Distributed Flight Routing and Scheduling in Air Traffic Flow Management

Yicheng Zhang1, Student Member, IEEE, Rong Su1, Senior Member, IEEE, Qing Li2,Christos Cassandras3 Fellow, IEEE, and Lihua Xie1 Fellow, IEEE

Abstract— Air traffic flow management is an important com-ponent in an air traffic control system and has significant effectson the safety and efficiency of air transportation. In this paper,we propose a distributed air traffic flow management strategy tominimize the airport departure and arrival schedule deviations.The scheduling problem is formulated based on an en-routeair traffic system model consisting of air routes, waypoints andairports. A cell transmission flow dynamic model is adopted todescribe the system dynamics under safety related constraintssuch as the capacities of air routes and airports, and theaircraft speed limits. Our air traffic flow management problemis formulated as an integer quadratic programming problem. Toovercome the computational complexity, we first solve a relaxedquadratic programming problem by a distributed approachbased on Lagrangian relaxation. Then a heuristic algorithmbased on forward-backward propagation is proposed to obtainthe final integer solution. Experimental results demonstrate theeffectiveness of the proposed scheduling strategy.

I. INTRODUCTION

Air traffic delays have been costing billions of dollarsto airlines each year [1]. Due to limited space for furtherinfrastructural expansion, improving efficiency of air trafficmanagement becomes critical for the aviation industry tocope with the expected demand surge, which consists ofair traffic control, aeronautical meteorology, air navigationsystems, Air Space Management, Air Traffic Services, andAir Traffic Flow Management (ATFM). In this paper we aimat improving efficiency of Air Traffic Flow Management(ATFM) by reducing the arrival and departure scheduledeviations in the air traffic system.

There are many techniques proposed in the literature andapplied in the current practice [2]. General Air Traffic FlowManagement [3] [4] considers the en-route capacities, whileundertaking air flow management. In terms of modeling,there have been several major types proposed in the liter-ature [5]. Lagrangian models [4][6] are used to describeflight trajectories of individual aircraft, which usually iscomputationally infeasible for a large network. Aggregatetraffic models and Eulerian models [7] are used to describeaverage behaviours of a group of aircraft, as described by the

This work is financially supported by the ATMRI-CAAS Project on AirTraffic Flow Management with the project reference number ATMRI:2014-R7-SU.

1Yicheng Zhang, Rong Su and Lihua Xie are affiliated with Schoolof Electrical and Electronic Engineering at Nanyang Technological Uni-versity, Singapore 639798. E-mails yzhang088@e.ntu.edu.sg,rsu@ntu.edu.sg, elhxie@ntu.edu.sg

2Qing Li is affiliated with Air Traffic Management Research Institute(ATMRI) at Nanyang Technological University, Singapore 639798. E-mail:liqing@ntu.edu.sg

3Christos Cassandras is affiliated with the Division of Systems Engineer-ing at Boston University, Brookline, MA 02446. E-mail: cgc@bu.edu

concept of flows. An aggregation approach typically providesa lower order, fixed-resolution model of the airspace, whilethe Eulerian approach provides a flexible resolution model[8]. A multicommodity Eulerian-Lagrangian large-capacitycell transmission model for en-route air traffic is proposed in[9], which adopts the basic idea of cell transmission modelsdescribed in [10] within a standard multi-commodity flowmodel equipped with origin-destination (OD) pairs to avoiddifficulties in describing flow merging and diverging.

In this paper we adopt a Eulerian-Lagrangian model simi-lar to [9], but with a richer set of constraint types than otherexisting flow models such as [9] [11] [12]. For example, weconsider a variety of aircraft types such as small, medium andlarge passenger/cargo aircrafts and unmanned aerial vehicles(UAVs) that are expected to be more popular in the nearfuture, and take the aircraft speed lower and upper limits intoaccount when applying air route capacity constraints on airroute volume dynamics. We formulate our ATFM problemas an integer quadratic programming (IQP) problem due tothe integral values of the decision variables of air route flightshifts within each discrete time interval, and aim to minimizethe total airport departure and arrival schedule deviationsin an air traffic network. To overcome the exponentialcomplexity in IQP, we first relax it into a standard QP, whichcan be solved by using some optimization solvers such as theoptimization toolbox in MATLAB or CPLEX [13]. Althoughan QP problem can be solved efficiently compared withother optimization formulations, for a large-scale system wemay still face a challenge of high computational complexitydue to an extremely large number of decision variables andconstraints, even though it is polynomial-time [14]. To re-duce the computational complexity, we propose a distributedscheduling strategy based on Lagrangian relaxation [15] andthe subgradient method [16] [17] [18], aiming for a goodtrade-off between the quality of scheduling and the compu-tational complexity. After solving the relaxed QP problem,we then propose a novel heuristic forward-backward prop-agation algorithm to achieve integral solutions. Comparedwith existing flow-based ATFM approaches, we have madethe following contributions: (1) an IQP ATFM formulationwith an Eulerian-Lagragian flow model and more types ofconstraints, (2) a Lagrangian Relaxation based distributedoptimization approach to solve a QP-relaxed ATFM problem,and (3) a heuristic forward-backward propagation algorithmto obtain an integral solution.

The rest of the paper is organized as follows. An ATFMproblem is formulated as an IQP problem in Section II. Adistributed air traffic flow routing and scheduling strategy is

Fig. 1. Simplified system model

proposed in Section III, which solves a QP-relaxed problem.A heuristic propagation algorithm is presented in SectionIV, which aims to derive an integral solution to the originalIQP problem. Experimental results are shown in Section V.Conclusions are drawn in Section VI.

II. AN ATFM PROBLEM FORMULATION

A. Introduction of an en-route air traffic network

An en-route air traffic network is depicted as a directedgraph shown in Figure 1. We consider an air traffic networkconsists of pre-defined routes. Each air route consisting ofa sequence of waypoints (or control points), and any twoconsecutive waypoints are connected by one link, which isone segment of an air route. The interesting point is thatthere is no direct connection between the departure pointand the arrival point in each airport. Although in the picturethere is at most one directed link between any two nodes(i.e., waypoints), it is possible to use two links with oppositedirections to denote one air route allowing bi-directionalflights but with safe vertical separations.

It is interesting to note that an air traffic managementsystem is usually distributed by nature, where multipleArea Control Centers (ACCs) running in parallel to controlindividual aircraft in their own responsible airspace, whereascommunicating with other neighboring ACCs to relay im-portant messages. This natural distributed infrastructure willlater facilitate our distributed strategy to lower the computa-tional burden for real-time operations.

B. Air route segmentation

We adopt a discrete-time cell transmission link dynamicmodel in this framework. Within a pre-chosen samplingperiod ∆T , the distance LC that an aircraft is able to coveris determined by the minimum acceptable cruising speed(denoted as S) and the maximum cruising speed (denotedas S), i.e., S∆T ≤ LC ≤ S∆T . In this work we choose LCas the distance that an aircraft can cover within ∆T withan economic cruising speed, and call LC one link segment.For different types of aircraft with possibly different cruisingspeeds, their segment lengths are different. In this case wechoose LC for a specific link to be the largest segment lengthamong all under consideration, and partition the link into aset of whole segments with possibly one fractional segment,which covers two consecutive links, as shown in Fig 2, wherethe link A − J consists of three segments and a fractionalsegment, whereas the link B − J has four segments and afractional segment.

The choice of ∆T needs to ensure that all aircraft that flyinto a segment during the period t will fly out of the segment

Fig. 2. Fractional Segments in air-routes

during the period t+1. This assumption of “aircraft hoppingamong segments” essentially rules out the possibility ofdealing with multiple types of aircraft with large speeddifferences. Fortunately, in practice aircraft with significantlydifferent speeds fly at different cruise altitudes. Thus, theycan be considered separately. In other words, in this paperwe deal with aircraft with similar speed limits.

To estimate the flow entering and exiting a fractionalsegment, the aircraft are required to be uniformly distributedin each segment. This assumption is roughly true when thelength of segment is short enough. In our model, the segmentlength is related to the sampling time, thus by adjusting thesampling time we can achieve this assumption via a suitableair traffic control strategy, which is nevertheless outsidethe scope of this paper. With this assumption of uniformdistribution, the exiting flows from one whole segment i toanother whole segment j via a fractional segment k can becalculated as follows.

fij =LC − LkLC

fik. (1)

The fractional segment model is introduced to obtainreasonable partitions in each air link. However, it may leadto a high concentration of aircraft at the entrance of thedownstream segment next to these fractional segments, e.g.,the air route J − C shown in Fig 2, even though the totalnumber of aircraft in the downstream segment still satisfiesthe segment capacity, i.e., the merging of aircraft temporarilycreate nonuniform distribution in the downstream segment.To ensure that this temporary concentration of aircraft willnot cause any problem for the air traffic control part, weimpose a constraint that the density at the entrance ofthe downstream segment should not surpass the maximumdensity of the downstream segment, which is shown in thefollowing section.

Comparing with the air traffic system model descriptionproposed in [12], our model description has some advantageswhich make this model more realistic for an en-route airtraffic system. Firstly, our model takes the minimum flightspeed into consideration which is not considered in [12]. Theminimal speed is an important character to ensure the flightsafety. Secondly, we consider RNP navigation, which allowsall concerned aircraft to fly through a fixed en-route network.In contrast, [12] considers sectors and only the entrance andexit of the sectors are fixed.

C. Statement of air traffic flow routing and scheduling prob-lem

1) Notations: Before we describe our air flow routingand scheduling problem, some necessary notations are listed

below.N,R+ – The sets of natural numbers and nonnegative real

numbers, respectively.G = (V,E) – The directed graph is to describe the

air traffic network, where vertices and edges denote way-points (together with all segmentation points) and air linksrespectively. From now on, we will use link and segmentinterchangeably.A – The set of all concerned airports, where each airport

a ∈ A. We use fa,Pin (φ, t) to denote the incoming (or arrival)flow of aircraft type φ with the OD pair P , and fa,Po (φ, t)for the outgoing (or departure) flow of aircraft type φ withthe OD pair P at interval t. For each airport, the scheduledarrival rate ra,P (φ, t) of aircraft type φ with the OD pairP and the departure rate sa,P (φ, t) of aircraft type φ withthe OD pair P are assumed known, which are essentiallythe airport arrival and departure handling capacity at t.i := (v, v′) ∈ E ⊆ V × V – denotes the directed air link

departing from waypoint v to waypoint v′. Some parametersfor each air link include the length, Li, the capacity, Ci(t)and the upstream links connected with i = (v, v′) at v, Ui,and the downstream links connected with i = (v, v′) at v′,Di.S(φ), S(φ) – The lower and upper speed limit of type

φ ∈ F aircraft, where F is the set of aircraft types.Hp ⊆ N – The set of labels of all discrete intervals. By

default, Hp := {1, 2, · · · , |Hp|}.C := A × A – The set of origin-destination (OD) pairs.

For each P = (a, a′) ∈ C, let P [1] = a and P [2] = a′.NPi (φ, t) – The link volume or the number of aircraft with

type φ in link i with OD pair P at time interval t.fPij (φ, t) – The flight shift of aircraft with type φ in link

i towards link j with OD pair P at time interval t.2) Constraints: We consider the following constraints: the

network dynamics constraints, the link capacity constraints,the flight shift limits constraints, which are described below.C1-Network dynamics constraints: The network dynamicsdescribes the relationship between the air traffic flight shiftsand the air link volumes. For all t ∈ Hp, i ∈ E, φ ∈ F , andP ∈ C,

NPi (φ, t+ 1) = NP

i (φ, t) + fPi,in(φ, t)− fPi,o(φ, t) (2)

where fPin(φ, t) and fPo (φ, t) denote respectively the totalincoming and outgoing flight shifts of type φ aircraft in airlink i with OD pair P at time interval t. More explicitly,

fPi,in(φ, t) =∑

j∈Ui∪Ui,F

fPji(φ, t), fPi,o(φ, t) =

∑k∈Di∪Di,F

fPik(φ, t),

where the set Di,F denotes all downstream whole segmentsconnecting with segment i via some fractional segments.C2-Link capacity constraint: Due to the head-and-tail sep-aration requirement imposed on all aircraft in the network,each link (or segment) has its own capacity. In general,different links may have different separation requirements,captured by a variable msep(t), which also suggests thatthe separation distance is time variant, due to possibly thetime variant weather conditions. This time variant separationdistance function is assumed known in advance in this paperfor a routing and scheduling purpose, from which the linkcapacity can be defined as Ci(t) = Li

msep(t), where Ci(t) is

the capacity of link i at t and Li is the length of the link i.The total number of all types of aircraft in link i should notbe greater than the link capacity at any time interval t, i.e.,

(∀t ∈ Hp)∑P∈C

∑φ∈F

NPi (φ, t) ≤ Ci(t). (3)

C3-Flight shift limits constraints: With the constraintsof the cruising speeds for different types of aircraft, theoutgoing link flight shift should also be bounded by theproduct of the link density and the speed limits, i.e., for allt ∈ Hp, i, j ∈ E, φ ∈ F , P ∈ C,

NPi (φ, t)

LiS(φ) ≤

∑j∈Di

fPij (φ, t) ≤ NPi (φ, t)

LiS(φ), (4)

where NPi (φ,t)Li

denotes the link density for aircraft of typeφ and OD pair P with a uniformly distribution.C4-Flight density limits constraints for fractional seg-ments: As mentioned in the network segmentation, thedensity of the merging air flows from the fractional seg-ments should be no bigger than the entrance density of thedownstream link. Given a link i ∈ E, let Ui,F ⊆ Ui be theset of all upstream links, whose lengths are fractional withrespect to a given whole segment length LC . Then we havethe following:

∑j∈Ui,F

∑φ∈F,P∈C

fPji(φ, t)

LC − Lji≤ CiLi. (5)

3) Objective function: Our objective is to minimize thetotal deviation from the original arrival and departure sched-ules as well as the chances of landing in airports differentfrom the originally planned ones. Based on the aforemen-tioned functions, the objective function can be formulated asfollows,

min∑

t∈Hp,φ∈F,P∈C

{∑a∈A

[fa,Pin (φ, t)− ra,P (φ, t)

]2+∑a∈A

[fa,Po (φ, t)− sa,P (φ, t)

]2+M

∑a∈A:P [2] 6=a

fa,Pin (φ, t)}(6)

where in the third term M is a very large positive constant,denoting the extremely high penalty on landing aircraft to anairport different from their originally planned destinations.The first term in the cost function denotes the total deviationfrom the original arrival schedules, and the second term forthe total deviation from the original departure schedules.

Now we summarize what we have developed so farand state the Air Flow Routing and Scheduling Problem(AFRSP). Teh AFRSP is to minimize objective function(6) subject to fractional segment flow equation (1), C1-network dynamics constraints, C2-link capacity constraints,C3-flight shift limits constraints and C4-flight density limitsconstraints for merging. The AFRSP is an IQP problem. Dueto the complexity involved in solving this IQP problem, wewill first relax it into a QP problem, which will be solvedby a distributed algorithm based on Lagrangian relaxation,and then use a heuristic algorithm to obtain a final integralsolution.

III. DISTRIBUTED AIR TRAFFIC FLOW MANAGEMENT

We first relax all integral decision variables in the AFRSPinto reals. This converts the AFRSP into a standard QPproblem. We partition the whole air traffic network into sub-networks. Each airport or waypoint only belongs to one sub-network, and so does each air link, except for a few insidethe network which are shared by two sub-networks. Formallyspeaking, we consider the network as a directed graph G =(V,E), where the vertex set is V contains one special nodecalled ext denoting the external of the entire network, andthe edge set is E ⊆ V × V − {(ext, ext)} denoting theset of all directed air links, i.e., each air link (v, v′) ∈ Erepresents an air traffic flow either from one waypoint v toanother waypoint v′, or from the external source v = extto a waypoint v′ (which represents an incoming boundarylink), or from waypoint v to the external source v′ = ext(which represents an outgoing boundary link). Let S be apartition of v − {ext}, i.e., each waypoint belongs to onesub-network, and let L(S) denote all air links belonging toS. We now make the following modification to the networkgraph G: for each link (v, v′) ∈ E with v ∈ S ∈ S ,v′ ∈ S′ ∈ S and S 6= S′, we add a node bv,v′ to V , whichrepresents a boundary of S and S′, and replace (v, v′) ∈ Eby two new edges (v, bv,v′) and (bv,v′ , v

′), which denotestwo disjoint air segments, whose union is the original link(v, v′), and (v, bv,v′) is placed in S and (bv,v′ , v

′) belongsto S ′. After the modification, let B be the collection of allsuch boundary nodes, and E′ be the new edge set. Thenthe new network graph is G′ = (V ′ = V ∪ B,E′), whereE′ ⊆ (V ∪ B) × (V ∪ B) − ({(ext, ext)} ∪ B × B). Forany two different sub-networks, they can only share someboundary points in B and, of course, the external node ext.

Suppose the whole network is partitioned into nk sub-networks denoted as {Sk ∈ S|k = 1, · · · , nk}. The boundaryconstraints are the consistent constraints for the air trafficflow on the boundary air-routes, i.e., at any time intervalt ∈ Hp, the incoming flow should be equal to the outgoingflow on the boundary air links, f inbv,v′ ,v′ = fov,bv,v′ , where(v, bv,v′) ∈ L(S) and (bv,v′ , v

′) ∈ L(S′) denote the bound-ary air segments in two adjacent sub-networks. Based on theprevious terminologies, the proposed ATFSP can be writtenin the formulation as follows,

min∑S∈S

J(S) (7a)

subject to Φ(S) (7b)

∀((v, bv,v′), (bv,v′ , v′) ∈ E′)f in(bv,v′ ,v′) = fo(v,bv,v′ )

(7c)

where Φ(S) is the set of constraints associated with sub-network S, and the sector objective function J(S) is definedas follows,

J(S) = min∑

t∈Hp,φ∈F,P∈C

{ ∑a∈A(S)

[fa,Pin (φ, t)− ra,P (φ, t)

]2+

∑a∈A(S)

[fa,Po (φ, t)− sa,P (φ, t)

]2+M

∑a∈A(S):P [2] 6=a

fa,Pin (φ, t)}

and Φ(S) denotes the constraint set mentioned in the lastsection. By adopting Lagrangian relaxation, we can remove

the boundary constraints in formulation (8) and obtain thefollowing Lagrangian dual problem.

max{λb

v,v′≥0|bv,v′∈B}min

∑S∈S

J(S) + λbv,v′ (fo(v,bv,v′ ) − f

in(bv,v′ ,v′))

(8a)subject to Φ(S), ∀S ∈ S (8b)

Let λ be the vector consisting of all {λbv,v′ |bv,v′ ∈ B} anddefine H(λ, S) as follows:

min J(S)−∑

bv,v′∈B:v′∈S

λbv,v′ fin(bv,v′ ,v′)

+∑

bv,v′∈B:v∈S

λbv,v′ fo(v,bv,v′ )

(9a)

s.t. Φ(S) (9b)

Then the Lagrangian dual problem (9a)-(9b) can be rewrittenin the following separable form,

maxλ≥0

∑S∈S

H(λ, S) (10)

Problem (10) can be solved by the following standarditerative subgradient method.

Algorithm 1 – Subgradient Algorithm for Lagrangian Dual Problem (10)

1) Pick λ0 ≥ 0;2) In round k ≥ 0, solve each H(λk, S) (S ∈ S) in parallel;3) Update λ(r+1)

bv,v′as follows,

λ(r+1)bv,v′

= max{0, λrbv,v′ + αrbv,v′ (fin(bv,v′ ,v′) − f

o(v,bv,v′ ))}

where the constant αrbv,v′≥ 0 is the step size at round r;

4) Iterate on r until λrbv,v′converges.

Because the QP-relaxed AFRSP is convex, Algorithm 1converges in polynomial time. In addition, the duality gap iszero because the Slater condition holds. Thus, all boundaryequalities will hold.

IV. FORWARD-BACKWARD PROPOGATION FORINTEGRAL SOLUTIONS OF AFRSP

It is well known that finding a globally optimal integralsolution is NP-hard. Thus, we aim to use a heuristic ap-proach called forward-backward propagation algorithm togenerate an integral solution. Suppose the QP solution ofAFRSP is xo, which consists of two parts: the link volumes{NP

i (φ, t) ∈ R+|i ∈ E∧P ∈ C∧φ ∈ F} and the flight shifts{fPij (φ, t) ∈ R+|i ∈ E ∧ j ∈ Di ∪Di,F ∧ P ∈ C ∧ φ ∈ F},where R+ denotes positive reals. The forward-backwardpropogation algorithm is shown as follows.

Algorithm 2 terminates finitely due to the termination ofstep (3.d) in iterations. The Step (2) can be done in O(|A| ·|E|2), where | · | denotes the size of a set, because for eacha ∈ A the shortest path problem within a directed graph canbe solved in O(|E|2), where we treat each link of G as anode and nodes of G as links. For each t ∈ Hp, φ ∈ F ,P ∈ C, and a link i ∈ E, backward propagation associatedwith solving inconsistency for one link in the worst casemay need to traverse all links co-reachable from the original

Algorithm 2 – Forward - Backward Propagation1) Input: Let the link volumes

{NPi (φ, t) ∈ R+|i ∈ E ∧ P ∈ C ∧ φ ∈ F}

and the flight shifts

{fPij (φ, t) ∈ R+|i ∈ E ∧ j ∈ Di ∪Di,F ∧ P ∈ C ∧ φ ∈ F}

be the solution from the QP-relaxed problem.2) Initialization: For each air port a, set the backward and forward

tier numbers ξb(i, a) and ξf (i, a) for each link i respectively.3) Iteration: For each time interval t = 1, 2, · · · , |Hp|

a) Round up all flight shifts fPij (φ, t) by the floor function,i.e.,

fPij (φ, t) = floor(fPij (φ, t))

b) For each link i ∈ E, update air link volumes as follows,

NPi (φ, t+ 1) = NP

i (φ, t) + fPi,in(φ, t)− fPi,o(φ, t)c) Check the feasibility of the obtained volumes. If

(∀i ∈ E)NPi (φ, t+ 1) ∈ [0, Ci(t+ 1)],

go to Step (3.f); elseif

(∃i ∈ E)NPi (φ, t+ 1) > Ci(t+ 1),

go to Step (3.d); otherwise, go to Step (3.e).d) Pick j ∈ Ui ∪Ui,F (j ∈ Di ∪Di,F ) based on procedure

2 and reduce fPji(φ, t) to lower the incoming (outgoing)flight shift fPi,in(φ, t) (fPi,o(φ, t)). Continue the selectionuntil the gap is absorbed for link i. Go to Step (3.b)

e) If all the air link volumes at time t + 1 are feasible, sett := t+ 1 and go to Step (3).

4) Output: The integral values of link volumes and flight shifts.

airport P [1]. Thus, the complexity is O(Kb|E|), where Kb isthe maximum adjacent upstream links for each link. For anydirected graph without self-loops, Kb ≤ |V | − 1, where theequality holds when the graph is complete. Thus, the worstcase complexity for backward propagation is O(|V | · |E|). Asimilar argument is applicable to each forward propagation.

V. EXPERIMENTAL RESULTS

The layout of the air traffic network in our case study isshown in Fig 3. The airports are connected to the networkas the boundary nodes of the air traffic grid. For a 8-to-

Fig. 3. A 8-8 air traffic system

8 air traffic system with the prediction horizon |Hp| = 4,it includes 16 airports, 80 waypoints and 176 air-routes. Atotal of 307904 decision variables exist in this system, whichis hard to be treated as a centralized system.

The optimization problem is solved by CPLEX basedon MATLAB on a PC with an Intel Core(TM) i7-4770@3.40GHz CPU and RAM 8GB. The sampling time ischosen as 5 minutes and the prediction horizon is chosenas |Hp| = 12, 24, 48, · · · , 288, respectively. The predictionhorizon |Hp| = 72 refers to 6 hours. The maximum predic-tion horizon |Hp| = 288 refers to 24 hours. We choose sev-eral different air traffic grid scales, nh = nv = 4 or 8, wherenh and nv denotes respectively the numbers of individualcells horizontally and vertically, where each cell consists of4-5 waypoints and 0-2 airports depending on whether it is aninternal cell or a boundary cell. After applying the centralizedIQP solver CPLEX, the experimental results are shown inTable I.

TABLE ICOMPARISON OF PROCESSING TIME WITH CENTRALIZED APPROACH

nh = nv |Hp| IQP Solver Relaxed QP4 12 0.03s * 0.02s *4 24 0.06s * 0.02s *4 48 0.23s 0.05s *4 72 1.44s 0.34s4 144 1.84s 0.64s4 288 3.06s 1.36s8 12 2351.5s 0.17s8 18 ** 0.26s8 24 - -

1) * The computational time may not be accurate for these small scaleproblem because of the minimal clock cycle of the operating system.

2) - The centralized problem cannot be complied by the complierbecause of the huge memory consumption.

3) ** The computational time is more than 2 hours.

From Table I, the centralized approach solving by IQPsolver is applicable when the system scale is small andthe prediction horizon is short. However, the computationaltime increases significantly with the increasing of systemscale. The processing time for solving a 8-by-8 system withprediction horizon equals to 1 hour cost 2351.5s, which is faraway beyond the real-time requirement. Solving the relaxedQP problem requires much less processing time when thesystem is large comparing to solving the IQP problem.

We revisit the same case study by applying our proposeddistributed air traffic flow routing and scheduling strategytogether with the heuristic propagation algorithm (Algorithm2) with the same PC configuration mentioned before. Theresults are shown in Table II. All air traffic networks in thistest are divided into four small scale sub-networks.

TABLE IICOMPARISON OF PROCESSING TIME WITH DISTRIBUTED APPROACH

nh = nv |Hp| Relaxed QP Propagation Total8 12 0.20s 0.13s 0.33s8 18 0.45s 0.16s 0.61s8 24 3.64s 0.17s 3.81s8 36 13.20s 0.85s 14.05s8 48 16.43s 1.02s 17.45s8 60 133.26s 1.43s 134.69s

From the result shown in Table II, the distributed ap-

TABLE IIITHE QUALITY COMPARISON BETWEEN THE DISTRIBUTED APPROACH

AND CENTRALIZED IQP APPROACH

nh = nv |Hp| Total flights DC DD %4 12 936 286 296 3.5%4 18 1368 432 440 1.9%4 24 1800 553 600 7.8%4 36 2664 882 902 1.1%4 48 3528 1183 1214 2.6%4 60 4392 1468 1532 4.4%

proach can solve larger scale problem with longer predictionhorizons. The processing time for solving a 8-by-8 systemwith |Hp| = 60 is around 2 minutes, which is shorterthan the sampling time 5 minutes and can be regarded asa real-time process. The results for some larger scale airtraffic networks with longer prediction horizons, e.g., a 8-by-8 network with |Hp| = 60, show that the processing timefor this distributed solution is related to the processing timefor solving the local optimization problem associated witheach sub-network. Thus, how to obtain a relatively shortprocessing time for each sub-network will be the next step ofthis study. Some evolution algorithms may be adopted here toovercome this computational issue for large scale problems.

To evaluate the quality of solutions attainable from ourdistributed approach in comparison with the solutions deriv-able from solving the centralized IQP, we consider a networkwith nh = nv = 4, and 12 ≤ |Hp| ≤ 60, and the numberof aircraft in the network is about 1.5 time of the networkcapacity, i.e., the network experiences a heavy traffic ”jam”,leading to delays of about 1/3 flights. The experimentalresults are shown in Table IV, DC denotes the deviationfrom the centralized IQP approach and DD denotes thedeviation with distributed approach, which indicate that thedifference between the outcomes of the centralized approachand our distributed approach are close to each other, andthe largest difference is less than 8%. This suggests thatour distributed approach has achieved a tremendous gainin reducing computational complexity with an acceptabledegree of quality degradation.

The processing time for the forward backward propagationalgorithm for these three scenarios are similar. Each of themtakes from 0.13s to 1.43s to be solved. As a conclusion,the time for solving the forward-backward propagation issignificantly less than the time for solving the QP-relaxedproblem, which certainly takes much less time than solvingthe centralized IQP problem. Thus, the total time con-sumption to obtain an integral solution via our distributedapproach, which combines the Lagrangian relaxation and theheuristic propagation algorithm (Algorithm 2), is viable forsolving a large-scale air traffic flow routing and schedulingproblem.

VI. CONCLUSION

In this paper we have proposed a realistic ATFM modelingmethod and then formulated an ATFM problem as an IQPproblem, which aims to minimize the deviation from the

originally planned departure and arrival flows. The key ideaunderlying the problem formulation is to use airport grounddelays (via adjusting the departure flight shifts) and en-route flight rerouting to minimize the networkwise scheduledeviation. To overcome the computational complexity, wehave proposed a distributed approach, which first relaxes theoriginal IQP probem into a convex QP problem, solvable by adistributed strategy based on Lagrangian relaxation, and thenfeeds the outcome of that QP-relaxed problem into a heuristicpropagation algorithm to derive a final integral solution.Our experimental results have indicated that this distributedstrategy can solve fairly large flow routing and schedulingproblems. We have been applying evolution algorithms inlocal computation to improve the computational efficiencyof our approach even further.

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