Integrated Taxiing and Take-Off Scheduling for Optimization of Airport Surface Operations

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Integrated Taxiing and Take-Off Scheduling for

Optimization of Airport Surface Operations

H.-S. Jacob Tsao, Wenbin Wei, Agus Pratama and Suseon Yang

College of Engineering, San Jose State University, San Jose, California, USA

Abstract

Just like surface transportation systems around the world, air transportation systems

experience growing congestion. Airports are the primary bottlenecks of an aviation

system, and runways are the primary bottlenecks of an airport. However, traditional

capacity expansion approaches like adding runways no longer suffice, due to lack of

space, noise and environmental constraints, etc. Therefore, airport operational efficiency

becomes more and more critical. Many operational functions have been defined and

studied individually in the literature, e.g., runway configuration, runway assignment,

take-off sequencing and scheduling, taxiway routing and scheduling, etc. However, a

salient feature of airport operations is that these problems are heavily interdependent.

Another salient feature of airport surface operations is the vast amount of uncertainty.

After a brief discussion of an optimization architecture, we focus on the combined

problem of taxiing and take-off scheduling. We report an integrated formulation as a

mixed-integer mathematical programming problem and our numerical experience.

Key words: Airport Surface Operations, Departure, Optimization, Scheduling, Taxiing

1. Introduction

Just like surface transportation systems around the world, air transportation systems

experience growing congestion. Although the three-dimensional sky seems to have unlimited

capacity for accommodating air traffic, an aircraft must enter or leave the airspace through

one of the few available one-dimensional air routes at an airport. Moreover, air routes of an

airport may be intertwined with those of neighboring airports. These clearly point to the fact

that airports are the primary bottlenecks of an aviation system. However, traditional capacity

expansion approaches like adding runways no longer suffice, due to lack of space, noise and

environmental constraints, etc. As a result, operational efficiency of the current airports, in

particular the runways, becomes more and more critical. Consequently, some researchers

studied the efficiency of airport operations (e.g., Idris and Hansman, 2000).

Much has been published in the literature about mathematical algorithms that could be used to

improve the efficiency of airport surface operations. Many controller tasks and corresponding

decision-support functions have been defined and studied in the literature. A salient feature of

airport operations is that these problems are heavily interdependent. However, due to the

complexity of the problem and perhaps the space limitation, they understandably tend to be

focused on sub-problems, with the much larger issue of integration left not fully specified.

For example, Anagnostakis et al. (2001) proposed an optimization architecture for airport

surface operations, with a focus on departures and runway operations, before formulating and

solving the problem of departure sequencing and timing. Capozzi et al. (2004) proposed an

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initial architecture and core functions for a basic automated airport surface traffic planning

and control system and presented several algorithms and preliminary numerical results.

Another salient feature of airport surface operations is the vast amount of uncertainty. In

addition, the capacity of most large airports in the US is constrained by the runway capacity,

which is “perishable.” Based on these salient features, the authors of this paper (2008)

proposed an optimization architecture for airport surface operations and discussed its

justification. We also defined a set of interdependent decision-support functions for optimal

planning and control of airport surface operations, with the goal of fully utilizing the runway

capacity in the presence of much uncertainty.

After a brief discussion of an optimization strategy proposed by the authors (2008), we focus

on the integrated function of taxiing and take-off scheduling. Although this integrated

function is to be achieved through both operations planning, which deals with a longer time

frame and quantities more aggregated with respect to time, and control, which deals with a

shorter time frame and quantities less aggregated. Due to space limitation, we focus on the

control portion of the integrated function and report an integrated formulation as a mixed-

integer mathematical programming problem and our numerical experience in the solving the

problem. This paper is organized as follows. In section 2, we define the focus of this paper

after a brief discussion of an optimization strategy. The problem is formulated in Section 3.

In Section 4, we discuss our implementation of the formulation and numerical experience.

Concluding remarks are given in Section 5.

2. An Optimization Strategy and the Focus of this Paper

Some researchers characterize the operations of an airport as a set of “controlled queues.”

Among all these controlled queues, the runway queues, created by demand that cannot be met

immediately by the capacity, have been observed to be the biggest source of delay (Idris and

Hansman, 2000). In fact, a salient feature of airport surface operations is that runway

capacity is the most common root cause of all these queues. As pointed out in (Cheng, 2004),

as more and more runways are constructed on an airport, the traffic congestion on the

taxiways may become so serious that it could limit the airport’s ability to fully utilize its

runway capacity. When this is a likely result of adding another runway to an airport and the

taxiway capacity cannot be increased, either through deployment of advanced technology or

conventional expansion projects, the benefit of the addition should be questioned. In any

event, optimal utilization of runways is the primary goal. Another salient feature is that many

of the operational functions are interdependent. This primary goal is to be achieved by

integration of not only operational functions like taxiway and take-off scheduling but also

integration of operations planning and control. Another salient feature of airport surface

operations is the vast amount of uncertainty. Recurrent sources of uncertainty include aircraft

departure delay from the gate, congestion at runway entrances and on taxiways, variability in

performance of humans (including tower controllers and pilots), variability in aircraft

performance, etc. The first two sources are the two primary recurrent sources in current

operations. Other sources of uncertainty include non-recurrent sources, e.g., the impact of

inclement weather on airport capacity, etc.

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One way to deal with the various uncertainty sources is to distinguish control functions from

planning functions, which include operational, tactical and strategic planning functions.

Integrated taxiway and take-off scheduling involves both control and operational planning.

We first describe the approach, in terms of systems requirements.

Control refers to the collective instructions of tower controllers issued to the pilots of those

aircraft that are ready to depart from the gate. Effective control can also reduce or virtually

eliminate another source of uncertainty – congestion at runway entrances or on taxiways.

Therefore, effective control can be used as a strategy to deal with the two major sources of

uncertainty. The control problem is formulated as a deterministic optimization problem, with

some operational remedies in anticipation of variability of human ability. The overall

objective of the control problem is to minimize the total amount of time all aircraft spend on

the airport after pushback readiness, particularly the total amount of time all aircraft spend on

the tarmac of an airport. This objective helps avoid congestion on the airport surface by

keeping some aircraft waiting at their gates or parking aprons. Congestion avoidance not only

reduces delays but also enables smooth travel and avoids stopping or even slow-down, which

incurs unnecessary fuel burn, noise, emissions, etc. More importantly, congestion avoidance

significantly reduces and perhaps virtually eliminates a major source of uncertainty. This

overall objective helps achieve system optimality. But, to balance system optimality with

“individual optimality,” constraints to avoid excessive delay to individual aircraft are imposed

to ensure fairness.

The smooth travel is to be enabled even through all taxiway-to-taxiway intersections, but not

through runway-to-taxiway intersections, i.e., runway crossings, or at runway entrances. A

conflict at a taxiway crossing between two aircraft can be easily avoided or its impact easily

recovered by controlling the two speed profiles. However, a conflict at a runway crossing

often necessitates aircraft waiting because a runway when occupied for take-off or landing

may not be crossed for a significant amount of time. This necessitates a small queue for

runway crossings. Another kind of queues is necessary. The primary traffic bottleneck is the

runways. To avoid wasting the perishable runway time slots for departures, a small queue at a

runway entrance can be provided in order to deal with possible inability of the controllers or

pilots to adhere to the schedule developed by the deterministic optimization process. The

authors (2008) proposed the following optimization strategy:

• Deterministic Control of Movement Only for Aircraft Ready for Pushback

o Objective of minimum time on airport: Control movement of departing aircraft

that are ready for pushback and arriving aircraft so as to minimize in a

deterministic mathematical program the total amount of time departing aircraft

spend on the airport between pushback readiness and take-off plus the total

amount of time arriving aircraft spend between landing and parking.

o Smooth-travel constraint: Control aircraft movement on tarmac in the same

mathematical program so as to ensure smooth travel and no or little tarmac

congestion by imposing constraints on minimum speed or maximum travel time

between the gate and the runway and keeping departing aircraft at their gates or on

parking aprons as long as possible (subject to gate availability), and hence to also

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virtually eliminate one of the major sources of uncertainty - recurrent tarmac

congestion.

o Fairness constraint: Imposing fairness constraints in the same mathematical

program, which may take the form of requiring the delay to any aircraft to be no

greater than a specified percentage over the average delay to all aircraft.

• Operations Planning in Anticipation of Uncertainty or Deviation from the Deterministic

Control

o Queueing to maximize runway and related throughputs: Maintain a small queue in

the same mathematical program at a runway entrance and allow a small queue at a

runway-taxiway crossing in anticipation of deviation of actual traffic from the

deterministic control, due to human factors, and in order to avoid wasting take-off

slots, which are perishable. However, determine the queue sizes according to

aggregated forecast demand in a planning function.

o Reordering actual sequence of arrival at runway: Reorder the sequence in which

aircraft actually arrived at a runway so that the take-off sequence maximizes the

runway capacity, with wake vortex and departure fixes taken into consideration, if

the airport is equipped with a holding point enabling such reordering and

deterministically optimized sequence did not materialize due to uncertainty.

This paper focuses on the control function. It further focuses on airports that are equipped

with independent runways, so that the only air separation to consider is the separation

between two aircraft taking off at the same runway and not air separation of aircraft taking off

at different runways, and equipped with a holding space for each runway, so as to

accommodate a small queue. In addition, it focuses on airports that are equipped with

perimeter taxiways and do not use runway crossings, for safety purposes. The mathematical

formulation reported in this paper is a major extension to the formulation proposed by

Smeltink et al (2004) for the problem of taxiway scheduling. The extension includes:

integration with other functions of an optimization architecture, integration with take-off

scheduling, small queue at a runway holding space, encouraging aircraft to wait in the small

queue and at the gate, fairness to individual aircraft with respect to wait time at the gate,

accommodation of the Ground Delay Program at the national or inter-airport level, etc. We

have further extended the formulation reported in this paper to deal with more complex

situations, e.g., reordering aircraft sequence of arrival at runway for a different and more

efficient take-off sequence for airports equipped with the necessary infrastructure, but will be

reported separately.

3. Formulation of a Mixed Integer Programming Problem

3.1 Given Input and Parameters

3.1.1 Aircraft and Demand

• A : the set of all aircraft i=1,2,…,a.

• oA : the set of all aircraft plus one dummy aircraft 0.

• DepA : the set of all departing aircraft.

• ArrA : the set of all arriving aircraft.

• Dep

GDPA : the set of all aircraft that are subjects of the Ground Delay Program (GDP).

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• [ ii GDPLGDPE , ],Dep

GDPAi ∈∀ : the time interval within which aircraft i must take off from

this airport, as required by the Ground Delay Program (GDP), where the E and L signify

earliest and latest time of take-off.

• iARTAXI , ArrAi ∈∀ (arrival readiness for taxiing): the time at which an arriving flight i is

to have landed and reached the end of the runway and to be taxied off immediately.

• iDRPUSH , DepAi ∈∀ (departure readiness for pushback): the time at which departing

aircraft i is ready for pushback.

3.1.2 Airport Configuration

• R: the set of all departing runways at the airport, with the total number of departing

runways denoted as Γ .

• N: the set of all real nodes u, where a real node could be the intersection of two taxiways

or runways or could simply be an entrance to a taxiway or to a runway.

• },...,1|{ , rlr

q

rclqN =≡ , for all runway r=1,2,…, Γ : a sequential set of artificial or nodes

defined to represent the rc sequential qeueuing slots accommodated by the holding space

equipped at the entrance to the runway r, with 1,rq denoting the first queueing slot after the

entrance to the holding space and rcrq , denoting the actual physical entry point to runway

r, i.e., runway entrance. We refer to rc as the capacity of the small queue allowed for

runway r or the capacity of the holding space for runway r.

• q

rr

q NNΓ=

∪≡,...,1

• L: the set of all links l, where a link is any paved surface intended for the purpose of

aircraft taxiing.

• ),( LNNG q∪≡ : the taxiing network.

3.1.3 Surface Aircraft Operations

The given route for an aircraft consists of two parts: a sequentially ordered set iR of real

nodes, from the starting node on the network G to the entrance to the holding space of the

assigned runway, and another set of sequentially ordered set q

iR of artificial nodes, which

represent the sequential queueing slots of the holding space for the assigned runway.

• iR , for all aircraft i: physical taxi route for aircraft i , from its first real node to its last real

node, i.e., the entrance to the holding space of the assigned runway. There are two

representations. iR can be represented by an ordered set of ik nodes }...,,{ ,21

i

k

ii

iuuu starting

with the first node of the route, progressing through the intermediate nodes and ending

with the entrance to the holding space of the assigned runway, where ik denotes the total

number of real nodes involved in iR . It can also be represented by a set of ordered

1−ik links }...,,{ 1,21

i

k

ii

illl

−starting with the first link of the route, progressing through the

intermediate links and ending with the last link of the route, where 1−ik is the number of

links involved in iR and link i

jl , j=1,2,…, 1−ik , is defined to be the ordered set ),( 1,

i

j

i

juu

+.

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• q

iR , for all aircraft i: the artificial extension of the physical taxi route iR for aircraft i

beyond the entrance to the holding space of the assigned runway, from its first queueing

slot to its last queueing slot, i.e., the runway entrance. It also has two representations. For

all aircraft assigned to runway r, r=1,2,…, Γ , q

iR is represented as { }rcrrr qqq ,2,1, ,...,, . It

can also be represented by a set of ordered 1−rc links:

{ }),(),...,,(),,( ,1,3,2,2,1, rr crcrrrrr qqqqqq − . Note that, for convenience, 0,rq is used to denote the

entrance to the holding space of runway r. Note that this entrance is a real and physical

node and it has already been defined as a node in N and as the last node i

kiu of iR .

• )(iγ , for all aircraft i: the runway assigned to aircraft i.

• uA , for all real nodes u: the set of all aircraft whose taxi route pass through node u.

• 0

uA , for all real nodes u: the union of the set of all aircraft whose routes pass through node

u and the set of the dummy aircraft, i.e., }0{0 ∪≡ uu AA .

• ijS , for all ordered aircraft pair i and j: required safe separation time at take-off (calculated

from distance, if necessary) of aircraft i from its immediate trailing aircraft j.

3.1.4 Other Parameters Used in Formulation

• gW : The weight assigned, for calculating the objective function, to the time a departing

aircraft spends at the gate. This weight should be between 0 and 1, inclusively, in order to

keep an aircraft waiting at the gate, rather than on the tarmac.

• qW : The weight assigned, for calculating the objective function, to the time a departing

aircraft spends in the small queue before the runway entrance. This weight should be

between 0 and 1, inclusively, in order to encourage departing aircraft to fill up the small

queue, Also, qW should be no greater than gW .

• M: The big-M, used as a device to include appropriate constraints, in order to ensure

consistency between the values of binary variables and the values of real-valued variables.

3.2 Decision Variables

• iut (non-negative real variable), for all aircraft i and for all real or artificial nodes u on its

route: the time at which aircraft i reaches node u. Note aircraft does not slowdown, let

alone stop, and continues on after reaching a node, in order to achieve smooth travel,

unless the node is a runway entrance (or runway crossing). Also note that some of the

nodes may be the artificial nodes representing queueing slots at a runway holding space.

• r

it (non-negative real variable), for all departing aircraft i: time at which aircraft i is to be

released at a runway entrance (i.e., the last queueing slot at the holding space for a

runway) and to enter the corresponding runway for take-off. The (artificial) node at which

the aircraft takes off from is implicit; it is )(),( ici

qγγ .

r

it can also be denoted as r

iqici

t)(),( γγ,

when explicit reference to the runway entrance is helpful.

• ijux (binary variable, about immediate predecessor), for all nodes u and all aircraft i and j

such that 0

uAi ∈ , 0

uAj ∈ and ji ≠ : ijux = 1 if any only if aircraft i reaches node u

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immediately before aircraft j does; ijux = 0 otherwise. If aircraft j is the first aircraft to

reach node u, then set 10 =jux . If aircraft i is the last aircraft to reach u, then set 10 =uix .

We use a square binary matrix to specify these relationships between aircraft, with

the0

uA being the range for both the row index i and column index j. This matrix has a

unique feature in that all its column sums and all its row sums are 1.

• ijuy (binary variable, about predecessor), for all nodes u and all aircraft i and j such that

0

uAi ∈ , 0

uAj ∈ and ji ≠ : ijuy = 1 if any only if aircraft i reaches node u before aircraft j

does; ijuy = 0 otherwise.

3.3 The Objective Function

Min [ ] [ ]∑∑∈∈

−+−+−+arrdep i

ciii

AiAi

r

iqqiuiuARTAXItttWttDRPUSHtW iiuiuiuig i

iki

ik,1i

ik1

)()()-(γγ

(0)

3.4 Constraints

We impose 16 sets of constraints, labeled as C1 through C16. They are defined by16 sets of

equations, which are labeled as Equation (1) through Equation (16).

.C1: To make sure that an arriving aircraft starts taxiing off the runway exit immediately after

landing, we impose:

iARTAXI=i1iu

t arrAi ∈∀ (1)

C2: To make sure that the time at which a departing aircraft i reaches the first node of its route

is no earlier than its time of readiness for pushback.

iDRPUSH≥i1iu

t depAi ∈∀ (2)

C3: To satisfy the requirement imposed by air traffic control, e.g., the National Ground Delay

Program dictating a time window for departure of a flight in order to cope with congestion at

another airport or in the airspace, that a specific aircraft must take off at this airport within the

time interval [ ii GDPLGDPE , ], we need:

i

r

ii GDPLtGDPE ≤≤ , dep

GDPAi ∈∀ (3)

C4: To ensure smooth travel, we require that the speed of an aircraft be within a given range.

This requirement translates into a constraint, for each aircraft i and each link ),(1

i

j

i

j

i

juul

+≡ of

its route, about the minimum travel time minijil

T and the maximum travel time maxijil

T for the

aircraft to travel the link. It is equivalent to the following constraint about the maximum

differences between the times an aircraft reaches a real or physical node on its route and the next node on the route.

,maxmin

1ij

ij

ij

ij

ij iliuiuiliu

TttTt +≤≤++

ikjAi ,...,1, =∀∈∀ (4)

Note that his constraint also ensure that the time at which aircraft i reaches the (j+1)-th node

on its route is greater than or equal to the time at which it reaches the j-th node on the route.

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To ensure against overtaking, i.e., no one aircraft passing another on the same link, and head-on collision, i.e., two aircraft traveling on the same link but in opposite directions, we need to

impose additional constraints. However, some of these constraints are applicable if and only if an aircraft precedes another in reaching a node. Therefore, for each node u of the network

G, we defined earlier in this section precedence relationships ijuy , for all the aircraft i and j

whose routes do include the node u in their times of reaching node u. These relationships can

be defined via immediate-precedence relationships ijux . However, ijux must be defined in

such a way that they are consistent with the times iut and jut of aircraft i and j reaching node u.

Therefore, we must impose the following constraints. Note that since ijux relates two aircraft,

we use the concept of a dummy aircraft, to signify whether an aircraft is the first aircraft to reach a node (on its taxi route) as well as to signify whether an aircraft is the last aircraft to

reach a node (on its taxi route). The dummy aircraft is represented as aircraft 0, and the set of

this dummy aircraft and all real aircraft has been denoted as }0{0 ∪= AA .

C5: Definition of Immediate Predecessors:

10

=∑∈ uAi

ijux 0, uAjNu ∈∀∈∀ (5.1)

10

=∑∈ uAj

ijux 0, uAiNu ∈∀∈∀ (5.2)

Mxtt ijuiuju )1( −−≥ }{\,, iAjAiNu uu ∈∀∈∀∈∀ (5.3)

Mxtt ojujuiu )1( −−≥ }{\,, jAiAjNu uu ∈∀∈∀∈∀ (5.4)

Mxtt ioujuiu )1( −−≥ }{\,, iAjAiNu uu ∈∀∈∀∈∀ (5.5)

In addition, ijuy must be defined in such way that they are consistent with ijux . To ensure this

consistency, we must impose the following two sets of constraints:

C6: Definition of Predecessors:

ijuiju xy ≥ }{\,, 00 iAjAiNu uu ∈∀∈∀∈∀ (6.1)

ojujiu xy ≥ }{\,, jAiAjNu uu ∈∀∈∀∈∀ (6.2)

1≤+ ijuoju yx }{\,, jAiAjNu uu ∈∀∈∀∈∀ (6.3)

ioujiu xy ≥ }{\,, iAjAiNu uu ∈∀∈∀∈∀ (6.4)

1≤+ ijuiou yx }{\,, iAjAiNu uu ∈∀∈∀∈∀ (6.5)

1−+≥ kiuijukju yxy },{\},{\,, jiAkiAjAiNu uuu ∈∀∈∀∈∀∈∀ (6.6)

C7: In terms of ijux and ijuy , the following constraint prevents overtaking:

,0=− ijviju yy ji RRvuijAjAi ∩∈∀≠∈∈∀ ),(,,, (7)

C8: The following constraint prevents head-on collision of two aircraft in a link (u,v):

0=− ijviju yy ji RuvRvuitsvuijAjAi ∈∈∀≠∈∈∀ ),(&),()(..),(,,, (8)

It is critical to note that (u,v) and (v,u) should be treated as two separate cases although they represent the same physical link.

C9: Aircraft must be separated for safety. Not only must aircraft on the same link be separated, aircraft traveling on two links and approaching the same node must also be

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separated. It turns out that the requirement of separation of two aircraft on a link can be achieved by separating the two aircraft with respect to the end node of the link. This

requirement can be combined with the separation requirement for two aircraft traveling on two different links (with a common end node) to constitute the following “node-separation”

constraints:

Mystt ijuiuiuju )1( −−+≥ }{\,, iAjAiNu uu ∈∀∈∀∈∀ (9.1)

where ius is the separation time between any aircraft i traveling on a link (u,v) from any

following aircraft j that can reach the start node u of the link and therefore

)(),()/(),(

iuiv

sep

iuiv

sep

iu ttvuD

D

ttvuD

Ds −=

−= (9.2)

where sepD denotes the requirement distance separation between two aircraft and

),( vuD denotes the distance of link (u,v). Therefore, the constraint can be expressed as

MyttvuD

Dtt ijuiuiv

sep

iuju )1()(),(

−−−+≥ , }{\,, iAjAiNu uu ∈∀∈∀∈∀ (9.3)

C10: The small queue at a runway has a limited capacity, and the capacity can be modeled as

a sequence of virtual links that have zero length. Note that the sequence of aircraft reaching

the different sequential queueing slots does NOT change because the holding point is

provided only to allow a small queue, not to enable re-sequencing of aircraft for better runway

throughput. Therefore, we impose

Mxttrrkrk ijqiqjq )1(

01−−≥

+,

1,...,1,0,)()(..,,, −=∀==≠∈∈∀ r

DepDep ckrjitsijAjAi γγ (10)

where 0rq is a real node and is the entrance to the holding space of runway r. Recall that this

artificial node 0rq was defined for convenience.

C11: We impose the following constraint to ensure that the release time for departing aircraft

i is no sooner than when it reaches the runway entrance, which is the last queueing slot of the

assigned runway:

rrcrrc iq

r

iq tt ≥ , )(; irAi Dep γ=∈∀ (11)

C12: Departing aircraft must be safely separated in the air. Therefore, the following

constraint is critical:

MxSttirrcrrc ijqij

r

iq

r

jq )1()(0γ

−−+≥ , rjitsijAjAi DepDep ==≠∈∈∀ )()(..,,, γγ (12)

C13: If an aircraft is released for take-off at a particular time at the runway entrance, i.e., the

last artificial node (or queueing slot) of the assigned runway, its immediate follower cannot

reach the runway entrance any earlier. This is ensured by the following constraint:

Mxttrrrcrrc ijq

r

iqjq )1(0

−−≥ rjitsijAjAi DepDep ==≠∈∈∀ )()(..,,, γγ (13)

C14: To ensure that the time at which a departing aircraft i reaches queueing slot k+1 is not

earlier the time at which it reaches queueing slot k, we impose the following constraint:

rkkr iqiqtt ≥

+1, 1,...,1,0),(, −=∀=∈∀ r

Dep ckirAi γ (14)

C15: Finally, we impose the following fairness constraint to ensure that the amount of time any departing aircraft waits at the gate after readiness for pushback is within a specified

percentage of the average of such amounts across all the departing flights.

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dep

kkuAk

iiuA

DRPUSHt

PDRPUSHt

k

dep

i

)(

)1(1

1

−

+≤−

∑∈ , DepAi ∈∀ (15)

C16: Binary and non-negativity constraints: +∈Rtiu

q

ii

Dep RRuAi ∪∈∀∈∀ , and i

Arr RuAi ∈∀∈∀ , (16.1)

+∈Rt r

i DepAi ∈∀ (16.2)

}1,0{, ∈ijuiju yx ijAjAiNu uu ≠∈∈∀∈∀ ,,, 00 (16.3)

4. Implementation and Numerical Experience

We used ILOG-CPLEX to solve the problem on a Lenovo laptop computer with an Intel Core

Duo T7500 processor, which can allocate up to 2 Giga-bytes of RAM and has a

computational speed of 2.2 Giga-hertz. In this section, we report some numerical experience.

We used the two parallel runways on the east side of the Dallas-Fort Worth International

Airport (DFW), the three terminals on the east side of the six-terminal complex, and the

taxiways connecting the two runways and the three terminals as the physical test network,

with the runway closer to the terminals dedicated to departures and the one farther from the

terminals dedicated to arrivals. This pair of runways constitute approximately one quarter of

the capacity of DFW, which is one of the largest airports in the US and is the largest hub for

American Airlines. The taxiways are modeled with 41 nodes and 70 links. The size of the

small queue at the departure runway is 1. We first discuss the performance of the base case

and then discuss the impacts of changes in some key parameters on the performance,

particularly the execution time.

The base case includes the following parameter values: To encourage filling-up and waiting

in the small queue, qW is set to 0.5. To encourage waiting at the gate, gW is set to 0.75. The

speed is set at 20 feet per second (i.e., 13.64 miles per hour or 21.95 kilometers per hour),

which is a good representation of the nominal speed during taxiing in uncongested conditions

for all aircraft types. The demand consists of 10 departing flights and 5 arriving flights. We

used the shortest paths as their routes. A route may involve 3 links to 9 links. This demand

level is consistent with the day-time demand at DFW within a 30-minute planning window.

These result in a mixed-integer programming problem of 1101 binary variables, 132 real-

valued variables, 7538 functional constraints involving binary variables, and 219 functional

constraints involving only real-valued variables. (These do not include the non-negativity or

binary constraints.) The optimal solution is reached with an execution time of approximately

5 minutes. It is important to note that we also observed that a good integer solution with an

objective value within 0.5% of the optimal objective value was reached within 10 seconds.

When qW is set to a small value, e.g., 0.01, the execution is very fast, with the optimal

solution found in about 3 seconds. At this qW , the problem virtually becomes a taxiing

scheduling problem alone, without integration with take-off scheduling. This is because the

negligible weight leads to a negligible objective-function coefficient of 0.01 for aircraft take-

off (or release) times r

it . The two-orders-of-magnitude increase in execution time is the price

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for integrated optimization for both taxiway scheduling and take-off scheduling. The

improvement in the total amount time spent by the departing aircraft on the tarmac (i.e.,

between pushback and take-off) however is 9.27%, which is very significant.

We tested the performance of extensions to the base case, with a different value for one

parameter at a time. For example, we tested the formulation against higher demand levels:

125% and 150% of the base-case demand level. The execution time for the two levels are

approximately 3 hours, and 4.5 hours, respectively. It is evident that the execution time grows

very fast with respect to the demand level. This is because of the much larger number of

binary variables to “branch-and-bound” and, more importantly, the much larger number of

possible combinations of taxiing schedules and take-off schedules to search. We also observe

that higher the taxiing speed tend to lead to longer execution time. Also, the wider the

possible speed range, the longer the execution time. The cause, we believe, is the larger

number of possible choices for the combined problem of taxiway scheduling and take-off

scheduling. We also attempted tests against larger networks, e.g., another pair of runways at

DFW together with the three terminals on the west side of the six-terminal complex and the

relevant taxiways. However, the branch-and-bound processes were halted due to insufficient

RAM. This should not be a real problem for real-world implementation, which should occur

on a main-frame computer or special-purpose processors.

5. Conclusion

In existing literature, taxiing scheduling is typically formulated to minimize total time aircraft

spend on taxiways or total delays with respect to a given set of scheduled take-off times while

the take-off scheduling is typically formulated separately to maximize aircraft throughput,

instead of time-based measures. We are able to formulate and solve the integrated problem

with one consistent time-based performance measure. Space and time separations required

between two consecutive departing aircraft hinge upon the weight-class of the two aircraft,

due to wake vortex, and their departure fixes (i.e., the first control point of the flight plan),

due to proximity of air routes. We are able to sequence arrivals of departing aircraft at a

runway, for optimal utilization runway capacity, in an integrated fashion along with taxiway

scheduling.

Our numerical experience shows that the major control and optimization objectives can be

reached, in particular sequencing arrivals of departing aircraft at runways through pushback

and taxiing scheduling (to achieve integrated optimality for the combined problem of taxiing

and take-off scheduling), making departing aircraft fill up a small queue at the runway (to

avoid possible spoilage of runway capacity during busy hours and to anticipate inability of

human operators to follow the optimized control), holding departing aircraft at the gate (to

avoid tarmac congestion), ensuring smooth travel on the tarmac for all aircraft (to avoid

tarmac congestion and to avoid unnecessary fuel burn and emissions), taxiing all arriving

aircraft off away from the runway upon completion of landing (to avoid traffic backup onto

runways), minimizing of total time spent on the tarmac (to minimize total delay), and limiting

the wait at the gate for each departing aircraft (to ensure fairness to all aircraft). Due to the

nature of mixed-integer programming and, more importantly, the integrated nature of the

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ISDSI 2009

formulation, the execution time and the memory requirement grow quickly as the problem

size, including the size of the demand and size of the network, increases.

Numerical performance depends on the problem size and the hardware. Although real-world

implementations will involve much more powerful hardware, we are studying ways to speed

up the execution. For example, the integer values of all the binary variables, i.e., all the

sysx ijuiju 'and' , of this formulation are uniquely determined by the real-valued times at

which aircraft reach the relevant nodes, i.e., all the stiu ' . In addition, heuristics are being

developed to search for suboptimal solutions with short execution times. Armed with the

formulation and computer program reported in this paper, the performance of the heuristics

can be gauged accurately. We are also extending this formulation to include more complex

airport features.

Acknowledgment

The authors acknowledge gratefully the financial support of National Aeronautics and Space

Administration (NASA) of the US Federal Government through Grant No. NNX07AU49A.

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