design of an optimal route structure using heuristics

Design of an Optimal Route Structure Using Heuristics-BasedStochastic Schedulers

Seongim Choi

University Affiliated Research Center, NASA Ames Research Center, Moffett Field, CA 94035.

John E. Robinson III and Daniel G. Mulfinger

NASA Ames Research Center, Moffett Field, CA 94035.

Brian J. Capozzi

Mosaic ATM, Inc., Leesburg, VA 20175, USA

Abstract

The purpose of this study is to investigate theeffects of efficient route structure in the extendedterminal airspace area on arrival scheduling perfor-mance. This paper will provide reasonable guide-lines for optimal route topology in the extendedterminal area by considering the uncertainties presentin real operations. In a previous study, a Mixed In-teger Linear Programming (MILP)-based schedul-ing algorithm proved to generate more optimalscheduling results than a traditional First-come-First-Served (FCFS) scheduler. However, an ex-pensive computational cost associated with exten-sive search process limited its usage to a smallnumber of flights in a dense terminal environment.Heuristics based on FCFS scheduling were intro-duced to alleviate this computational limitation.However, that heuristic was not sufficient to ac-commodate the amount of traffic associated withdense terminal operations. In this study, we intro-duce a Genetic Algorithm (GA) as an alternativeheuristic for queuing aircraft and route assignmentto reduce the computational cost dramatically. Totake into account realistic operations, a dynamicplanner framework is constructed that integratesthe GA heuristics-based scheduler with a stochas-tic trajectory simulator. Uncertainty quantifica-tion and propagation along the routes are imple-mented in the trajectory model. The trajectorymodel is simulated based on the Scheduled Timesof Arrival (STAs) provided by the scheduler. As

a practical application of the proposed schedulerto the dense terminal environment, a design of anoptimal route structure is carried out for the termi-nal airspace represented in cartesian coordinates.The effects of airspace topologies on the schedulingperformance are investigated and numerous routestructures with different merge topologies are con-structed. An optimal merge topology is identifiedby comparing their scheduling performances andthe resulting optimal route structure is validatedby the dynamic planner framework. Finally, thesensitivities of the scheduling performance with re-spect to the uncertainty quantification and propa-gation modeling are discussed.

Introduction

A concept of advanced terminal airspace areaoperations is part of the Next Generation Air Trans-portation System (NextGen) efforts to accommo-date expected increase in the demand and providea higher level of throughput at the airports andwithin en route airspace [1, 2, 3]. Scheduling opti-mization problems in dense terminal airspace areaoperations are drawing interest [4], since the in-crease of airspace capacity is becoming more im-portant to accommodate large air traffic flows inthat environment. The use of scheduling algo-rithms more efficient than the traditional FCFSapproach is a way to increase a way to increasethroughput and efficiencies in congested terminalairspace compared to the FCFS scheduler. On the

other hand, it is known that the route topologiesin the extended terminal airspace play an impor-tant role in the scheduling performance. Efficientscheduling and route assignment directly affect im-portant performance metrics such as runway de-lays, throughput, fuel efficiency, and robustness touncertainties in operations. However, despite therealization of the importance of the route topology,few extensive studies of the problem have been re-ported.

The purpose of this paper is: 1) to investigateoptimal and efficient scheduling algorithms for adense terminal airspace operation that yield bet-ter performance compared to the traditional FCFSapproach, and 2) to design an optimal route struc-ture for the extended terminal area.

A scheduling algorithm based on MILP hasbeen successfully used in airport surface manage-ment due to its ability to optimize both the ar-rival/departure sequence and their scheduling [5].However, due to the expensive computational costattributed to the branch and bound search algo-rithm, only a limited number of flights, typicallyless than 40, were allowed in the scheduling. Theapplication of MILP-based scheduling algorithmsto the dense terminal airspace operation has beenless effective. Consideration of hundreds of flightsin the span of a couple of hours typical of thedense terminal airspace operation dramatically in-creases the size of the MILP formulation and thecorresponding computation burden easily exceedsthe available computing resources. In a previousstudy, a heuristics-based MILP formulation wasintroduced in order to reduce computational bur-den and applied it to metroplex operations dur-ing their busiest operations [6]. However, the costfor the branch and bound search is still expensivethough not as prohibitive as in the original MILPformulation.

The combination of GA-based heuristics and aLinear Programming (LP) method was proposedby Capozzi et al. [8] This is the method that isadopted in the current work. It explicitly separatesthe search for the optimum binary variables ofroute assignment and aircraft sequencing from thesolution procedure for the continuous schedulingvariables. GA-based MILP optimization schemeis able to find the optimal solution in significantlyless computational time on the example problemconsidered.

A central focus of the current work is the de-sign of an optimal route topology in the extendedterminal area using the GA-based MILP sched-uler. This work will also use with flight trajec-tory uncertainty which is different from the previ-ous work that held an unrealistic premise that de-tailed flight intent information including the tran-sit times is known a priori. This would assumethat one snapshot of the planning is sufficient topredict the scheduling performance. However, thisis not true in the real-time simulation where un-certainties are present in flight trajectories and theschedules of the following flights are dynamicallyupdated later in time based on the schedules of theprevious flights.

In order to overcome the above issues, we de-veloped a dynamic planner framework that period-ically updates the flight schedules to handle uncer-tainties. The dynamic planner consists of separatemodules: a planner and a simulator. Uncertaintyis implemented in the trajectory model of the sim-ulator to account for aircraft arrival time errors.The planner adds an extra separation buffer at thescheduling points to cope with these inter-arrivalerrors.

As a practical application of the heuristics-based,stochastic schedulers to dense terminal airspacesimulation, a design of an optimal route structurein the extended terminal airspace area is carriedout. Key design parameters are the number ofmerge points and their locations. First, we con-struct five distinct route structures with variousmerge topologies. The scheduling performance isevaluated for each topology using the stochasticFCFS heuristics-based scheduler and a dynamicplanner framework is applied to each topology inorder to validate the predicted scheduling perfor-mance. It is demonstrated that the largest sepa-ration amount required at all scheduling points isa dominant factor in scheduling performance.

Finally, more generalized airspace topologiesare considered. To investigate the sensitivities ofthe merge topology to the uncertainty modeling,three types of uncertainty distributions are con-sidered: a constant, linear and quartic incrementof the uncertainty per unit route length. The com-parison of the corresponding scheduling results showthat: 1) there exist optimal merge locations in theextended terminal airspace area, 2) the optimalmerge locations tend to be positioned where large

uncertainties are present so that pilot’s control ef-forts reduce the local uncertainties at the controlpoint.

The rest of this paper is organized as follows:First, the basic formulation of the original MILPalgorithms is explained, and the objectives andconstraints for our route assignment and schedul-ing problem are defined. Second, a number ofheuristics are introduced into the original MILPformulation as ways to reduce computational cost.Third, the dynamic planner framework with uncer-tainty modeling and extra separation buffers aredetailed. Finally, a design of optimal route struc-ture in terms of the merge topologies is discussedand followed by conclusions and future work.

Problem Formulation

We apply a MILP-based formulation to thescheduling problem in the extended terminal airspacearea, from the entry fixes to runway, and only thearrival portion of the scheduling will be considered.The plan consists of a time-constrained route foreach aircraft in the demand set, with STA specifi-cations at control points along each route, as dic-tated by separation rules, such that the resultingmovement plan for all aircraft in the demand setis conflict-free.

Basic Mixed Integer Linear Program-ming

A MILP scheduler has advantages in the schedul-ing problems over the traditional FCFS scheduler [9].The design variables can be freely chosen specifi-cally to a problem as forms of binary variables andcontinuous variables. However, the computationalcost of a branch and bound search algorithm is ex-pensive and its scalability with the problem sizeis rather poor and a number of heuristics to re-duce the computational burden will be explainedin later sections.

Initial Demand Set

Let F ,R, and P define a set of flights, availableroutes, and scheduling points, respectively, andNF , NR, and NP are the total number of flights,

routes, and scheduling points in the demand set.

F = {fj | 1 ≤ j ≤ NF } = {f1, f2, ..., fNF}

R = {rj | 1 ≤ j ≤ NR} = {r1, r2, ..., rNR}

P = {rj | 1 ≤ j ≤ NR} = {p1, p2, ...pNP}

We further presume the existence of functionsthat select the subset of routes from R that arefeasible for a given flight f ∈ F and the subset ofpoints from P that are feasible for a given router ∈ R.

Rf = {rk | all possible routes that flight (1)

f can fly, k ∈ {1, 2, ..., NR(f)}}

Pr = {pk | ordered set of scheduling points (2)

on the route r, k ∈ {1, 2, ..., NP (r)}}

where NR(f) is the number of all routes that flightf can fly, and NP (r) is the number of all thescheduling points on route r. Then, it can be eas-ily inferred that a set of total routes and the pointsare the superset of Rf and Pr, respectively, andthe relation is represented as follows:

R =⋃f∈FRf and P =

⋃r∈RPr

Decision Variables

Given the notation in the previous section, thedecision variables of interest for this problem canbe defined:

• Af,r - A binary route assignment variablethat takes on the value of 1 if flight f is as-signed to the route r and zero otherwise.

• Tf,r,p - A continuous variable representingthe time that flight f is scheduled to crossthe scheduling point p on route r, where f ∈F , r ∈ Rf , and p ∈ Pr.

• Sf,f ′ ,r,r′ ,p - A binary variable that takes onthe value of 1 if flight f on route r is se-quenced prior to flight f

′on route r

′at shared

scheduled point p, where f ∈ F , r ∈ Rf ,r′ ∈ Rf ′ , and p ∈ Pr ∩ Pr′ 6= ∅.

Objective and Constraints

For the purposes of this paper, the objectivefunction is defined so as to minimize the total timerequired for all flights to reach the end of theirroute, i.e., the runway threshold:

J =∑f∈F

∑r∈Rf

Af,rTf,r,pF (3)

The problem constraints are as follows:

• Assignment to Only One Route. Each flightcan be assigned to one and only one route.

• Crossing Time at Initial and Final Point. Ifa flight is assigned to a given route, then itsstart time on that route must be no earlierthan the earliest feasible time on that route,TEf,r, and no later than the latest feasible

time, TFf,r.

• Ordering Constraint at Potentially SharedScheduling Points. For each pair of flightsand each pair of route assignment optionsthat share a common scheduling point, theorder in which the flights are sequenced atthe common point must be uniquely speci-fied.

• Separation Constraint. At each common schedul-ing point, successive aircraft must be sepa-rated by a minimum time that is potentiallyintersection-dependent.

• Transit Time Constraints. In order to bephysically realizable, the travel time betweenscheduling points must be greater than theminimum possible travel time and should bebounded by the maximum “delayability” ofthe flight between the scheduling points.

Heuristics-Based Schedulers

One of the drawbacks of the MILP-based sched-uler is its prohibitive computational cost. How-ever, observation of the branch and bound searchprocedure indicates a large portion of search timeis spent on evaluating unrealistic combinations ofroute assignment and sequencing. Heuristics inthe route assignment and the queuing can helpeliminate some of the unrealistic search efforts and

reduce the computational burden. In our study,we tried two types of heuristics. First, heuristicsbased on the FCFS scheduling behavior can elim-inate a number of binary decision variables in theoriginal MILP formulation. Second, the branchand bound search algorithm can be replaced byGA-based heuristics.

Heuristics-Based Mixed Integer LinearProgramming

Based on the observations of the FCFS schedul-ing strategy, assumptions are made on sequenc-ing along certain route segments, such as mergeportions. The details of the following heuristicswere explained in our previous work on the metro-plex operations and only brief summaries are listedhere:

• Precedence constraint heuristic. Sequencesalong the common segments of the routes donot change and follow the queuing from theupstream of the merging segments.

• Windowing heuristics. A sequence changeis not allowed for a pair with their earliestcrossing times at the entry fixes separatedby more than a certain amount, i.e., win-dowing value. Using windowing heuristics,resulting schedules are planned locally andsubsequently corresponding to this value.

• FCFS heuristics. The ordering at all schedul-ing points can be predetermined based onunimpeded transit times to a specific schedul-ing point, a runway or a entry fix. Thisstrategy is equivalent to the FCFS schedul-ing ideas and the computational cost benefitsare maximum.

A scheduling performance almost equivalent to thatof the original MILP formulation was obtained atonly a fraction of the computation cost of the orig-inal MILP formulation. However, the cost for thebranch and bound search is still expensive thoughnot as prohibitive as in the original MILP formu-lation.

Heuristics of Genetic Algorithms

A GA is a stochastic search method widelyused in numerous optimization areas and has ad-

vantages in handling both discrete and continuousdesign variables [10]. GAs are simple in mathe-matical formulation but are typically expensive incomputation due to their stochastic search proce-dure.

The idea of adopting GAs in the schedulingproblem for determining the binary decision vari-ables in lieu of the branch and bound searches hasbeen suggested and used in surface managementwork [7] and metroplex operations [8].

Once the binary variables of the route assign-ment and the sequencing are determined throughGAs, computation of the STAs is carried out by anLP solver. Since the LP is very efficient in compu-tation and takes less than a couple of seconds toschedule hundreds of flights, the idea of combin-ing GAs with pure LP procedure is favored for thedense terminal airspace simulations. Advantagesof the GA-based MILP planner include:

• It allows for the solution to be seeded with agood initial guess, based on heuristics.

• All individual candidate solutions are, by def-inition, feasible solutions - thus a usable so-lution is available at all times.

• The solution tends to improve with compu-tational time.

• It naturally handles windowing heuristics.

Individual Candidate Solution Representation

Each individual candidate solution, or “an in-dividual” in short, consists of two vectors: an as-signment vector and a sequence vector. The lengthof each vector is equivalent to the number of flightsin the demand set. Each element of the assignmentvector represents a possible route assignment for aflight in the demand set. Given the structure ofthe route, a sequence of the scheduling points inthe route is prescribed, and the sequence vector isdefined at each scheduling point as a possible se-quence of the flights passing through that point.Route assignments and the queuing are randombased on the stochastic nature of the GAs. A con-straint of no passing in the sequence vector alongthe common route segment is enforced to furtherreduce the computational cost.

Mutation

To maintain the diversity of the individual fromone generation to the next, mutations are carriedout at a specified probability in each generationto both vectors of route assignment and sequence.Due to the coupling between the assignment mapand sequence map held within each individual, themutation operators are applied sequentially. Theassignment mutation simply consists of randomlyreplacing the route assigned for an individual withanother value from the feasible set of routes forthe flight. Then, the sequence map for each in-dividual is updated based on the mutated routeassignment map. A sequence mutation is appliedto each scheduling point at a given route assign-ment. Swapping sequence is determined via proba-bility. If a random number sample is less than thespecified probability of mutating sequence, thenthe number of swaps are chosen from a (0,Nmax),where Nmax is the maximum number of swaps. Arandom pair of flights that contain this schedulingpoint in their assigned routes swap their orderingrelative to the current ordering.

Fitness and Selection

Fitness of each individual is defined as an ob-jective function value and evaluated by solving thepure LP problem implied by its route assignmentand sequence. The definition of the objectivesand the imposition of the constraints are equiv-alent to those of the MILP formulation: earliesttransit time limit, lower and upper bounds of thetransit time via the specified speed controllability,and separation requirements specific to the aircrafttype at the scheduling points.

Once each individual in the population has beenassigned a fitness, the selection of individuals toform the basis of the next generation is performedusing a simple tournament selection scheme. Atournament scheme finds the best-performing P/2number of individuals, where P is the size of thepopulation, and those are selected as the basis ofthe next generation. These P/2 number of indi-viduals are then mutated to form the populationof size P to be evaluated. This cycle of fitness-selection-mutation is repeated until a specified num-ber of generations are completed.

Comparison of Computation Time andOptimality

The optimality and the computation times arecompared for the schedulers that were describedin previous sections. The terminal route structuretested for comparisons is a binary route topologywith double merges: 4 route options and 8 schedul-ing points (4 entry fixes, 2 merge points and 1runway). The route topology is shown in Figure 1with the entry fixes of WP31 through WP34 andthe runway of WP0. The number of flights variesfrom 6 to 100. Computation times with respectto varying number of flights and the average de-lays of 8 flights are compared in Table 1. The ex-pensive computational costs of the original MILPscheduler and the FCFS heuristics-based MILPscheduler prohibited computations for more than 8flights and 40 flights, respectively and their valuesare represented as N/A in Table 1. Although theFCFS heuristics allows scheduling up to 40 flights,the computation time is not still satisfactory forthe dense terminal airspace simulation. However,the speed-up of the computation time for the GA-based MILP planner is considerable even with 100flights. It is observed from the additional schedul-ing of a larger number of flights using GA-basedMILP planner that it is able to handle hundreds offlights in a few minutes. Therefore, it is concludedthat the GA-based MILP scheduler is faster thanthe others and is more appropriate for the schedul-ing problem in the dense terminal airspace opera-tion.

Figure 1. Example route topology

Table 1. Computation times (in seconds) to sched-ule different number of flights

Number Average

of 6 7 8 40 100 delay

fights (8 flights)

MILP 6.7 98.3 1258.4 N/A N/A 8.23

MILP +

FCFS 2.1 33.2 419.3 2040 N/A 8.38

heuristics

MILP +

GA ≤ 1.0 1.0 2.2 23.4 50.0 8.58

heuristics

Dynamic Planner Framework

The schedulers described in the previous sec-tion are deterministic. The transit time of anyroute segment was a function of aircraft type andthe speed profile only, and the uncertainties alongthe routes were not considered in the planning.Even if the uncertainties are taken into accountin the planner, the aforementioned schedulers arebased on the premise that the uncertainties of eachaircraft are known prior to the planning along anyroute segments. However, the STAs in the real-time simulation should be updated in a dynamicmanner corresponding to the varying situations ofweather, wind and off-nominal scenarios. A key tothe realistic scheduling is a dynamic update of theSTAs in a real-time trajectory model in consider-ation of the uncertainties in flight simulation.

A dynamic planner framework is developed inour study by interactively integrating the trajec-tory simulation and the schedule planning for STAupdate. The framework consists of two compo-nents:

• Simulator: This module is responsible for ad-vancing time. It manages the creation oftargets at specified location and time, andconstructs the demand snapshot at a giveninstance of time. The module delegates toa trajectory model that handles the actualmovement of flights along their most recentplan and blends motion between successiveplans. Uncertainties in the transit time alongthe route segment and their propagations areimplemented in the simulation module, whichwill be explained in later section.

• Planner: This module is responsible for con-

structing conflict-free plans for each aircraftin a given demand set. Each motion planconsists of a sequence of waypoints with anassociated STA. Although any type of thescheduler can be used in the dynamic plan-ner framework, the GA-based MILP sched-uler is integrated into the current framework.Furthermore, to take into account the uncer-tainties in the trajectory simulation, an extrabuffer additional to the desired separation isadded to mitigate effect of uncertainties sothat the resulting schedule remains conflict-free. The details of the additional buffer arediscussed in later section.

A specified amount of controllability is allowed inspeed profile to maximize the scheduling perfor-mance and is implemented in both the planner andthe trajectory model simulator.

−40 0 40

−40

0

40

Runway1

Waypoint1

Fix1

(a) Route topology

400 500 600 700 800 900 1000780

800

820

840

860

880

900

920

940

960

980

1000

Simulation time (secs)

Cros

sing

time

at th

e ne

xt c

ontro

l poi

nt

Waypoint1

Runway1

STA ATA

(b) Trajectories of STAs and Actual Time of Ar-rival (ATA): STAs in solid lines and ATA as thelast point of the line.

Figure 2. Example of dynamic planner framework

Trajectory Model Simulation

Flight simulation of the trajectory model ismade via subsequent communications with the plan-ner. First, the dynamic planner starts from thepre-planning of the initial demand set. Given thespeed profile and the ETA of each aircraft to thefirst schedule point, the initial STAs are computedby the scheduler at all scheduling points on theassigned route. The STAs are predicted such thatthey satisfy the constraints of the transit time boundson the route segments and the desired separationat the scheduling points. The trajectory modelperiodically computes the distance from the cur-rent position to the next scheduling point. For ourwork, the update period is 60 seconds. With dis-tance to the next scheduling point and the STApredicted by the planner, the target speed is cal-culated and checked whether it is bounded by thespeed range specified in the original speed profile.Once the target speed is determined, then the tra-jectory simulator advances the flights by an updateperiod. After the simulation, the aircraft positionis recalculated and the earliest time to the schedul-ing point is updated. A subsequent planning cy-cle updates the STAs based on the most recentsimulation results . This cycle of simulation andplanning is iterated and advanced in time by theupdate period until all the flights arrive at the run-way. An example of the cycles of simulation andplanning is shown in Figure 2. Given a simpleroute structure shown in Figure 2(a), the STAs atthe scheduling points of ”Waypoint1” and ”Run-way1” are updated at each update period of 60seconds. Their convergence history is plotted inFigure 2(b), and STA updates are shown by thetriangles. Unlike the static planner, the STA val-ues are updated at each update period and finallycoincide with the Actual Time of Arrival (ATA)values at the scheduling points. The speeds alongthe routes change correspondingly in each updateperiod.

Controllability

To delay or expedite an aircraft on its way tothe next scheduling point, the controllability onany route segment is modeled such that it allows±10 % speed variation. The corresponding transittime bounds are computed. This controllability

is derived from the statistics of the aircraft flyingwith modern avionics and the onboard precisionsystem [11].

Uncertainty Modeling and Propagation

Accurate prediction of uncertainties along anentire aircraft’s trajectory is not trivial. It is acomplicated function of space and time, which re-quires precise understanding of where and how muchof the uncertainties are present and how they af-fect individual aircraft operations. However, theuncertainties in the runway arrival times are quan-tifiable from the statistics of the runway arrivaltimes observed in a given duration of time. We canmodel the aircraft arrival time prediction errors atthe runway by a normal distribution with the time-invariant mean and standard deviation values. Onthe other hand, the uncertainty at the intermedi-ate control points and route-merge points requirea mathematical model of the uncertainty propa-gation mechanism along the routes. A simple lin-earized form is introduced in our simulation mod-ule: variance at each scheduling point is assumedto be proportional to the variance of the runwayarrival time prediction errors when there are nocontrol effort in between the scheduling point andrunway. Uncertainty amount at any point on theroute is scaled by the ratio of the intermediateroute segment length to the entire route lengthfrom the entry fix to the runway. This presumesthat the uncertainties grow longer along the longerroutes since the flight is associated with longertransit time without control efforts.

Based on the central limit theorem, we assumethat the position error of aircraft at the schedul-ing point is approximately normally distributed.Then, the corresponding inter-arrival error of anypair is also normally distributed, and the mean andvariance of inter-arrival errors are estimated by thefollowing basic relationships: if X and Y are inde-pendent random variables that are normally dis-tributed, then X + Y is also normally distributed,i.e., if X ∼ N(µ, σ2) and Y ∼ N(ν, τ2) and X andY are independent, then aX+b ∼ N(aµ+b, a2σ2)and X + Y ∼ N(µ + ν, σ2 + τ2). The means andthe variances of the X and Y are the µ and ν, andσ and τ , respectively.

The validity of the above relations holds bestwhen the independence of two variables, X and Y ,

is relatively well guaranteed. The inter-arrival er-ror of a pair of aircraft is a complicated function ofmany factors such as precision errors in navigationand weather including wind. We assume for sim-plicity in our trajectory simulation that the windeffect during the traffic simulation is rather con-stant. Direction and magnitude of the wind arerelatively non-changing on each aircraft through-out the whole simulation, then we can treat thewind effect as a constant that is freely addable/ subtractable to/from the standard deviation ofthe position error of each aircraft at all schedul-ing points, and the above relation holds relativelywell.

Additional Separation Buffer due to Un-certainties

Aircraft arrival time errors and the correspond-ing inter-arrival error in a pair are likely to causeviolations of the desired separation. In order toensure desired separation is maintained in spite ofarrival time error, extra buffers are added to theoriginal desired separation [12]. The amount of theadditional buffers is determined from the probabil-ity of the inter-arrival errors and its normal distri-bution shown in Figure 3. If we choose a value,

Figure 3. Probability distribution of position errors

Z, for an additional buffer such that the cumula-tive probability corresponding to Z coincides witha specified confidence level, 90% for the currentwork, then we can say that the separation require-ment in any pair will be satisfied under uncertaintywith 90% confidence and be violated with 10% tol-

€

f1,σ1

€

f2,σ 2

Figure 4. Uncertainties of position errors in a pairat the merge point (σ1=σleader and σ2=σfollower.)

erance. A brief graphical explanation is shown inFigure 3. This sets the additional buffer value as1.645σ, where σ is the standard deviation of theinter-arrival error distribution.

The above can be expressed mathematically asfollows. Standard deviations of the arrival timeerrors of a leader and a follower in a given pairare denoted as σleader and σfollower, respectively asin Figure 4. The amounts are scaled from runwaystandard deviation in proportion to the ratio ofthe local route segment to the entire route fromrunway. Assuming the probability of the positionerrors of a leader and a follower are independentof each other, the standard deviation of the inter-arrival error is assumed to be

√σleader + σfollower.

A corresponding additional buffer is set as 1.645√σ2leader + σ2follower for a 90 % confidence interval

based on the normal distribution of the inter-arrivalerrors. A simple mathematical formulation of totalamount of buffers is expressed in following Equa-tion.

tsep tot = tsep desired + tsep σ

=dsep desired

V+ 1.645

√σ2leader + σ2follower ,

where V is airspeed and tsep tot represents totalamount of separation requirement. Terms oftsep desired, dsep desired and tsep σ represent a desiredseparation in time, a desired separation in dis-tance, and an uncertainty-related, additional buffer,respectively.

Application: Optimal Route Struc-ture Under Uncertainty

A practical application of the developed sched-ulers and the dynamic planner framework is shownin this section. A design of an optimal route struc-ture under uncertainty in the extended terminalairspace area is carried out to improve schedulingperformance and, thus, to best utilize the limitedairspace resources.

First, key parameters for a route structure de-sign are identified. A total of five example routestructures are constructed with varying design pa-rameters. A static FCFS heuristics-based MILPplanner is used to analyze the scheduling perfor-mance of each notional route structure, but the re-sults are validated by using the dynamic plannerframework. Based on the analyses of the schedul-ing performance of the notional route structures,more general cases of various merge topologies areconsidered subsequently.

Parameterization

A route structure consists of such parametersas the location and number of entry fixes, run-ways and merge points as well as route segmentlengths. A demand set is also critical in schedul-ing performance. The demand set defines rele-vant flight information including the total num-ber of flights, arrival time at entry to the routestructure, and traffic duration time. In fact, thescheduling performance is very sensitive to the de-mand set. A fully saturated demand set, i.e., onewhich has no periods of low demands, is used toisolate the scheduling performance from the effectsof the route structure alone. A fully saturated traf-fic flow is consistent with dense terminal airspaceoperation whereby demand exceeds capacity forextended periods of time. In this way, the run-way capacity is always exhausted and the numberof runways becomes no longer a parameter of theairspace topology. An entry fix topology, i.e., thetotal number of entry fixes and their locations, isalso assumed to be given to facilitate the fully sat-urated traffic flow. Thus, the main parameter inour design study is the merge point topology, i.e.,the location and number of the merge points.

Numerical Test I

First, a numerical experiment is performed onfive route structures having different topologies withvarying numbers and locations of the merge points.Figures 5 and 6 have a single merge point whereasFigures 7 through 9 have two merge points. Thelocations of the merge points are moved in order tovary the ratios of route segment lengths in a giventopology and thus vary the uncertainty distribu-tion along the route. These topologies can alsobe defined by the parameters of the route segmentlength and the merge angle between two routes.For example, Figure 7 through 9 can be defined byvarying the parameters of a, b, c, α, and β as shownin Figure 1, where a, b and c represent the routesegment length, and α and β represent angles be-tween two merging routes.

A FCFS heuristics-based MILP is used for com-puting the scheduling performance of each topol-ogy. In the cost comparison of the schedulers ex-plained in previous sections, a demand set of 80flights is not trivial in scheduling even for the heuristics-based MILP scheduler. Total computation timegrows very quickly, especially for a stochastic casewhere hundreds or thousands of Monte Carlo sim-ulations are performed. Thus, for this numericalexperiment, the route is pre-assigned for each air-craft and the orders at the merge points are prede-termined based on the the unimpeded transit timesfrom the entry fix to the runway. For a stochasticscenario, an uncertainty model is directly imple-mented in the scheduler, and we do not employa dynamic planner for this preliminary numericalexperiment. However, a more realistic validationof these five airspace topologies is carried out bythe dynamic planner and analysis results will beshown in the following section.

Initially, a total of 80 flights pass through theentry fixes equally divided into four streams andfollow their pre-assigned routes during a short pe-riod of 100 seconds. This short duration timeensures fully saturated air traffic. Four types ofweight class categories are used: “heavy”, “B757”,“large”, and “small”. A majority of aircraft in ourdemand set, more than 90%, belong to the “large”type, indicative of today’s operations.

The total amount of separation required at eachscheduling point is the sum of the minimum de-sired separation and the extra additional buffer to

Table 2. Desired separation (nmi)

leaderPPPPPPPPP

followerheavy B757 large small

heavy 4 5 5 5

B757 3 3 3 4

large 3 3 3 4

small 3 3 3 3

mitigate uncertainty. The quantification of thisamount is done using the formulation of Equa-tion 4. The amount of the desired separation isbased on the weight classes of the leader and thefollower in a given pair, and the values are specifiedin Table 2. Airspeed is assumed to be linearly de-creasing along the routes in the extended terminalairspace area and its value at each scheduling pointis summarized at the tables in Figure 5 through 9.Airspeeds at the entry fixes are fixed at 250 kts.Additionally, the desired separation of the entryfix was 5 nmi to model the separation required foren route airspace. Previously collected data in-dicates that a standard deviation of uncertainty(in the aircraft arrival time errors) at the entryfix is found to be approximately 30 seconds. Theadditional uncertanty-related separation buffer re-quired at the entry fix in order to mitigate thisuncertainty is calculated in a similar way for themerge points and runway.

Given the prescribed uncertainty and its corre-sponding separation buffer, the FCFS heuristics-based MILP scheduler simulates both the deter-ministic and stochastic scenarios for all airspacetopologies shown in Figure 5 through 9. A to-tal of 500 Monte Carlo simulations were carriedout for the stochastic simulations. The separa-tion buffer and total separation are computed foreach case and summarized in the tables in Figure 5through 9. The resulting average delays are shownin Table 3. The average delay is calculated as adifference between unimpeded transit time and ac-tual transit time. The value of average delay is anartifact of the fully saturated traffic scenario. Thekey result is the performance improvement.

First, for the deterministic scenarios where noconsideration of uncertainty is made, the third col-umn in Table 3 implies that the scheduling per-formance is relatively independent of a particu-lar airspace topology and its merge point loca-tions. Although a slight improvement is shown in

waypoint WP21 WP1 WP0

air speed (kts) 160 130

tsep desired (sec) 67.5 83.1

tsep σ 29.13 6.063

tsep tot 96.63 89.14

average spacing 99.33 99.43

Figure 5. Case 1: single late merge. −30 −20 −10 0 10 20 30

0

10

20

30

40

X−coordinate (nmi)

Y−co

ordi

nate

(nm

i) WP31

WP32 WP33

WP34

WP1

WP0


air speed 200 130

tsep desired 54 83.1

tsep σ 21.3 17.0

tsep tot 75.3 99.2


Figure 6. Case 2: single early merge.−30 −20 −10 0 10 20 30

0

10

20

30

40


Y−co

ordi

nate

(nm

i) WP31

WP32 WP33

WP34

WP1

WP0


air speed 200 160 130

tsep desired 54 67.5 83.077

tsep σ 14.8 14.76 5.653

tsep tot 68.48 82.26 88.73


Figure 7. Case 3: double merges, equally distanced. −30 −20 −10 0 10 20 30

0

10

20

30

40


Y−co

ordi

nate

(nm

i) WP31

WP32 WP33

WP34

WP22WP21

WP0

WP1


air speed 225 160 130


tsep σ 8.968 20.64 5.32

tsep tot 56.97 88.142 88.397


Figure 8. Case 4: double early merges. −30 −20 −10 0 10 20 30

0

10

20

30

40


Y−co

ordi

nate

(nm

i) WP31

WP32 WP33

WP34WP21 WP22

WP1

WP0


air speed 180 160 130


tsep σ 16.157 7.144 5.76

tsep tot 76.157 74.64 88.837


Figure 9. Case 5: double late merges. −30 −20 −10 0 10 20 30

0

10

20

30

40


Y−co

ordi

nate

(nm

i) WP31

WP32 WP33

WP34

WP21 WP22WP1

WP0

Table 3. Predicted average delays (seconds).

Average Performance Average Performance

Topology Cases delay improvement delay improvement

(Deterministic) (w.r.t Case 1) (Stochastic) (w.r.t Case 1)

single Case 1 3363.4 1.0 % 3834.8 1.0 %

merge Case 2 3370.8 -0.2 % 3918.2 -2.1 %

double Case 3 3348.1 0.45 % 3571.6 6.8 %

merge Case 4 3343.6 0.6 % 3564.8 7.0 %

Case 5 3352.0 0.3 % 3579.7 6.7 %

the double merge cases, differences in the averagedelays among all cases are very minor, less than1%. This can be explained from the formulation inEquation 4, where the amount of total separationis solely a function of airspeed alone when thereis no uncertainty. As airspeed gradually decreasestowards the runway, the runway always requiresthe biggest separation of all the scheduling points.This makes the scheduling performance largely in-sensitive to the particular upstream route struc-ture. The controllability of each aircraft does notaffect the scheduling performance either, as mostof the aircraft have to absorb delays propagatedfrom the preceding aircraft.

Second, for the stochastic scenarios, an im-provement of 6.7 % is shown in Case 5, when com-paring the single merge and the double merge topolo-gies. Uncertainty creates perturbations in the tran-sit times, and some of the aircraft can exploit theircontrollability to fill the gaps in a pair created bythe uncertainty.

The results from both deterministic and stochas-tic cases in Figure 5 and 6 are interesting and in-formative. The actual, average spacings betweenall pairs are extracted from the scheduling resultsfor both cases and shown in blue in tables in Fig-ure 5 and 6. A careful comparison of five casesof the resultant average spacings and the requiredseparation indicates that the average spacing ateach scheduling point is dominated by the largestseparation required along the route. This is shownin red in tables in Figure 5 and 6. This observa-tion suggests that in a fully saturated traffic flow,the traffic becomes steady with acceleration anddeceleration adjusted by the controllability that isallowed in each route segment and incurs approxi-mately the same amount of spacings in any pairs.

Validation Using Dynamic Planner

The scheduling results in Section of Numeri-cal Test I are validated using the dynamic plan-ner framework. A GA-based MILP is employedfor the planner of the framework for a comparisonwith the FCFS heuristics-based MILP. Unlike thestatic planner used in Section of Numerical Test I,a dynamic planner involves iterative interactionsbetween the planner and the simulator as time ad-vances. An update period of 60 seconds was usedin our validation. In each dynamic planning cycle,the simulator tries to track the STAs provided bythe planner at all scheduling points, and the plan-ner creates new schedules as a result of updatedaircraft positions and ETAs from the simulator. Arealistic demand set is critical in the dynamic plan-ner so that the trajectory model can be simulatedbased on the operationally reasonable and realisticschedule plans. For this reason, rather than usingfully saturated traffic as in the Numerical Test I,initially well-separated traffic flows, by 5 nmi inany pairs with random deviation ranging from -20% to +20%, are used for the dynamic planner.This results in hour-long traffic flows for the samenumber of flights. Also the initial departing pointfor all aircraft is fixed at about 100 nmi away fromthe runway and is almost aligned with the freezehorizon.

The validation results are summarized in Ta-ble 4. An average delay value is, again, defined as adifference between the unimpeded transit time andthe actual transit time from the entry fix to therunway. Compared to the values in Table 3 whichused a fully saturated traffic flow, average delayvalues become more reasonable with the maximumvalue less than five minutes.

It should be noted that the dynamic planner ismore computation-intensive than the static schedul-ing planner, as the specified plan update periodcannot be set too large in a real-time simulation. Awall-clock CPU time per dynamic planning takesabout an hour with 60 seconds update period. Thedynamic planning requires hundreds of cycles ofsimulation and planning for aircraft to travel theentire route of 100 nmi in length. Thus, the com-putation time of the dynamic planner for the stochas-tic case becomes very expensive with Monte Carlosimulation. The results for the stochastic casesshown in Table 4 are obtained from only 100 MonteCarlo simulations. More simulations are plannedas part of the future work.

It can be concluded from the average delay re-sults shown in Table 4 that Case 5 is the best-performing airspace topology and it has a perfor-mance improvement, compared to Case 1, of ap-proximately 50% in the deterministic case and ap-proximately 30% in the stochastic scenario. Al-though the sample size of the Monte Carlo simula-tions in the stochastic scenarios is not big enoughto make the conclusion more trustworthy, the stan-dard deviation of the 100 Monte Carlo samplesare as little as 15 seconds resulting in 5% toler-ance that is far less than our percentile perfor-mance improvement. It should also be noted thatfor a less saturated traffic flow, an improvementin the scheduling performance is more dramaticcompared to the 6.7% in Table 3. This can be ex-plained by the fact that in fully saturated trafficflow, all the aircraft quickly exhaust their control-lability and are assigned their slowest speed profile.That is not the case with less saturated air traffic,and the benefits from the controllability are moredramatic in this case. It should also be noted thatthe same trend of the static planning in Numer-ical Test I is shown in the validation results: anairspace topology with small separation require-ments at the scheduling points is more favorablein the scheduling performance.

Numerical Test II

Based on the results in the previous sectionsthat the scheduling performance is heavily depen-dent on the amount of maximum separation at thescheduling points, a simple numerical experimentis carried out to investigate more general variations

of the airspace topology. Relationships among thecomponent separation amounts at the schedulingpoints are analyzed when the merge points movearound in the extended terminal airspace area. Theairspace topology is simplified to a circle with 40nmi radius as shown in Figure 10. The points,WP1 and WP2, represent the merge locations thatcan move circumferentially at a relatively constantradial distance away from the runway, and theroutes can be merged at these locations. Radialdistances, x and x + y, are also allowed to varywithin radius bounds: 0 < x < R and 0 < y <R − x, where R is the radial distance from therunway to the entry fix. The movement of themerge points, WP1 and WP2, are shown in Fig-ures 10(a) and 10(c), and the corresponding exam-ple airspace topologies are plotted in Figures 10(b)and 10(d), respectively. As WP1 and WP2 moveabout in the extended terminal airspace area, theentire route length in a particular airspace topol-ogy from the entry fix to the runway and the corre-sponding transit time may change. However, anylarge variations were excluded in the transit timesby locating the entry fix such that the the entireroute from the runway to the entry fix does notdeviate much from the original radial lines.

For the airspace topology with a single mergeas in Figures 10(a) and 10(b), total separationamounts at the runway and the WP1, are com-puted from Equation 4 and plotted as solid lineswith symbols in Figure 11(a). The two dashedlines represent the desired separation amount us-ing the nominal speed profile, and the uncertainty-related, extra separation buffers are plotted as dot-ted lines. The traces of the maximum separationwith respect to the varying merge locations areshown in Figure 11(b). Figure 11(b) implies thatthere exists an optimal merge location somewherearound 5 or 6 nmi away from the runway.

The case of double merges inside the extendedairspace area as in Figures 10(c) and 10(d) showssimilar trend as the single merge case. At the givendownstream merge point location, x, which canmove from the runway to the entry fix, the up-stream merge location, y, also traverses betweendownstream merge point and the entry fix. Theplot of the maximum separation buffer of all schedul-ing points, i.e., runway, downstream and upstreammerge points, are almost identical to the plot ofFigure 11(b) and is omitted in the paper; how-

Table 4. Average delays (seconds) validated by dynamic planner framework for stochastic case.

Topology Cases Update Deterministic Performance Update Stochastic Performance

cycles scenario improvement cycles scenario improvement

(w.r.t. Case 1) (w.r.t. Case 1)

single Case 1 304 249.1 1.0 % 350 291.0 1.0 %

merge Case 2 257 235.0 5.6 % 329 283.0 2.7 %

double Case 3 228 107.26 56.9 % 294 228.2 21.6 %

merge Case 4 232 108.31 56.5 % 295 213.5 26.6 %

Case 5 247 109.17 56.0% 294 199.3 31.0 %

runway

fix

x

WP1

(a) Merge topology with sin-gle mere point.

(b) Example airspace topol-ogy with single merge point.

runway

fix

x

yWP1

WP2

(c) Merge topology withdouble merge points.

(d) Example airspacetopology with double mergepoints.

Figure 10. Simplified airspace topology of the ex-tended terminal area with varying merge point lo-cations

ever, this indicates that the separation at the run-way typically requires the biggest amount and inother cases the downstream merge point requiresa larger buffer than the upstream merge point.

The exact optimal locations of merge pointsare less meaningful in our analysis, as they canvary depending on the underlying airspeed profile,uncertainty quantification and propagation model,and the assumed additional separation buffer dueto uncertainty. Possible changes to the optimalmerge locations can be inferred from Figure 11(a).

Desired separation represented as the dashed linesin blue and red are rather non-changing as theseare solely determined by the airspeed topology alone,and the assumption of monotonically decreasingairspeed in the extended terminal airspace area isreasonable. On the other hand, the modeling ofuncertainty and its propagation in the extendedterminal airspace area is still an area of active re-search that requires extensive studies on how theuncertainties are distributed or propagated alongthe routes. How temporal or spatial deviationfrom its predicted trajectory behavior is quanti-fied over time and distance and translated into thetime-based scheduler is a difficult and yet very im-portant topic in the scheduling and real-time sim-ulations.

Various Uncertainty Propagation Mod-els

A brief sensitivity study of the optimal mergelocations is carried out with respect to the differentmodels of uncertainty distribution and propaga-tion. The results in previous sections assumed thatthe increment of the uncertainty per unit routesegment is constant and the resultant uncertainty

0 10 20 30 400

20

40

60

80

100

120

x

buf to

t= b

uf m

in+

buf s

ig, (s

ec) total runway

separaiton

location of merging point

S1 S3 S2

wp1 wp2 fix 0 (rwy)

δσ

-  - - - desired separation at runway . . . . uncertainty-related buffer at runway

total merge point separation

t sep_

tot =

t sep

_des

ired+

t sep_σ

(sec

onds

)

Figure 12. Constant distri-bution of uncertainty

0 10 20 30 400

20

40

60

80

100

120

x

bu

f tot=

bu

f min

+b

uf s

ig,

(se

c)

location of merging point wp1 wp2 fix 0 (rwy)

S1 S3 S2

t sep_

tot =

t sep

_des

ired

+ t se

p_σ

(sec

onds

)

Figure 13. Linear distribu-tion of uncertainty

260 10 20 30 40

0

20

40

60

80

100

120

x

buf to

t= b

uf m

in+

buf s

ig, (s

ec)

location of merging point wp1 wp2 fix 0 (rwy)

S3 S2 S1

location of optimal merge point

t sep_

tot =

t sep

_des

ired

+ t se

p_σ

(sec

onds

)

Figure 14. Nonlinear (quar-tic) distribution of uncer-tainty

0 10 20 30 400

20

40

60

80

100

120

x (nmi)

t sep_

tot=

t sep_

desir

ed+t

sep_!, (

seco

nds)

tsep_tot at WP1

tsep_desired at WP1

tsep_! at WP1

tsep_tot at runway

tsep_desired at runway

tsep_! at runway

(a) Buffer amount w.r.t. merge location (x rep-resents the distance from the runway: 0 at therunway and 40 at the entry fix.)

0 10 20 30 4020

40

60

80

100

120

x (nmi)

t sep_

tot=

t sep_

desir

ed+t

sep_!, (

seco

nds)

tsep_tot at merging point

tsep_tot at runway

(b) Max buffer amount w.r.t. merge location

Figure 11. A total amount of separation with re-spect to varying merge locations.

is linearly proportional to the route segment length.The plot of constant uncertainty increment andthe corresponding optimal merge location is plot-ted in Figure 12. The maximum of the two solidlines with the symbols represents the traces of thelargest separation amount of all the scheduling points.The x-axis represents the location of the down-stream merge point. Once the downstream mergepoint is located at the predicted optimum point,the separation amount at the upper merge pointappear to be smaller than the one at the down-stream merge point.

On the other hand, if we put more weights inthe uncertainty towards the entry fixes, differentoptimal merge locations are predicted. An as-sumption of a linear increment as in Figure 13moves the optimal merge location slightly towardsthe entry fix, about 10 nmi away from the runway.If we put more weights in the entry fix boundaryarea as in Figure 14, at a quartic increase rate forexample, the optimal merge location falls in the re-gions farther from the runway, about 23 nmi away.

From these simple models of the uncertaintyquantification and propagation, it is concluded thatthe optimal location of the merge point is where alarge amount of uncertainties are present, and themerge point plays a role in reducing the uncertain-ties in that region by enforcing the pilot’s controlefforts to meet the suggested STA at that point.

Conclusions

MILP-based optimization algorithms were usedin our scheduling and route assignment problem,and a number of heuristics were introduced intothe original formulation to keep its computationalcost realizable in the dense terminal airspace oper-ations. FCFS-heuristics and GA-based heuristicswere adopted to reduce the computational burden,and the resultant computational costs and schedul-ing results were compared with the original MILPsolutions. The GA-based MILP planner appearsto be very efficient without loss of optimality. Totake into account uncertainty propagation in theroute structure, a dynamic planner framework isdeveloped. The dynamic planner consists of themodules of the planner and the trajectory simula-tor. The STAs are updated in a dynamic mannervia the interaction of the planner and the simulatorthroughout the whole simulation. An uncertaintymodel is implemented in the trajectory model ofthe simulator and the extra separation buffers areadded at the scheduling points in the planner. Asa practical application of the proposed schedulers,an investigation of the optimal route structure un-der uncertainties in the extended terminal airspacewas carried out. A constant uncertainty incre-ment was assumed along the route, ensuring theuncertainty amount grows linearly in proportionto the route segment length. After analyzing theairspace topologies with varying merge topologies,a route structure having the least maximum sepa-ration at the scheduling points has shown the bestscheduling performance. These results were val-idated with a dynamic planner framework witha more reasonable demand set. Finally, airspacetopologies with various uncertainty distribution mod-els were tested: a constant, linear and quartic unit-increment of the uncertainty along the route seg-ment. It is shown that the optimal merge pointshould be positioned to bound the growth of theuncertainty-related separation buffer such that themaximum total separation at any point along theroute is minimized. This fact indicates that thereexists a likely optimal merge topology. The opti-mal merge topology is still tuned for a range ofuncertainty propagation models while the exacttopology did vary.

References

[1] Swenson, H., Barhydt, R., and Landis, M., “Next Gen-eration Air Transportation System (NGATS) Air TrafficManagement (ATM)-Airspace Project,” NASA Tech. Re-port , Ver. 6.0, Moffett Field, CA, 2006.

[2] Prete, J., Krozel, J., Mitchell, J. S. B., Kim, J., and Zou,J., “Flexible, Performance-based Route Planning for Super-Dense Operations,” AIAA Guidance, Navigation, and Con-trol Conference, Honolulu, HI, Aug. 2008.

[3] Isaacson, D. R., Robinson III, J. E., Swenson, H., andDenery, D. G., “A Concept for Robust, High Density Termi-nal Air Traffic Operations,” 10th AIAA Aviation Technol-ogy, Integration, And Operations (ATIO) Conference, FortWorth, Texas, Sep. 2010.

[4] Clarke, J. P., Ren, L., Schleicher, D., Thomson, T.,Cross, C., Lewis, T. B., “Characterization of and Conceptsfor Metroplex Operations,” NASA/CR-2010-000000 , 2010.

[5] Rathinam, S., Montoya, J., and Jung, Y., “An Optimiza-tion Model For Reducing Aircraft Taxi Times at the Dal-las Fort Worth International Airport,” 26th InternationalCongress of the Aeronautical Sciences, 2008.

[6] Atkins, S., Capozzi, B., and Choi, S., “ Towards Opti-mal Routing and Scheduling of Metroplex Operations ,”9thAIAA Aviation Technology, Integration, and OperationsConference (ATIO), Sep. 2009, Hilton Head, South Car-olina.

[7] Garcia, J. Berlanga, A., Molina, J. M., Casar, J. R., “Op-timization of airport ground operations integrating geneticand dynamic flow management algorithms,” Al Communi-cations, Vol. 18, N. 2, pp. 143–164, 2005.

[8] Atkins, S., and Capozzi, B., “ Hybrid Optimization Ap-proach To Traffic Management ,”10th AIAA Aviation Tech-nology, Integration, and Operations Conference (ATIO),Sep. 2010, Fort Worth, Texas.

[9] Kaufman, D. E., Nonis, J., and Smith, R. L., “A MixedInteger Linear Programming Model for Dynamic RouteGuidance,” Transportation Research: Part B, Methodology ,Vol. 21 pp. 431–440, 1998.

[10] Goldberg, D-E., “Genetic Algorithms in Search, Opti-mization and Machine Learning,” Addison-Wesley , 1989.

[11] Barmore, B., Abbott, T, and Krishnamurthy K, “Airborne-Managed Spacing in Multiple Arrival Streams,”24th International Congress of the Aeronautical Sciences,Sep. 2004, Yokohama, Japan.

[12] Meyn, Larry. A., and Erzberger, H., “ Airport ArrivalCapacity Benefits Due to Improved Scheduling Accuracy”, AIAA Aviation, Technology, Integration and OperationsConference (ATIO), Arlington, Virginia, September 2005,AIAA 2005-7376.

[13] Callantine, T. J., “TRAC Trial Planning and ScenarioGeneration to Support Super-Density Operations Studies,”AIAA Modeling and Simulation Technologies Conference,Chicago, IL, August 2009, AIAA 2009-5836.

design of an optimal route structure using heuristics

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