (( f, f @)3F if c flights f and f @ are performed by the same aircraft in succession); K "capacity of airport z at time t, for all z 3 Z and z,t for all t 3 ¹; z "arrival airport for flight f, for all f 3 F; f r "desired arrival time for flight f, for all f 3 F; f c "cost of one unit of delay suffered by flight f ; f s "slack time, i.e., maximum delay for flight f withf,f{ out causing any delay on flight f @ (( f, f @)3F ); c * "delay suffered by flight f, for all f 3 F; f * "maximum allowed number of delay periods per .!9 flight; ¹ "set of time intervals in which flight f may land f (¹ "Mr , 2 , r #* N). f f f .!9

integrality:

¹ t Z z F f F c

6. Ground-holding exact algorithms As was mentioned in Section 3, in recent years the ground-holding problem has received a lot of attention, both from the aviation authorities (FAA, Eurocontrol, etc.) and from the scientific research community. (Andreatta et al. 1993) gives an overview of the models related to the problem where congestion may only arise at a single airport. The multi-airport ground-holding problem was introduced in Vranas et al. (1994). The problem in the European network was recently discussed in Matos and

capacity: assignment:

coupling:

a limited number of planes may land at a given airport in any period of time; every flight must arrive at the destination airport sooner or later (more specifically, in ¹ ); f every flight must wait for its preceding flight to arrive (plus turnaround time) before taking off; every flight must arrive at the destination airport in ‘one and only one’ period of time.

In the integer programming model proposed in Vranas et al. (1994), the generic variable x is 1 if flight f arrives ft at time t, and 0 otherwise. The value of the objective function + c * is obtained by considering the identity f|F f f * "[+ t(x )!r ); assume, for example, that flight f t|Tf ft f f is assigned to arrive at time t and one wants to 1 1 compute its delay * : since x is 1 only when flight f1 ft f arrives at time t, the sum reduces to t (x )!r 1 f1t1 f1 "t !r , i.e., the delay imposed on flight f is equal to 1 f1 1 the assigned arrival period minus the desired arrival period of the flight. The coupling constraints are modeled by considering the relation * ** !s ∀( f, f @)3F , f{ f f,f{ c i.e., the delay of flight f @ must be greater than or equal to the amount inherited from its preceding flight. In the model (Andreatta and Tidona, 1994) the coupling constraints are dropped and the set of variables is substituted by a larger one that assigns to each pair (or triplet) of consecutive flights, a set of variables. In other words, if f and f are consecutive flights, the generic 1 2 variable x is 1 if flight f arrives at time t and f1t1/f2t2 1 1 f arrives at time t , and 0 otherwise. The advantage of this 2 2 model is that by relaxing the coupling constraints a transportation problem is obtained; the disadvantage is that the number of variables becomes very big (there is a variable for every combination of t 3¹ and t 3¹ ), making large 1 f1 2 f2 instances hard to solve because of memory requirements The key idea in the model (Bertsimas and Stock, 1994) is the meaning attributed to the binary variables x : ft x is 1 if and only if flight f arrives by time t, i.e., if it ft arrives at time t or earlier. The constraints and the objective function must be stated following this idea. The value of the corresponding objective function + c * is f|F f f

G. Andreatta et al. / Control Engineering Practice 6 (1998) 727—733

obtained, for example, by considering the identity * "[+ t(x !x )!r ), since t(x !x )"t t|Tf ft f ft~1 f ft ft~1 only when x "1 and x "0. Please refer to Bertft ft~1 simas and Stock (1994) and Andreatta and Brunetta (1998) for more details. The three models mentioned are computationally compared in Andreatta and Brunetta (1998); the BS model is the one that performs best among the three. The Andreatta Brunetta Guastalla Exact (ABGE) model (Andreatta et al., 1996) follows the idea of the model proposed by Vranas, Bertsimas and Odoni (VBO), i.e., the generic variable x is 1 if flight f arrives at time t, ft and 0 otherwise. The representation of the coupling constraints is different from that of the VBO model. The two formulations are equivalent when all variables are integers but the ABGE one is stronger when the integrality constraints are relaxed. Since the ABGE model is currently the one that produces an exact solution in the shortest computational time (Andreatta et al., 1996), it is the model that is going to be presented here in some detail. Using the notation of Section 5, the Andreatta Brunetta Guastalla Exact (ABGE) model is:

C

D

min + c + tx !r , subject to: f ft f f|F t|Tf x )K + ft z,t f f (f,t)/z /z,t|T + x "1 ∀f3F ft t|Tf

∀z3Z ∀t3¹

(1) (2)

+ x * + x ∀( f, f @)3F ∀(t , t )3¹ ]¹ fi f{i c 1 2 f f{ rf{)i)t2 rf)i)t1 with t "r #t !r !s (3) 2 f{ 1 f f,f{ x 3M0, 1N ∀f3F; ∀t3¹ . (4) ft f The capacity constraints are expressed by Eq. (1): no more than K planes may land at airport z during the z,t period of time t. Eqs. (2) and (4) model the fact that every flight ‘must’ arrive at the destination airport during ‘one and only one’ of the time intervals in which a flight is allowed to land (assignment and integrality constraints). The coupling constraints given by Eq. (3) are needed only for pairs of connected flights f and f @: they state that flight f @ cannot take off unless flight f has arrived and a proper amount of time (turnaround time) has elapsed. Notice that there is a constraint for every time t between 1 r #s and r #* (for smaller values, t N¹ ). f ff{ f .!9 2 f{ 7. A heuristic algorithm This section presents one of the heuristic algorithms mentioned in Section 3: the Andreatta Brunetta Guas-

731

talla Heuristic (ABGH) algorithm. A more extensive description of the procedure can be found in the original paper (Andreatta et al., 1997). The procedure is based on a specific priority rule: each priority rule chooses, for any two flights, which one has priority over the other. A priority rule can be specified by a matrix with as many columns as there are flights and as many rows as the possible number of delay periods, which assigns to every flight f, that has suffered a delay * , a priority index n . f f,*f Since it is conceivable that some flights will share the same priority index, in the ABGH algorithm flights are grouped dynamically into a manageable number of classes (drastically reducing the amount of memory required to store the priority rule). A Priority ¹able has as many columns as the number of classes considered, and as many rows as the possible number of delay periods. The priority indexes are then used in the algorithm for scheduling the departures of the various flights. Table 1 shows an example of a priority table utilized in Andreatta et al. (1997). There, the maximum allowed delay for a single flight f is 4 intervals (of 15 minutes length) for all the instances considered, (i.e., * "4), .!9 and the same aircraft might perform an itinerary of at most 5 legs or, in other words, every flight has from 0 to 4 successors (0—succ, 2 , 4—succ in Table 1). The rationale behind this table is to favor flights with successors. Thus, for example, flights with successors and no delay have higher priorities than flights with no successors and 1 or 2 units of delay. The priority of a flight is increased according to the number of its successively connected flights. In the third, fourth and fifth columns, higher priority is given to flights with a larger number of successors, rather than to flights with a larger delay. For example, a flight with 4 successive flights and 2 units of delay is scheduled to depart before a flight with 2 successors that has already suffered a delay of 3 units. By feeding different priority rules to Procedure ABGH, one obtains different heuristics, with different performances. As a matter of fact, Procedure ABGH does not even guarantee to provide a feasible solution, because it will not automatically satisfy the constraints * )* f .!9 for all f3F; this depends on the choice of the priority table. In the test cases solved in Andreatta et al. (1997), the objective function value often coincides with the

Table 1 A Priority table

* "0 f * "1 f * "2 f * "3 f * *4 f

0 — succ

1 — succ

2 — succ

3 — succ

4 — succ

0 10 20 60 110

30 40 50 70 110

33 43 65 80 110

35 45 75 90 110

37 47 85 100 110

732

G. Andreatta et al. / Control Engineering Practice 6 (1998) 727—733

optimal one (in 8 out of 39 cases), and is 2.17% higher on the average. The ABGH algorithm uses queues ordered according to priorities. The queue Q(z, t) is composed of all flights scheduled to land at airport z and with ‘current’ arrival time t (i.e., t"r #* ), ordered by descending values f f of n . f,*f In the following description of the ABGH algorithm, the symbol g is used to denote the inherited delay of f{ flight f @ (g "maxM0, * !s N, ( f, f @)3F ). f{ f f,f{ c

ABGI is implemented in C, using the routines of the Cplex Callable Library 3.0 for the integer programming solver. In Andreatta et al. (1996) it is shown that the ABGI algorithm leads to an optimal integer solution in a much shorter computational time than ABGE or BS. Furthermore, the ABGI algorithm often finds the optimal integer solution by simply solving the linear programming relaxation of the ABGE model, or by exploring a very few branch-and-bound nodes.

Procedure ABGH

9. Computational results on test cases based on OAG data

For every flight f without predecessors, place f in the queue Q(z , r ); f f For every time t, for every airport z, complete the following steps: M

In this section, computational experience obtained on test cases that are based on the real OAG data for the days of January 13th and July 3rd, 1993, is reported. The two data sets comprise respectively 63 455 and 55 106 flights, on a network with more than 100 airports. For each flight, the data include the official flight number, the departure/arrival airports, the departure/arrival times, and the equipment (i.e., the aircraft type). The information on connections, kept confidential by the airlines, is not available. For this reason, a C program to generate connections was implemented. The program takes as input the entire OAG schedule, a list of airports of interest and two thresholds: the minimum and maximum connection times. Two flights are considered to be connected if (i) the second flight departs from the arrival airport of the first, (ii) they are performed with the same kind of equipment and by the same carrier, and (iii) the time of departure of the second flight is between the time of arrival of the first one plus the minimum connection time, and the time of arrival of the first flight plus the maximum connection time. For these instances c was assumed to be a constant f equal to 50. The minimum and maximum connection times were set to 20 and 40 minutes respectively. In order to increase the connection percentage (given by 100D F D%D F D), first the constraint about the equipment c was relaxed. Then the constraint about the carrier was relaxed. Thus three different instances for each day were obtained. The list of airports of interest was the same for all the examples. It comprises the 23 airports that in year 1993 exceeded 20 000 hours of annual aircraft delay (FAA 1994). Examples from January comprise 22 522 flights (with 69%, 72% and 77% of connections), while the July examples comprise 20 220 flights (with 67%, 71% and 77% of connections). In Table 2 the computational results of ABGI on these six instances are reported. In the first column the test case names are reported, in the second the CPU time spent by the heuristic (CPU H), and in the third the final CPU time (CPU ILP). The fourth (H V) and fifth (ILP V) columns give the optimal values of the objective function provided by the heuristic algorithm and by the integer model. For example, Jan93.69 identifies the instance of

Assign the arrival slots to the ‘first’ K flights in the z,t queue Q(z, t) (the actual number of assigned slots being k"minMK , DQ(z, t) D N). Place all the remaining flights z,t in the queue Q(z, t#1); For each flight of the k selected, try a ‘tentative exchange’ with flights previously assigned to earlier times; if a successor f @ exist ((f, f @)3F ), compute its c inherited delay g and place it in the queue f{ Q(z , r #g ); f{ f{ f{ N In the ‘tentative exchange’ step, the exchange of the time slot of the current flight f with that of a previously scheduled flight f * is tried. If the exchange is convenient, and feasible, then it is accepted. Since the total sum of delays does not change, the exchange is convenient whenever the slot currently assigned to flight f results in an infeasible solution (i.e., t!r "* '* ), or when it f f .!9 would imply an inherited delay to its successor f @ (i.e., t!r "* 's , ( f, f @)3F ). f f f,f{ c Notice that, if the coupling constraints are disregarded (i.e., no flights have successors), this heuristic algorithm becomes exact by virtue of the Monge property (Andreatta et al., 1996).

8. A new exact approach: the ABGI algorithm The ABG Integrated Algorithm (ABGI) is nothing else than the integration of the heuristic procedure ABGH with the exact method ABGE. The key idea is to provide ABGE with a feasible starting solution, which should be basic and good, so that ABGE can quickly drive to an optimal solution. This good initial solution is produced by ABGH; the interesting part is that this solution happens to be basic, so that it can be fed in directly to ABGE.

G. Andreatta et al. / Control Engineering Practice 6 (1998) 727—733

References

Table 2 Results of ABGI applied to six instances (OAG data) Test

CPU H

CPU ILP H V

ILP V

LP IT

Jan93.69 Jan93.72 Jan93.77 Jul93.67 Jul93.71 Jul93.77

0.28 0.26 0.32 0.25 0.30 0.30

616.49 673.39 1009.92 556.13 594.52 823.86

644,450 645,100 648,900 615,100 615,850 620,800

5466 6177 9277 6032 6462 8681

666,700 674,750 709,350 639,250 643,850 671,350

733

January 1993 with 69% of connections. The integer programming models have about 50 000 rows and 110 000 variables. No branch-and-bound nodes are needed. The CPU time spent by the heuristic (CPU H) is always less than 1 second, while the overall CPU time (CPU ILP) is between 10 and 20 minutes. These instances were also solved with the BS model and ABGE. The CPU time for ABGE ranges from 1 733 to 2 025 seconds, and for BS from 6 493 to 12 719 seconds. The branch-and-bound nodes are at most 4 for both algorithms.

10. Conclusion This paper first outlines the difference in air traffic management between United States and Europe; then the ground-holding policy and the ‘free flight’ concept, with their similarities, are presented. The new exact ABGI approach, based on the integration of the heuristic procedure ABGH with the exact method ABGE, is discussed. ABGI can solve at optimality all the large instances that have appeared in the previous literature, in a much shorter computational time than other exact approaches, and it can solve at optimality very large instances based on real OAG data in less than twenty minutes. ABGI is a procedure that could be integrated into a pre-tactical decision support system.

Acknowledgments The research by the first author was partially supported by two research grants from Progetto Finalizzato ¹rasporti 2 of the Italian National Research Council (contracts CNR/94.01335.PF74, CNR/96.00005.PF74). The research work of the second author was partially supported by a NATO research grant (contract NATO CRG 971550). The research by the third author was partially conducted at the Charles Stark Draper Laboratories in Cambridge, MA, USA.

Adams, M., Kolitz, S., Milner, J., & Odoni, A.R. (1996). Evolutionary concepts for decentralized air traffic flow management. Air ¹raffic Control Quarterly 4(4), 281—306. Andreatta, G., Brunetta, L., & Guastalla, G. (1997). Multi-airport ground-holding problem: A heuristic approach based on priority rules. In L. Bianco, P. Dell’Olmo & A.R. Odoni (Eds.), Modeling and Simulation in Air ¹raffic Management (pp. 71—89). Springer Verlag, Berlin. Andreatta, G., & Tidona, G. (1994). A new formulation for the multi-airport ground-holding problem. Technical report. Dip. di Matematica Pura e Applicata, University of Padova. Padova, Italy. Andreatta, G., & Brunetta, L. (1998). Multi-airport ground-holding problem: A computational evaluation of exact algorithms. Operations Research 46, 57—64. Andreatta, G., Odoni, A.R., & Richetta, O. (1993). Models for the ground-holding problem. In A.R. Odoni & L. Bianco (Eds.), ¸argeScale Computation in Air ¹raffic Control (pp. 125—168). Springer Verlag, Berlin. Andreatta, G., Brunetta, L., & Guastalla, G. (1996). From groundholding to free flight: A new exact approach. Technical report. Dip. di Matematica Pura e Applicata, University of Padova, Padova, Italy. Bertsimas, D.J., & Stock, S. (1998). The air traffic flow management problem with enroute capacities. Operations Research (to appear). Delahaye, D., Durand, N., Alliot, J.M., & Schoenauer, M. (1994). Genetic algorithms for air traffic control system. Technical report. Centre d’E¨tudes de la Navigation Ae´rienne — Sofre´avia. Orly Ae´rogare Cedex, France. FAA, Federal Aviation Administration (1994). Aviation capacity enhancement plan 1994. Technical report. Office of System Capacity and Requirements. Washington, D.C. Jenny, M. (1997). Free flight. Air ¹raffic Control Quarterly. Matos, P., & Ormerod, R. (1995). The application of operational research to european air traffic flow management — understanding the context. Technical report. Warwick Business School, Warwick, U.K. Matos, P., Chen, B., & Ormerod, R. (1996). Optimisation models for re-routing air traffic flows. Technical report. Warwick Business School, Warwick, U.K. Mugis, L. (1995). Mathematical programming for the air traffic flow management problem with enroute capacities. Technical report. Centre d’E¨tudes de la Navigation Ae´rienne — Sofre´avia, Orly Ae´rogare Cedex, France. Navazio, L., & Romanin-Jacur, G. (1995). Multiple connections multiairport ground-holding problem: Models and algorithms. ¹ransportation Science (to appear). Odoni, A.R., Deyst, J., Feron, E., Hansman, R.J., Khan, K., Kuchar, J., & Simpson, R.W. (1997). Existing and required modeling capabilities for evaluating atm systems and concepts. Technical report. International Center for Air Transportation, Massachusetts Institute of Technology, Cambridge, MA. Odoni, A.R. (1987). The flow management problem in air traffic control. In A.R. Odoni, L. Bianco & G. Szego (Eds.), Flow Control of Congested Networks (pp. 269—288). Springer Verlag, Berlin. Richetta, O. (1995). Optimal algorithms and a remarkably efficient heuristic for the ground-holding problem in air traffic control. Operations Research 43, 758—770. Vranas, P.B. (1996). Optimal slot allocation for european air traffic flow management. Air ¹raffic Control Quarterly 4(4), 249—280. Vranas, P.B., Bertsimas, D.J., & Odoni, A.R. (1994). The multi-airport ground-holding problem in air traffic control. Operations Research 42, 249—261.