A Multi-stage Centralized Approach to Formation Flight Routing

and Assignment of Long-haul Airline Operations

Malik Doole and Hendrikus G. Visser

Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB, Delft, Netherlands

Keywords: Formation Flight, Flight Routing, Fleet Assignment, Fuel-efficiency.

Abstract: This paper describes the development of an optimization-based multi-stage centralized planning system for

the efficient routing and assignment of extended flight formations in commercial airline operations. In an

extended formation, where aircraft are longitudinally separated by 5-40 wingspans, a trailing aircraft can

attain a reduction in induced drag at fixed lift, and consequently in fuel burn, by flying in the upwash of the

leading aircraft’s wake. To organize the assembly of flight formations on a network-wide scale essentially

two distinct approaches can be taken, viz., a centralized approach and a decentralized approach. Both

approaches have distinct advantages and disadvantages. In this study a novel multi-stage method for flight

formation assignment is proposed that combines the advantages of the decentralized approach (fast

computation and reduced vulnerability to flight delays) with the main benefit of the centralized approach (a

near-global optimum in terms of fuel savings). The multi-stage centralized approach that we propose is

validated and subsequently demonstrated in a case study involving a wave of 267 eastbound transatlantic

flights. In the case study fuel savings of 6.8% are recorded (relative to flying “solo”), while flying in

formations comprising up to 16 aircraft.

1 INTRODUCTION

Over the past decades, a variety of studies has been

devoted to exploring the use of extended formation

flight as a means to improve the fuel efficiency in

commercial aviation. In an extended formation,

where aircraft are longitudinally separated by 5-40

wingspans, a trailing aircraft can attain a reduction

in induced drag at fixed lift by flying in the upwash

of the leading aircraft’s wake (Bower et al, 2009).

Evidently, a reduction in induced drag translates

directly into a reduction in fuel burn for aircraft

trailing in a formation. An experimental study

involving a formation flight of two military C-17

transport aircraft revealed that fuel savings of 5-10%

for the trailing aircraft are possible, increasing with

mission length (Flanzer&Bieniawski, 2014).

With the drag reduction mechanism in formation

flight now relatively well understood, the focus in

research on flight formation has shifted in recent

years into the direction of the organization and

planning of flight formations of commercial aircraft

operations on an airline (alliance) network-wide

scale (Ning et al, 2014).

To organize the assembly of flight formations on

a network-wide scale essentially two distinct

approaches can be taken, viz., a centralized approach

and a decentralized approach (Visser et al, 2016). In

a centralized approach formations are determined

pre-flight through concurrent optimization of the

routing and assignment of formation flights for an

entire fleet, whilst in a decentralized (agent-based)

approach the assembly of flight formations is

conducted in-flight through local coordination of

flights. Both approaches have distinct advantages

and disadvantages. The main benefit of using a

centralized formation planning method is the ability

to provide globally optimal (fuel-saving) solutions.

However, a centralized approach is computationally

expensive compared to the fast local solutions that

are obtained via a decentralized approach, in a

realistically sized network. The huge computational

burden of a centralized approach is a direct result of

the combinatorial complexity of the associated (NP-

hard) assignment problem. Another disadvantage of

a centralized approach is that it does not readily

accommodate schedule disruptions, thus providing a

global optimum only in the absence of flight delays.

In contrast, decentralized approaches make it

Doole, M. and Visser, H.

A Multi-stage Centralized Approach to Formation Flight Routing and Assignment of Long-haul Airline Operations.

DOI: 10.5220/0006661800470058

In Proceedings of the 4th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2018), pages 47-58

ISBN: 978-989-758-293-6

possible to manage delayed flights by simply

searching for alternative formation flight partners.

However, decentralized approaches typically exhibit

a number of shortcomings as well, notably the sub-

optimal fuel savings due to the local nature of the

employed coordination method.

Excellent examples of a centralized approach

concern the studies presented by Kent & Richards

(2015) and by Xu et al (2014). In both studies, the

centralized approach taken to formation routing and

assignment concerns a so-called two-stage (or bi-

level) method. In the first stage, the routing/mission

problem is considered for each candidate set of two

or three long-haul origin/destination flights that

might join in, respectively, a two or three aircraft

formation. The first stage routing problem

essentially deals with locating the rendezvous and

splitting points for the flights involved in each

potential formation and with scheduling the

associated altitude/speed profiles such that the

overall mission (fuel) cost is minimized. Assessment

of the mission fuel burn is based on the Breguet-

range equation for given speed and altitude. The

second stage concerns the assignment problem, in

which the network is optimized by selecting the best

subset of formation and solo missions given the

complete set of all possible combinations of

individually optimized formation and solo missions

obtained in the first stage.

Both studies involving centralized approaches

(Kent & Richards, 2015; Xu et al, 2014) were faced

with the necessity to deal with the highly-

combinatorial assignment process, while keeping the

computational burden within check. The

combinatorial problem emerges when enumerating

the number of possible combinations of aircraft

formations for a set of n (unidirectional) O/D flights,

to be grouped into formations of size m. Therefore,

given a formation of size m and a set of n flights, the

number of possible formation combinations N

can

be computed from the binomial coefficient:

()

mn m





−



(1)

The number of possible formations N

grows

rapidly with increasing values of either m or n. For

example, when considering a fleet size n = 500, the

number of possible formation combinations is equal

to N

= 124,750 for a formation size m = 2, while N

= 2,573,031,125 for a formation size m = 4. Since all

possible combinations need to be evaluated in the

first stage of the flight scheduling process, i.e., the

routing/mission analysis problem, it is readily clear

that introducing formations of large size renders the

scheduling problem computationally intractable. In

order to keep the computational burden in check, in

both studies (Kent & Richards, 2015; Xu et al, 2014)

flight formations up to size 3 only are considered,

while other simplifying measures have been

introduced as well. In Kent & Richards (2015) it is

proposed to partition the global lists of flights in a

number of ways (e.g. by airline or alliance) to keep n

relatively low. Moreover, a computationally efficient

geometric method for formation routing and mission

analysis is employed in the first stage, which makes

it possible to evaluate a large number of

combinations very quickly. In contrast, in Xu et al

(2014) a relatively high-fidelity, computationally

expensive routing/mission analysis is employed in

the first stage of the scheduling process; here the

computational savings are obtained through the

introduction of heuristics that allow the number of

formations to be considered in the first stage of the

scheduling process to be reduced, by filtering out all

non-viable flight formation combinations upfront.

Motivated by concerns regarding the

combinatorial complexity of the assignment of large

fleets in a centralized approach, an alternative

decentralized (agent-based) approach to the coupled

problem of formation flight routing and assignment

was conceived in (Visser et al, 2016). In this

approach formation flight is not anticipated pre-

flight but rather treated as an in-flight option based

on local coordination. More specifically, this study

implements a `proposal-marriage' type greedy-

algorithm for decentralised assignment of flights

into formations, similar to the one proposed in

(Ribichini&Frazzoli, 2003) for formation scheduling

of UAVs. However, the coupled routing and mission

analysis required for each formation combination

option is treated differently in both studies. While

the study reported in (Visser et al, 2016) relies on

the - slightly adapted - geometric formation flight

routing method as proposed by Kent & Richards

(2015), in combination with the Breguet-range

equation for fuel-burn assessment, the work in

(Ribichini&Frazzoli, 2003) implements a graph

search over possible rendezvous and splitting points

to find the optimal routing for each formation

combination, in conjunction with the assumption of

constant fuel-burn rates.

Both studies (Ribichini&Frazzoli, 2003; Visser

et al, 2016) clearly demonstrate that by resorting to

an agent-based, decentralized approach the inherent

weaknesses, notably their vulnerability to delayed

flights and the huge computational burden resulting

from the highly-combinatorial assignment process,

VEHITS 2018 - 4th International Conference on Vehicle Technology and Intelligent Transport Systems

can essentially be eliminated. In (Visser et al, 2016)

it is clearly shown that in the proposed decentralized

approach flight (departure) delays can be readily

accommodated, as delayed aircraft are simply able

to search alternative formation flight partners within

the network. Another beneficial feature of the

approach proposed in (Visser et al, 2016) is that it

permits building formations of (arbitrarily) large

size. This is achieved by considering two aircraft

that have just joined in a two-ship formation as a

single “entity” that then becomes eligible for further

assignment in subsequent down-route stages.

Despite its ability to overcome the typical

weaknesses of centralized approaches, there is also a

major downside associated to a decentralized

approach, in the sense that the global fuel savings

potential of flight formation is not fully achieved, due

to the greedy nature of the flight formation assembly

decision making process. Indeed, the numerical

experiments conducted in (Doole, 2016) show that the

decentralized approach presented in (Visser et al,

2016) results in fuel burn savings that are only

fraction of the global fuel burn savings potential (50

to 60% in large-scale transatlantic scenarios).

In this paper a novel method for flight formation

routing is proposed that aims to combine the advant-

ages of the decentralized approach (fast computation

and reduced vulnerability to flight delays) with the

main benefit of the centralized approach (a near-

global optimum in terms of fuel savings). The main

aim of this study is to evaluate the fuel saving

potential of this novel multi-stage centralized

approach to formation flight routing, which combines

some elements of the decentralized approach

presented in (Ribichini &Frazzoli, 2003) with the

geometric formation flight routing method as

proposed by Kent & Richards (2015) in their bi-level

centralized approach. The sections that follow provide

a problem formulation and the proposed operational

concept. Next, this paper discusses the employed

routing method along with the modelling of fuel burn.

This is then followed by a description and validation

of a multi-stage fleet assignment model. Finally, a

transatlantic case study is presented, from which the

conclusions of this paper originate.

2 OPERATIONAL CONCEPT

Inspired by (Visser et al, 2016) and (Kent &

Richards, 2015), a novel multi-stage centralized

approach has been conceived for formation routing

and assignment. The new approach that is proposed

is based on a multi-stage assignment algorithm that

updates the assignment of flight formations each

time a certain event occurs. An event in this context

is defined as: a departure, joining, or splitting of a

flight within the network. As soon as one these

events occurs, a flight formation assignment is made

involving all eligible en-route flights, while

considering the current aircraft positions as the

“origin” nodes of the flights. The key feature of the

multi-stage centralized approach is that in each

assignment stage, only formations of size two are

assembled, reducing the combinatorial complexity

of the assignment in each step.

The initial assignment evaluation is made as

soon as the first two aircraft that have departed to

their destinations (using great circle routes) reach

cruise altitude. The most likely outcome of this first

assignment evaluation is that an assembly of a flight

formation is not favourable and the two flights will

continue on their solo routes towards their

destination. As soon as the next aircraft reaches its

cruise level, a new assignment evaluation is

triggered, which includes the new flight. This

process is repeated until at a certain point, the first

two-ship flight formation is assigned. The two

aircraft involved in the formation assignment then

change course and speed so as to rendezvous at the

calculated joining point. During their transition

towards this joining point, the two flights involved

are not eligible for any subsequent assignment.

However, once these two aircraft have joined, the

resulting two-ship formation is regarded as a single

“entity” that then again becomes eligible for

assignment, thus allowing the assembly of larger

formations in subsequent assignment stages.

3 FUEL BURN ASSESSMENT

In this study we consider all aircraft operating in the

network to be of the same type The parameters of

the particular aircraft model employed in this study

relate to a Boeing B777, and have been extracted

from (Visser et al, 2016). The transatlantic routes are

modeled in this study as great circle paths from

origins to destinations. Moreover, it is assumed that

the entire route is flown in cruise at a single constant

altitude and at constant speed. The fuel consumption

along the routes is estimated using the well-known

Breguet-range equations for flight at constant

altitude and constant speed (Vinh, 1993), assuming

the absence of wind:

max

tan tan

md md

DLLt

CCCc

−−



=−





(2)

A Multi-stage Centralized Approach to Formation Flight Routing and Assignment of Long-haul Airline Operations

where :

;,1,2

md i

CC i

== =

(3)

are, respectively, the values of the lift coefficient for

minimum drag, and for the initial and final weight of

the aircraft (at the considered altitude and speed).

The parameter :

max







(4)

in Eq.(2) represents the maximum lift over drag

ratio. The above relationships assume a traditional

drag polar of the form C

CKC=+ The

parameter

c in Eq.(2) represents the Specific Fuel

Consumption (SFC) of the aircraft type considered.

Equation (2) can be resolved for the aircraft

weight at the end of the flight stage, W

, given an

initial weight W

tan

md md





−







 

 



 



 



(5)

where :

max







(6)

Alternatively, given a final weight W

, the required

initial weight W

needed to fly a distance R can be

estimated using Eq.(6). Note that Eq.(6) can be used

to assess fuel burn in both solo flight legs and

formation flight legs. To allow for the formation

drag reduction, a discounting factor λ is used in the

induced drag coefficient K of a trailing aircraft in a

formation, i.e. K

trailing

= λ K. In the present study, a

discount factor λ = 0.867 has been adopted (Visser et

al, 2016), which results in a fuel flow rate reduction

of about 10% for each trailing aircraft in a

formation. It is noted that several aerodynamic

studies related to formation flight point out that the

formation induced drag reduction increases as the

number of aircraft in the formation string increases

(Ning et al, 2014; Xue&Hornby, 2012). However, in

this study this particular effect is ignored and each

trailing aircraft in a formation essentially enjoys the

same induced drag discount factor, regardless of the

size of the formation string. Note that organizing the

traffic flow into larger formation flights can still be

very rewarding in the sense that the number of

formation flight leaders that are needed (and that

don’t enjoy any drag reduction benefit) can be

reduced in this way.

The constant speed that is adopted along a route

is generally taken as M=0.82 in this study. This

Mach number is slightly lower than the typical

cruise value for the aircraft type considered, in order

to mitigate compressibility effects in a formation

which may impede aircraft safety (Ning et al, 2014).

For reasons of simplicity, this Mach number is also

adopted in all solo flight legs, with one exception.

This exception relates to aircraft flying the shortest

flight leg towards the joining point in a formation

rendezvous. In order to rendezvous at the joining

point, the aircraft flying the shortest leg needs to

reduce its speed relative to its partner aircraft flying

the longer leg. Note that the lowest speed in a

synchronization flight leg is the maximum

endurance speed, which is the speed at which the

lowest fuel consumption per unit of time is attained

(Vinh, 1993). Assuming a drag polar as defined in

Eq.(4), the maximum endurance speed corresponds

to the speed for minimum drag and is given by :

1/4







(7)

The actual take-off weight for a specific (great

circle route) flight is calculated using the Breguet-

range equation assuming that at destination the

weight of the aircraft is equal to the zero-fuel weight

plus the weight of the reserve fuel. Given the aircraft

weight at destination, and the distance covered along

the route, the aircraft weight at the origin can be

assessed using the Breguet-range equation, for the

assumed speed and altitude. To allow for the fact

that aircraft may have to fly detours and thus longer

routes in order to engage in flight formation, the

initial fuel load is increased by 10%; the take-off

weight of the aircraft is increased accordingly.

4 FORMATION ROUTING

To generate routes for the assembly of formation

flights, a routing method was used based on (Kent &

Richards, 2015). In this approach, a formation flight

route for two aircraft is obtained through the

minimization of weighted distance. The time-free

routing method from Kent & Richards was extended

VEHITS 2018 - 4th International Conference on Vehicle Technology and Intelligent Transport Systems

in (Visser et al, 2016) to ensure rendezvous of timed

flights in the formation assembly process.

4.1 Geometric Routing Method by

Kent & Richards

In (Kent & Richards, 2015), a simple geometric

method to construct a formation flight routing is

presented based on a classical mathematical problem

posed by Fermat in the 17

century. The problem,

illustrated in Figure 1, is posed as follows: given a

triangle ABC (Fig.1.a), find a point P such the that

the sum of the distances

|| ||

|| ||BP

and

|| ||CP

is minimized. The geometric approach to

construct the solution to this problem is illustrated in

Fig.1.b.

The method shown in Fig.1.b is based on

constructing outwardly three equilateral triangles

along the sides AB, BC and CA. Then the lines from

the outer vertex of each new triangle to its opposite

vertex of the original will intersect at a single point,

which is the desired point P. Equivalently, the point

P can be found as the intersection point of the

circumscribed circles of each of the three new

equilateral triangles.

Fermat’s problem provides a good analogy to

the formation flight assembly problem, if it is

assumed that fuel consumption is proportional to the

distance covered. However, it is readily clear that

the fuel consumption per unit distance along the solo

arcs

|| ||

and

|| ||

differs from that on the

formation flight arc

|| ||

. To resolve this issue,

Kent & Richards formulated a weighted-arc version

of the problem, where the arc weights reflect the

different fuel consumption per unit distance. More

specifically, to represent the cost of flying a unit of

distance, the arc weights w

, w

, and w

are

introduced for the segments AP, BP and PC,

respectively. Note that the value of w

is typically

set equal to the combined values of w

and w

, while

applying some discount factor to represent the fuel

savings due to induced drag reduction of the trailing

aircraft.

Thus, in the modified problem the location of

the joining point P has to be selected such that the

total cost of distance, expressed by Eq.(1) is

minimized:

( ) || || || || || ||

ABC

Minimize

PwAPwBPwPC=⋅ +⋅ +⋅

(8)

Following Kent & Richards, the location of point P

that minimizes Eq.(9) must satisfy the vectorial

equilibrium condition expressed by Eq.(2):

0||

|| || || || || ||

ABC

AP BP PC

www

AP BP PC

⋅+⋅+⋅ =

(9)

(a) (b)

Figure 1: Geometric construction to locate the optimal

joining point P (Kent & Richards, 2015).

Application of the law of cosines to Eq.(9) yields

expressions for the intersection angles ∠APB,

∠APC, and ∠BPC. Since the angle ∠APB represents

the intersection angle between the two solo legs AP

and BP, it is referred to as the “formation angle”.

Equation (10) gives the expression for the resulting

formation angle

222

cos |

ABC

www

−



−− +





(10)

Note that the formation angle

only depends on the

routing weights w

, w

, and w

. The formation angle

is illustrated in Figure 2. It is noted that as long as

the weights w

, w

, and w

are not altered, also the

formation angle

remains unaffected. As a result,

the point C can be shifted freely along the line PC,

without altering the solution.

Figure 2: Illustration of formation angle.

The method for locating point P can be extended to a

scenario in which the two flights do not have a

common destination C. Figure 4 illustrates two solo

A Multi-stage Centralized Approach to Formation Flight Routing and Assignment of Long-haul Airline Operations

routes connecting origins A and B to destinations C

and D, respectively. The joining point J and the

splitting point S are to be determined. Since the

formation angle condition must be satisfied at both J

and S in order to minimize the weighted distance,

one can draw two circular arcs from A to B and from

C to D along which the formation angle is constant

and equal to the value obtained from Eq.(11). These

arcs are displayed in Figure 4.

Figure 3: Illustration of example solo routes AC and BD.

Figure 4: Geometric construction of the formation route.

The (back vertex) point X

in Fig. 4 is obtained by

making use of the fact that Eq.(11) holds in triangle

ABX

||:| |:| | ::|

CAB

BBX XAwww= (11)

Mirroring the described steps at the destinations C

and D provides the (back vertex) point Y

. The

locations of J and S that minimize the weighted

distance from A to C and from B to D are obtained

from the intersections of the line from X

to Y

and

the arcs of constant formation angle.

4.2 Accommodating Synchronization in

the Basic Routing Method

The geometric routing procedure presented in

Section 4.1 is inherently time-free, essentially

assuming that the flight schedules are synchronized

such that the two flights joining in formation are

able to rendezvous at the calculated joining point.

However, the approach proposed herein considers

flight formation to be an in-flight option, with the

current aircraft positions serving as the origin nodes

of the flights. Therefore, in order to realize

formation flight, the route must be constructed such

that the two aircraft are able to arrive at the joining

point simultaneously. For solo flight legs to the

joining point that are not too dissimilar in size,

synchronization is typically accomplished by

slightly slowing down the aircraft that is flying the

shortest leg. However, if the formation flight route

does not permit synchronization in this way, the

joining point must be relocated to enable a stretch of

the connecting flight legs. It is conceivable that even

relocating the joining point may not be sufficient to

ensure a rendezvous. In this case, excess delay time

needs to be absorbed by holding one of the aircraft

at the joining point. When the latter situation occurs,

usually the formation option turns out to be less

favorable then flying solo. Details regarding the

synchronization process can be found in (Visser et

al, 2016).

4.3 An Extension to Larger Formation

Sizes

Kent&Richards (2015) demonstrated that the

geometric routing approach developed for the

assembly of two-ship formations could be extended

to formations of any size. Here, we will illustrate

this extension for the assembly of a four-ship flight

formation. Figure 5.a shows the assembly of two

two-ship formations; in the first two-ship formation

Flights 1 and 2 are joined, while in the second two-

ship formation Flights 3 and 4 are joined. These

two-ship formations are now regarded as two

“pseudo flights” (respectively, Flight

and Flight

with, respectively, the back vertices

flight

X and

flight

serving as origin nodes and the back vertices

flight

and

flight

as destination nodes. Next, as

shown in Fig.6 the two pseudo flights Flight

and

Flight

join in formation, and a joining and splitting

point are computed in the same way as for the

assembly of a two-ship formation. The resulting

four-ship formation can then then regarded as a new

pseudo flight (Flight

1234

). It is noted that the four-

ship formation assembly sketched in Fig.6 presumes

a certain order of joining in formation (e.g., Flight 1

first joins with Flight 2) and therefore represents just

one of the options in the four-ship formation

assembly process for these four flights. All

alternative joining order options (e.g., Flight 1 first

joins with Flight 3, or Flight 1 first joins with Flight

4) need to be explored as well, including the

possibility of three-ship formations (pseudo flights)

joining with single aircraft flights. It is readily clear

that the assembly of larger formations represents is

also problem that is highly combinatorial in nature.

In (Doole, 2016) the time-free four-ship assembly

VEHITS 2018 - 4th International Conference on Vehicle Technology and Intelligent Transport Systems

process proposed by Kent & Richards has been

extended to include flight synchronization.

Figure 5: Assembly of two two-ship formations (pseudo

flights, Flight

and Flight

) from four solo flights.

4.4 Formation Leaders

When two, or more, flights have successfully

completed their rendezvous, one of the aircraft

involved needs to be assigned as the formation

leader. A flight that does not lead a formation, is

referred to as a ‘trailing’. In the present set-up,

where only aircraft of the same type are considered,

it is readily clear what the best choice is from a

collective perspective: the least heavy aircraft of the

two is designated as the lead aircraft, as the heavy

aircraft can benefit relative more from an induced

drag reduction. It is noted that the formation leader

does not gain any direct benefit from flying in an

extended formation; however, in this study it is

assumed that formation partners will somehow share

the overall economic benefits.

Figure 6: Assembly of a four-ship formations (pseudo

flight, Flight

1234

) from two two-ship formations.

4.5 Azimuthal Equidistant Projection

Method

The geometric approach presented herein is

inherently planar in nature, however it can be

extended to hold for problems on a sphere (Kent &

Richards, 2015). This latter option has not been

pursued in this study; rather the great circle routes

connecting the various O/D pairs were projected on

a plane by means of the so-called azimuthal

equidistant projection method (Tobler, 1962), so that

the original planar geometric approach can be

retained. The azimuthal equidistant projection

method has the property that all distances from the

center are rendered correctly to scale and that all

points on the map are at the correct azimuth

(direction) from the center point. By using this

particular projection method, the relative locations

of the origins and destinations, as well as the route

lengths and relative flight headings remain

reasonably well preserved in the transformation.

Since these route characteristics are most relevant

for the success of formation flight implementation,

the selected projection method is considered

appropriate for this study.

5 FLEET ASSIGNMENT

PROBLEM

5.1 Basic BMILP Formulation

The fleet assignment model that has been adopted

selects a compatible set of possible flight formations

in order to achieve global minimum fuel burn. The

optimization problem used to solve the formation

flight assignment is formulated as a Binary Mixed-

Integer Linear Program (BMILP), using a similar

formulation as adopted by Kent & Richards (2015).

Given a fleet of size n = N, and flight

formations up to size m = M (including solo flights,

which are regarded as flight formations of size one),

the total possible number of flight formations N

given by:

(, ) ()

NNM N N



, (12)

where N

is the binomial coefficient defined in

Eq.(1) that gives the number of possible formation

combinations of size m. To give an example,

suppose we have a fleet of N = 5 aircraft and we

consider formations up to size M = 3, we then have:

5N = ;

10N = ;

5N = ;

1N = ,

and thus N

(5,3) = N

+ N

= 31

possible formation combinations are found.

The variables and (cost and pairing) coefficients

that are used to define the fleet assignment problem

are as follows:

A binary decision variable that has the value

one if configuration j is selected in the

solution and zero otherwise.

A real-valued cost coefficient that represents

the total incurred fuel cost of formation j.

A Multi-stage Centralized Approach to Formation Flight Routing and Assignment of Long-haul Airline Operations

A binary pairing coefficient that indicates

whether flight i is included in formation j

= 1) or not (p

= 0).

The BMILP formulation that is used to optimally

assign each flight to a formation is then given by :

minimize

=⋅



, (13)

subject to :

{

}

1 , 1,...,

ij j

px i N

⋅= ∀∈



(14)

{

}

, 1,...,



=∀∈





(15)

The number of binary variables is equal to the

number of possible flight formations N

, which

increases dramatically with the number of flights N,

and especially with the maximum formation string

size M. It is readily clear that for larger problems (in

terms of maximum formation string size M), the

combinatorial complexity of the MILP becomes

significant and a huge computational effort is

required. In the multi-stage centralized assignment

approach the number of decision variables is kept

low in each assignment step basically by restricting

the formation string size to just two. In this case the

number of potential formation combinations

amounts to:

(1)

−

(16)

5.2 Multi-Stage Fleet Assignment

Model

Figure 7 presents a schematic diagram of the

developed multi-stage fleet assignment model. A

new assignment is made every time an event (i.e., a

departure, joining, or splitting) occurs. When an

event takes place, the location of each eligible flight

(or pseudo flight) that is en-route is determined and

subsequently all potential flight formation missions

for the eligible fleet are assessed in terms of routing

and fuel burn performance, and the associated cost

and pairing coefficients are determined. Next, the

BMILP problem is formulated and solved (using

CPLEX) and the resulting formations are assigned;

the aircraft (strings) involved in the assignment are

declared ineligible for further assignment until they

have joined. Note that when such a joining takes

place a new event is triggered.

6 VALIDATION OF THE

MULTI-STAGE FLEET

ASSIGNMENT MODEL

To prove the effectiveness and efficiency of the

proposed multi-stage centralized approach to flight

formation routing and assignment a validation study

was undertaken. In this validation study the

(computational and fuel burn) performance of the

newly developed multi-stage fleet assignment

approach is compared against that of the original bi-

level centralized approach of Kent & Richards

(2015), albeit that both approaches are somewhat

modified in order to provide a proper basis of

comparison. More specifically, since the centralized

approach of Kent & Richards is essentially time-

free, it has been augmented with the rendezvous

synchronization mechanism for formation flight

routing developed in (Visser et al, 2016), so that it

can be used in conjunction with a given flight

schedule. Since the centralized approach of Kent &

Richards is computationally expensive for large

problems, a relatively small fleet assignment

problem is considered in the validation scenario.

More specifically, in the validation scenario the

assignment problem is limited to formations

comprising strings up to four aircraft (M = 4) only,

whilst considering a fleet of 50 aircraft (N = 50) that

seek to traverse the North Atlantic. Even for this

relatively small problem, the enumeration time is

significant as the number of potential formations is

already quite large. Using Eq.(1), it is readily

established that there are 1,225 two-ship, 19,600

three-ship, and 230,300 four-ship formation

combination possibilities, in addition to the 50 solo

flights. Therefore, aside from the 50 solo flights,

there are 251,125 possible formation combinations

in total, each requiring individual formation routing

and fuel burn predictions. It is recalled that the

routing/mission analysis itself is also plagued by

combinatorial complexity issues, leading to

computational times that increase dramatically with

the maximum size of the formation string.

The multi-stage fleet assignment model

developed herein is run in a “degraded” mode of

operation in the validation scenario. In order to

avoid the assembly of formations in excess of size

four, an assignment step is not made after each

occurrence of an event, but rather in just two

discrete stages. First of all, in the validation scenario

it is assumed that all 50 flights depart at exactly the

same time; at this given departure time the first

assignment evaluation is made. A second assignment

VEHITS 2018 - 4th International Conference on Vehicle Technology and Intelligent Transport Systems

step is then made as soon as all formations that were

assigned in the first stage have actually joined up.

Evidently, in the first stage only formations up to

size two can emerge, while in the second stage

formations up to size four can be assembled.

Clearly, in the first stage there are only 1,225

possible (two-ship) flight formations, in addition to

the 50 solo flights. In the second assignment stage,

the “fleet” to be considered is substantially smaller

than in the first stage. The reason for this is that it

turns out that in the first stage 24 two-ship

formations were assembled (involving 48 aircraft),

so that in the second stage the fleet only comprises

24 “pseudo flights” and 2 solo flights, for a total of

26 entities. The total number of possible formations

in the second stage (for formations ranging from size

1 up to size 4) is therefore just 302.

Figure 7: Flow chart of the operational multi-stage fleet

assignment concept.

The performance characteristics of both

assignment approaches are summarized in Table 1.

The centralized approach by Kent&Richards (2015)

is denoted here as the “single-stage” approach,

whilst the multi-stage fleet assignment approach

proposed herein is labeled as the “two-stage”

approach. In Table 1, the number of potential

formation combinations (and thus routing/mission

analysis evaluations) that is assessed in each

approach is indicated, along with the required CPU

time (on a standard 2.1 GHz personal computer),

and the resulting fuel savings of formation flight in

absolute terms and in relative terms (as compared to

executing all flights solo).

A close inspection of the results in Table 1

reveals that the two-stage approach performs almost

equally well as the single-stage approach in terms of

fuel savings, but at only a fraction of the

computational cost. Overall, the differences in the

routing and assignment solutions between the two

approaches turn out to be marginal.

7 TRANSATLANTIC CASE

STUDY

The proposed multi-stage fleet assignment model is

applied in a case study involving 267 eastbound

transatlantic flights. It is assumed in this case study

that each flight can potentially join in formation with

any other flight in the network (regardless of airline

or alliance membership). The routes included in the

case study, shown in Figure 8, are obtained from an

available real-life data set (Lith et al, 2014) by

means of selecting the longitude/latitude coordinates

of all origins and destinations. However, in order to

make the case study more challenging, it is assumed

that all 267 flights depart at the same time.

Also in the transatlantic case study, an alternative

fleet assignment approach is considered to obtain a

basis of comparison for the multi-stage centralized

approach. Given the size of the assignment problem,

the agent-based decentralized approach reported in

(Visser et al, 2016) is applied in the case study to

provide the basis for comparison.

In the simulation of the decentralized formation

flight planning method a local search for potential

partners is conducted once every 5 minutes. The

employed communication radius for formation

assembly negotiation is set at 250 km; hence only

flights within this radius are considered eligible for

formation assembly.

The results for the decentralized approach bear

out that the largest observed formation is a sixteen-

ship formation. To establish a fair basis of

comparison, a restriction on the maximum size of a

formation is enforced in the multi-stage centralized

approach as well, in the sense that no formations is

excess of size 16 are allowed to be assembled. To

this end, the multi-stage assignment model is again

run in “degraded mode”, limiting the assignment

updates to just four discrete stages.

A Multi-stage Centralized Approach to Formation Flight Routing and Assignment of Long-haul Airline Operations

Table 1: Performance comparison between single-stage and two-stage fleet assignment solutions.

Approach Formation

combinations

Computation

time (mm:ss)

Fuel savings

(%)

Fuel saved

(kg)

Single-Stage 251,175 58:43 5.72 184,381

Two-Stage 1,577 00:14 5.66 182,480

Figure 8: Route set used in the case study (created using

Great Circle Mapper at www.gcmap.com).

The results for the multi-stage assignment model

are summarized in Figure 9, which shows the

fraction of total time spent in formations ranging in

size 1 to size 16. It is readily clear that the majority

of the time is spent in formations of size 4. Indeed,

no less than 58 four-ship formations are assembled

in this transatlantic case study, half of which later

merge into larger formations. Although ultimately

four sixteen-ship formations are assembled, the total

time spent in these largest formations is relatively

small. No formations of size 5 or 7 were assembled.

Moreover, the sixteen-ship formations are the only

formations in excess of size 8. Only two flights do

not engage in flight formation and remain solo. It is

Figure 9: Formation size usage when the multi-stage fleet

assignment approach is employed.

noted that the results might be somewhat tainted by

the fact that the multi-stage fleet assignment model

has been run in “degraded mode”. This will be

explored in future research.

Figure 10 shows the corresponding results for the

agent-based decentralized approach. It can be seen

that in the decentralized approach considerably less

time is spent in formations of size four. Moreover, a

wider variety in the size of the formations that are

formed is observed.

Figure 10: Formation size usage when the decentralized

fleet assignment approach is employed.

In Table 2 the computational and fuel burn

performances of the two approaches are compared.

When comparing the results, it is evident that the

multi-stage centralized approach is vastly superior is

terms of fuel savings. Indeed, the savings in the

multi-stage centralized approach are almost twice

that of the decentralize approach. Although the total

computational time for the multi-stage approach

features a significantly higher computational burden

relative to the decentralized approach, in absolute

terms the computational time is still modest and

remains well within the requirement for real-time

application.

VEHITS 2018 - 4th International Conference on Vehicle Technology and Intelligent Transport Systems

Table 2: Performance comparison between multi-stage centralized and decentralized assignment solutions.

Approach

Computation

time (mm:ss)

Fuel savings

(%)

Fuel saved

(kg)

Multi-stage Centralized

Decentralized

09:24 6.88 1,202,000

03:23 3.90 681,200

8 CONCLUSIONS

This study proposes a multi-stage centralized

cooperative planning system for the efficient routing

and scheduling of flight formations. In the proposed

concept, the global fleet assignment is updated every

time a certain event occurs en-route, where an event

is either the emergence of a new flight, the

emergence of a newly assembled formation, or the

emergence of a disassembly of a formation. In each

assignment update, only the assembly of formations

involving two flights are considered, where a flight

can either be an actual solo flight or a “pseudo

flight”, representing a prior assembled formation. By

limiting the formation size to just two (pseudo)

flights in each assignment step, the combinatorial

complexity of the overall assignment process can be

kept in check.

In a validation scenario involving flight

formations up to size four, it was demonstrated that

the proposed multi-stage fleet assignment model is

capable of generating a near-optimal global solution,

at a fraction of the computational cost required for

the single-stage assignment of formations up to size

four. In a subsequent large-scale transatlantic

scenario, it was demonstrated that the multi-stage

fleet assignment offers a vast improvement over a

decentralized approach in terms of the fuel savings

that can be achieved.

In view of the results obtained in this study, the

proposed concept holds out great promise for further

development towards real-world application. In

future studies, many of the simplifying assumptions

taken in the present study will need to be removed

for this purpose. In particular, the no-wind

assumption that underlies the current study has a

profound impact on the solution behavior and more

realistic wind scenarios will need to be considered in

lieu of the no-wind assumption in future studies.

Although in an extended formation aircraft fly at

safer separation distances than in close formation

flight, aircraft still fly in relatively close proximity,

exceeding current regulated separation minima.

Many aviation regulations and standards, including

the reduction of aircraft separation limits, therefore

will need to be adapted. The technical, operational

and regulatory challenges associated to introducing

extended formation in the civil aviation domain need

to be investigated in future research.

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VEHITS 2018 - 4th International Conference on Vehicle Technology and Intelligent Transport Systems