A QUBO Model to the Tail Assignment Problem
Luis N. Martins
1 a
, Ana Paula Rocha
2 b
and Antonio J. M. Castro
2 c
1
Department of Informatics Engineering (DEI), Faculty of Engineering (FEUP), University of Porto, Porto, Portugal
2
Artificial Intelligence and Computer Science Lab (LIACC), University of Porto, Porto, Portugal
Keywords:
Tail Assignment Problem, Quantum Annealing, Scheduling Problem.
Abstract:
Tail Assignment is the problem of allocating individual aircraft to a set of flights subject to multiple constraints
while optimising an objective function, such as operational costs. Given the enormous amount of possibilities
and constraints involved, this problem has been a case study over the last decade. Many solutions have emerged
using classical computing, but with limitations. Quantum Annealing (QA) is a heuristic technique to solve
combinatorial optimisation problems by finding global minimum energy levels over an energy landscape using
quantum mechanics. In this study, Tail Assignment Problem was framed as a Quadratic Unconstrained Binary
Optimisation (QUBO) model and was solved using a classical and two hybrid solvers. The considered hybrid
solvers made use of the D-Wave 2000Q quantum annealer. Tests were run based on extractions from real-
world data, analysing, empirically, the performance of the implementation in terms of quality (i.e., the lowest
operational costs) of the obtained solutions. We concluded that, for the considered datasets, there was a higher
probability of obtaining better solutions for this problem using one of the hybrid solvers when compared with
a classical heuristic algorithm such as Simulated Annealing (SA).
1 INTRODUCTION
Due to a large number of routes, aircraft and crew,
scheduling and operational management are the most
complex and challenging tasks that airlines need to
face every day. Airline scheduling is a complex pro-
cess composed of a sequence of tasks, starting from
defining which airports and routes the airline com-
pany will operate, until the moment an individual air-
craft and crew members are assigned to each specific
flight. The Tail Assignment Problem is part of this
process and aims to efficiently allocate individual air-
craft (also called tails) to flights minimising a certain
objective function, such as operational costs, while
ensuring some constraints (Gr
¨
onkvist, 2005). Quan-
tum Annealing (QA) is a heuristic technique used to
find a global minimum of an objective function de-
fined over an energy landscape using quantum me-
chanics (Kadowaki and Nishimori, 1998; Ray et al.,
1989). It operates by taking advantage of the natural
evolution of a quantum system changing an Hamilto-
nian representing the initial quantum state to a final
state representing the minimum final value of the op-
a
https://orcid.org/0000-0001-5032-049X
b
https://orcid.org/0000-0002-8129-9758
c
https://orcid.org/0000-0001-8121-1974
timisation problem it aims to solve (Albash and Li-
dar, 2018). Such technique has been used to solve
complex optimisation problems with a very large set
of possibilities aiming to find global minimum solu-
tions (Neukart et al., 2017; Ikeda et al., 2019). Given
the definition and complexity of finding a proper so-
lution to the Tail Assignment Problem, as well as
the good results when applying QA to optimisation
problems, the main goal of this study is to verify the
possibility of solving the Tail Assignment Problem,
considering the operational restrictions and the oper-
ational costs, using QA in order to evaluate the use-
fulness of this kind of technique in such a complex
domain. The rest of this paper is organized as fol-
lows, Section 2 presents some literature review con-
cerning the Tail Assignment problem and Quantum
Annealing approaches to scheduling problems. Sec-
tion 3 presents the definition of the problem, includ-
ing the constraints to be considered as well as the op-
timisation criteria. Section 4 details the modelling of
the problem to be solved using a quantum annealing
approach. Experiments and results are discussed in
Section 5. Finally, conclusions and future work are
presented in Section 6.
Martins, L., Rocha, A. and Castro, A.
A QUBO Model to the Tail Assignment Problem.
DOI: 10.5220/0010259608990906
In Proceedings of the 13th International Conference on Agents and Artificial Intelligence (ICAART 2021) - Volume 2, pages 899-906
ISBN: 978-989-758-484-8
Copyright
c
 2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
899

2 RELATED WORK
The Tail Assignment Problem has been studied over
the last decade with multiple approaches being pre-
sented (Gr
¨
onkvist, 2005; Ruther et al., 2017; Froy-
land et al., 2013; Montoito, 2016; Yadav, 2017).
In (Gr
¨
onkvist, 2005) the authors studied an hy-
brid approach, using constraint programming and a
branch-and-price algorithm. By defining the Tail As-
signment Problem as a set partitioning problem based
on pre-calculated routes, it starts by modelling the
problem as a Constraint Satisfaction Problem (CSP).
Such modelling is important for generating an ini-
tial feasible solution that is later optimised using a
branch-and-price algorithm solving the problem for
a fixed period of time and taking into account some
specific activities and irregular schedules. The com-
bination of both algorithms allows improving initial
solutions that respect operational constraints such as
maintenance, turnaround times and pre-assigned ac-
tivities.
Following also a branch-and-price algorithm,
in (Ruther et al., 2017) the authors present a solution
for the Tail Assignment Problem together with the
crew pairing. This approach is thought to be executed
only few days before the operation, creating sched-
uled routes for a specific aircraft instead of generic
ones.
In (Montoito, 2016) the author presents a solution
for the Tail Assignment Problem using SA with an
adaptive neighbourhood local search approach. Start-
ing by obtaining an initial feasible solution using a
First-In-First-Out (FIFO) approach, it made use of a
SA algorithm to minimise the operational costs.
To solve problems using a QA approach, as for us-
ing D-wave 2000Q quantum annealer, multiple stud-
ies formulate it as a specific Binary Quadratic Model
(BQM) called QUBO model. By doing this it is pos-
sible to set the problem aimed to be solved as the fi-
nal Hamiltonian of the system Hamiltonian. In fact, a
QUBO model may be expressed as in equation 1:
E
qubo
(a
i
,b
i, j
;q
i
) =
∑
i=1
a
i
q
i
+
∑
i< j
b
i, j
q
i
q
j
(1)
where q
i
and q
j
correspond to qubits, a
i
represents the
linear coefficients of the qubits and b
i, j
are quadratic
coefficients representing the strength between each
pair of connected (coupled) qubits.
Although presented here in a scalar notation, the
biggest difference between both notations is that
QUBO model can also be redefined using an upper-
diagonal matrix whereas Ising model cannot.
In fact, a QUBO model can be expressed as a min-
imisation problem presented in equation 2 where x is a
vector of binary variables and Q is an upper-triangular
matrix of constants that represent the biases and cou-
pling strengths of the considered binary variable.
minimise y = x
t
Qx (2)
Due to the few research developments in this area, no
studies were found applying QA to the Tail Assign-
ment Problem. Therefore, a wider analysis was per-
formed searching for another kind of problems that
could have an implementation using this approach.
In (Venturelli et al., 2016) the authors propose a so-
lution for the Job Shop scheduling problem using a
QA approach. They concluded that the usage of hy-
brid approaches and meta-heuristics were fruitful and
should exist further investigation on such area. Fur-
thermore, they also concluded that pruning unneces-
sary variables is an important step as it may highly
reduce the size of the problem. Moreover, in (Stol-
lenwerk et al., 2019) the authors implemented a flight
gate assignment minimising the total time for tran-
sit passengers. Converting this NP-hard problem to a
QUBO model, they tested how a quantum annealer,
in this case a D-Wave 2000Q, could solve this prob-
lem. Due to hardware limitations, only a small por-
tion of the real data was considered, concluding that
extracting problem instances from data can lead to
a QUBO model with distributed coefficients, reduc-
ing the success probability. Another study made use
of a D-Wave 2000Q quantum annealer, but this time
for solving the Nurse Scheduling Problem with hard
constraints (Ikeda et al., 2019). It concluded that
satisfiable solutions could be achieved using QA but
only for small samples due to hardware constraints.
Some other QUBO models were also defined for well-
known problems such as the graph partitioning prob-
lem (Ushijima-Mwesigwa et al., 2017), the maximum
clique problem (Chapuis et al., 2019), the traveling
salesman problem (Papalitsas et al., 2019) or the min-
imum multi-cut problem (Cruz-Santos et al., 2019).
Furthermore, in (Lucas, 2014) the author proposes
a formulation for multiple well-known NP-hard and
NP-complete problems, that can be run on a quantum
annealer, and in (Tran et al., 2016) the author pro-
posed a hybrid approach for multiple complex prob-
lems.
When taking into account other quantum ap-
proaches rather than QA, it is possible to find one
study presenting a partial implementation of the Tail
Assignment Problem using a Quantum Approximate
Optimisation Algorithm (QAOA). In fact, in (Vikst
˚
al
et al., 2019) the authors had as main goal to find a
feasible solution considering some constraints. Due
to the limited capabilities of the used quantum com-
puter, a limited number of pre-calculated routes (us-
ICAART 2021 - 13th International Conference on Agents and Artificial Intelligence
900

ing branch-and-price) was considered, ensuring that
one of these routes was a solution with all the flights
assigned. It also considered that all the aircraft would
be used. Furthermore, it was able to find a solution
satisfying all the constraints disregarding costs. Re-
garding the application of QA to the Tail Assignment
Problem, no studies were found so the present study
aims to fulfil such a gap.
3 PROBLEM STATEMENT
The Tail Assignment Problem consists of assigning
individual aircraft to flights, while ensuring multiple
constraints and minimising a certain objective func-
tion. Due to the complexity of this problem, in this
paper it was considered a simplification of it, for il-
lustration purposes. Therefore, we considered the fol-
lowing constraints and optimisation criteria.
3.1 Assignment Constraints
Assignment constraints are responsible for ensuring
that no activities are left unassigned and that each ac-
tivity is attributed only to one tail. Following this con-
straint, it is possible to guarantee that each and every
flight is allocated exclusively to one aircraft.
3.2 Connection Constraints
Connection constraints are the most basic ones and
can be as simple as ensuring that two activities may
connect one another, i.e., can be operated in sequence.
Such constraints can be thought as a graph where
each node represents an activity and two connected
nodes are representative of two activities that can be
assigned to a same aircraft in sequence, taking into
account the turnaround time, i.e., the minimum time
needed for an aircraft to park in a gate between two
consecutive activities.
3.3 Maintenance Constraints
Maintenance constraints ensure that each aircraft per-
forms the required maintenance tasks. In fact, main-
tenance tasks can be divided in two major groups
namely Minor Maintenance Tasks and Pre-assigned
Maintenance Tasks. The first group happen every
flight or up to once a week and take up to few
hours. The second group of maintenance tasks are
considered as long-term tasks and take from one day
up to months to be completed, and happening more
sparsely in time, are normally planned ahead in time.
In this study we just take into account Pre-assigned
Maintenance Tasks, that specify the aircraft that has
to perform it as well as the exact time and location it
must happen.
3.4 Flight Restriction Constraints
Flight Restriction Constraints aim to ensure that each
flight is operated by a valid tail. In fact, each flight
may require a certain fleet, minimum number of seats,
among others. In this study we considered that each
flight has a minimum fleet required as well as a min-
imum number of seats corresponding to the expected
demand of the flight. Therefore, tails belonging to
a smaller fleet or not having the required number of
seats were considered as being unfit to perform the
flight.
3.5 Optimisation Criteria
The optimisation criteria is considered as the min-
imisation of an objective function (OF) which com-
prises the costs that represent the bigger impact on
the schedule. Those are the costs related to the execu-
tion of the flights by the different tails. In fact, due to
the variability of the tickets price and lack of informa-
tion of the real tickets price, we decided to take into
consideration only the biggest costs that really affect
the sustainability and profitability of the airline com-
panies. For a matter of simplicity, we divided them
into three parts: Ground costs, Flying costs, and Take-
off/Landing costs.
Therefore the objective function is defined as in
equation 3, where F represents the set of all flights
that must be performed.
OF =
F
∑
f
(Ground costs
f
+ Flying costs
f
+ Takeo f f /Landing costs
f
)
(3)
4 TAIL ASSIGNMENT PROBLEM
AS A QUBO MODEL
Modelling the Tail Assignment Problem as a QUBO
model is set in three sequential steps, described below.
4.1 Formulate the Objective and
Constraints
Starting by Formulate the Objective and Constraints,
we specify and systematize the optimisation criteria
and groups of constraints.
A QUBO Model to the Tail Assignment Problem
901

Objective. To minimise the considered operational
costs of the schedule, i.e., to get a solution such that
each flight is performed by the tail with the lowest
execution cost.
The group of constraints to be considered are
specified as follows:
• Assignment Constraints: each flight must only
be performed by one tail;
• Connection and Maintenance Constraints: a
tail must have a valid schedule regarding arrival
and departure locations and times of activities.
This group of constraints can be divided into three
subgroups as follows:
– Impossible Pairing Activities: activities that
have no valid path between them and therefore
cannot be performed by the same tail.
– Impossible Flights Due to Maintenance: as
maintenance is obligatory, tails cannot perform
flights that have no valid path for all their main-
tenance tasks;
– Activity Path Consistency: all consecutive ac-
tivities performed by a tail must follow a valid
path;
• Flights Requirement Constraints: since each
flight has a pre-defined minimum fleet and a min-
imum number of seats required, a tail that meets
such requirements is needed to perform the flight.
4.2 Redefine the Problem into a Binary
Concept
To Redefine the Problem to a Binary Concept it is nec-
essary to convert it from an optimisation problem into
a decision problem. We state it as follows: ”Should
flight X be performed by tail Y?”. Each variable of
the problem is then represented as the possible assign-
ment of a certain flight to a specific tail.
Considering a set of F flights labeled as f =
1,...,F and a set of T tails labeled as t = 1,...,T , we
define the binary variables q
f ,t
as the variables of the
problem’s domain.
4.3 Transform the Problem into a
QUBO Model
The QUBO model is set as the sum of six different
terms that represent both the constraints and the ob-
jective function as presented in equation 4.
H = γH
A
+ ηH
B
1
+ λH
B
2
+ τH
B
3
+ φH
B
4
+ ψH
C
(4)
From this equation γ, η, λ, τ, φ and ψ represent posi-
tive real-valued numbers used to tune the relative im-
portance of each term in the global QUBO model. H
A
is related to the assignment constraints. On its turn,
terms H
B
1
, H
B
2
, H
B
3
and H
B
4
assure connection and
maintenance constraints. Finally, H
C
is the term re-
sponsible for setting a lower energy value to solutions
with minimum value of the objective function. To as-
sign values to the different tuning parameters, it is im-
portant to take into account that some of them may
incorrectly affect the others. For example, the value
of ψ must be chosen sufficiently small not to violate
any of the constraints of the problem. We considered
the following notation:
T : set of all tails
F : set of all flights
M : set of all maintenance tasks
T
f
⊂ T : subset of T that can perform flight f
F
t
⊂ F : subset of F that can be performed by a
given tail t
I
f
⊂ F
t
: subset of F
t
that cannot be assigned to-
gether with a given flight f
IC
f
⊂ F
t
: subset of F
t
that is indirectly connected
to a given f on the corresponding tail’s connection
network graph
M
t
⊂ M : subset of M that must be performed by
a given tail t
4.3.1 Assignment Constraints
Since it is desirable that all flights are assigned to ex-
actly one tail, a quadratic penalty is introduced for
schedules not meeting such condition. The imple-
mentation of this constraint follows equation 5.
H
A
=
F
∑
f
T
f
∑
t
q
f ,t
− 1
!
2
(5)
Such constraint is set by defining a negative bias to
each one of the individual variables q
f ,t
and a positive
coupling strength, for each pair of variables { q
f ,t
;
q
f ,t
0
}, where the latter corresponds to the assignment
of a same flight to different tails.
4.3.2 Connection and Maintenance Constraints
As previously described, connection and maintenance
constraints aim to guarantee that the obtained solution
is valid regarding sequences of flights while ensuring
all pre-defined maintenance tasks can be performed
by the correspondent tail.
ICAART 2021 - 13th International Conference on Agents and Artificial Intelligence
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Impossible Pairing Activities. This subgroup of
constraints is set to penalise the cases where two
flights ( f , f
0
) that have no valid path between them
are assigned simultaneously to the same tail (t). As
defined in equation 6, it penalises variables that rep-
resent non-pairable flights.
H
B
1
=
T
∑
t
F
t
∑
t
I
f
∑
f
0
q
f ,t
q
f
0
,t
(6)
Activity Path Consistency. To ensure that the cho-
sen schedule is valid regarding path consistency, an
extra penalty is added on paths that are not valid. It
can be divided in the following subgroups: Path con-
sistency between non-consecutive flights, Path con-
sistency between maintenance tasks and Path consis-
tency between flight and maintenance task
Path Consistency between Non-consecutive
Flights. To ensure path consistency between two
non-consecutive flights, equation 7 penalises all pairs
of indirectly connected flights ( f , f
0
) assigned to
a tail (t) that do not have a valid path assigned to
that same tail. It is done by penalizing cases where
p( f , f
0
,t), which corresponds to the penalty function
of the boolean expression p
b
( f , f
0
,t) in equation 8,
has a value different than 1.
H
B
2
=
T
∑
t
F
t
∑
f
IC
f
∑
f
0
(p( f , f
0
,t)− 1)
2
(7)
p
b
( f , f
0
,t) = ((q
f ,t
∧ q
f
0
,t
) ⊕ 1) ∨ w (8)
In equation 8, w = q
f
1
,t
∨ q
f
2
,t
∨ ... ∨ q
f
n
,t
and
[ f
1
,..., f
n
] ∈ PF where PF represents the set of flights
that can be reached from flight f and are directly con-
nected to flight f
0
, when both flights are assigned to
tail t.
Translating the desired boolean expression de-
fined in 8 to a penalty function, we use the conver-
sions defined in Table 1, where x
3
represents the out-
put auxiliary variables that take the value of the rela-
tionship between the two inputs (x
1
and x
2
). As pre-
viously presented, the first two penalty functions can
be found in the literature, whereas the third penalty
function was constructed for this study based on the
penalty functions of the trivial boolean relationships.
Table 1: Boolean penalty functions for the QA approach.
Classical Constraint Equivalent Penalty
x
3
⇔ x
1
∧ x
2
P(x
1
x
2
− 2(x
1
+ x
2
)x
3
+ 3x
3
)
x
3
⇔ x
1
∨ x
2
P(x
1
x
2
+ (x
1
+ x
2
)(1 −2x
3
) +x
3
)
x
3
⇔ (x
1
⊕ 1)∨ x
2
P(−x
1
x
2
− x
3
+ 2x
1
x
3
− 2x
2
x
3
− x
1
+ 2x
2
+ 1)
Path Consistency between Maintenance Tasks.
To guarantee a valid path between obligatory main-
tenance tasks, a new penalty is added for the cases
where none of the flights that guarantee such path is
assigned to the considered tail, as in equation 9.
H
B
3
=
T
∑
t
M
t
∑
m,m
0
(g(m,m
0
,t)− 1)
2
(9)
In this equation g(m,m
0
,t) corresponds to the penalty
function of the boolean expression g
b
(m,m
0
,t) pre-
sented in equation 10
g
b
(m,m
0
,t) = (q
f
1
,t
∨ q
f
2
,t
∨ ...
∨q
f
n
,t
) ∧ (q
f
0
1
,t
∨ q
f
0
2
,t
∨ ...∨ ...q
f
0
n
,t
)
(10)
where [ f
1
,. . . , f
n
] ∈ MPF and [ f
0
1
,..., f
0
n
] ∈ MNF. On
one hand, MPF represents the set of flights that are
part of a valid path between the two maintenance tasks
( m and m
0
) and are directly connected to m on the
tail’s connection network graph. On the other hand,
MNF represents the set of flights that are part of a
valid path between two maintenance tasks ( m and m
0
) and are directly connected to m
0
on the correspond-
ing tail’s connection network graph. Setting boolean
expression defined in equation 10 can be done using
the penalty functions from Table 1.
Path Consistency between Flight and Maintenance
Task. It is necessary to guarantee that any flight that
is not directly connected to an existent maintenance
task, is only assigned to the tail that must execute such
maintenance task if the valid path between both activ-
ities is part of the same assignment. Equation 11 sets
a penalty for solutions that do not follow such con-
straint,
H
B
4
=
T
∑
t
F
t
∑
f
M
t
∑
m
(r( f ,m,t) − 1)
2
(11)
where r( f ,m,t) corresponds to the penalty function of
the boolean expression r
b
( f , m,t) presented in equa-
tion 12,
r
b
( f , m,t) = (q
f ,t
⊕ 1) ∨ (q
f
0
1
,t
∨
∨q
f
0
2
,t
∨ ...∨ q
f
0
n
,t
)
(12)
where [ f
0
1
,. . . , f
0
n
] ∈ MCF, with MCF representing
the set of flights that are directly connected to main-
tenance task m and have a valid path to flight f on the
corresponding tail’s connection network graph.
Analysing all constraints as a whole, some redun-
dancies may appear. Therefore, to minimise the num-
ber of variables of the problem, while implementing
A QUBO Model to the Tail Assignment Problem
903

2,A
3,A 4,A 5,A 6,A 2,B 3,B 4,B 5,B 6,B 7,B 8,B 2,C 3,C 4,C 5,C 6,C 7,C 8,C
2,A -1 1 1 2 2
3,A -1 1 1 2 2
4,A -1 2 2
5,A -1 2 2
6,A -1
2 2
2,B -1 1 1 1 1 2
3,B -1 1 1 1 1 2
4,B -1 1 1 2
5,B -1 1 1 2
6,B -1 2
7,B -1 2
8,B -1
2
2,C -1 1 1 1 1
3,C -1 1 1 1 1
4,C -1 1 1
5,C -1 1 1
6,C -1
7,C -1
8,C -1
Figure 1: Partial QUBO model matrix for the illustrative
example: 2,A corresponds to the variable represented by
the assignment of flight 2 to tail A. The colored entries cor-
respond to the coefficients from Equations (5) and (6) and
empty entries correspond to coefficient 0.
each constraint, it is necessary to verify whether or
not that constraint is already assured.
Figure 1 shows a QUBO model of the illustrative
problem for some of the constraints, in the form of an
upper-triangular matrix. Furthermore, every coeffi-
cient is multiplied by the tuning parameter associated
to the constraint it represents.
Objective Function. Each variable q
f ,t
gets its bias
increased depending on how smaller is the operational
cost of the assignment it represents when compared
with the maximum cost possible for performing flight
f . Equation 13 represents the associated penalty func-
tion,
H
C
=
F
∑
f
T
f
∑
t
execcost
f ,t
maxcost
f
q
f ,t
(13)
where execcost
f ,t
represents the cost of performing
flight f on tail t, and maxcost
f
represents the maxi-
mum cost of performing flight f in any of the possible
tails.
Therefore for each flight, H
C
will take the value of
1 for the most expensive assignment.
5 TESTS AND RESULTS
To verify and analyse the proposed implementation it
was considered different sets of tests based on a real
dataset. Such dataset was provided by TAP Air Por-
tugal
1
and was composed by 1965 flights, 30 main-
tenance tasks to be performed by 42 different tails. In
fact, the considered dataset belongs to a well-known
airline company which usually operates in a hub-and-
spoke network.
1
https://www.flytap.com/
As this initial dataset was too large to perform the
desired tests, it was necessary to define some rules
selecting partial datasets data while keeping the pro-
portions of a real scenario. Therefore, this data was
selected based on the fact that for the original dataset
on average a tail should perform two flights per day.
To perform the tests, two different subsets were con-
sidered. On the one hand, dataset A was composed
by a group of 10 tails and a set of 266 flights. On the
other hand, dataset B, a bit larger that included 442
flights.
In Table 2 we present the values of the tuning pa-
rameters used during the performed tests. Since no
rules were possible to set on the values of these pa-
rameters, they were found empirically.
Table 2: Tuning parameters’ values for QUBO model.
γ η λ τ φ ψ
5 8.5 3 4 4 0.3
The main goal of the tests was to compare the so-
lutions obtained by solving the proposed implemen-
tation using three different solvers, one classical and
two hybrid, all part of D-wave’s toolchain. The cho-
sen classical solver was the SimulatedAnnealingSam-
pler
2
whereas the hybrid solvers were D-Wave Hy-
brid Solver Service (HSS)
3
and KerberosSampler
4
.
Both hybrid solvers made use of a D-Wave 2000Q
quantum annealer, composed by 2048 qubits, which
QPU is organized in a Chimera topology.
For doing such analysis we run for each test case
(i.e, each dataset and each solver), 10 shots to have a
considerable accuracy on the obtained results.
5.1 Finding a Feasible Solution
As presented in Table 3, the probability of each one of
the solvers to find a valid solution is high. However
for the biggest dataset such probability becomes re-
ally low when solving the problem using SASampler
and Kerberos.
Table 3: Probability of finding valid solutions.
Prob. valid solutions
Dataset SASampler Kerberos HSS
A 1 1 0.8
B 0.2 0.3 1
2
https://docs.ocean.dwavesys.com/projects/neal
3
https://docs.dwavesys.com/docs/latest/
doc leap hybrid.html
4
https://github.com/dwavesystems/dwave-
hybrid/blob/master/hybrid/reference/kerberos.py
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5.2 Solutions’ Distribution
Since the main goal of solving the Tail Assignment
Problem is not just to get a valid solution but also a
solution with minimum cost, it is also important to
analyse the quality of the the obtained solutions from
the three solvers. When considering complex prob-
lems, the possibility of finding a good but still non-
optimal solution is relevant. Therefore, we analysed
the distribution of the feasible solutions obtained for
each one of the datasets when solved using the three
solvers aforementioned.
Regarding dataset A, as presented in Figure 2, for
SASampler and Kerberos some outliers were identi-
fied. In fact, the majority of the valid solutions ob-
tained had similar OF costs and, therefore, the solu-
tion with minimum OF cost obtained for SASampler
and the solution with maximum OF cost obtained for
Kerberos were considered outliers. For HSS, the solu-
tions found shown a higher variability with 25% of the
solutions having an OF cost of less than 29.334.216
units of cost (UC). It is also relevant to note that, for
HSS, the most expensive solutions found have an as-
sociated OF cost considerably higher when compared
with the most expensive solution obtained by any of
the other two solvers. Summing up, HSS has a bigger
probability of finding cheaper solutions when com-
pared with the other solvers used. Nonetheless, this
solver also has some drawbacks as it found a big num-
ber of expensive solutions.
Figure 2: Box plot of the OF costs regarding the obtained
solutions from the three solvers for dataset A.
Finally, regarding dataset B, we tried to verify if the
same previous conclusions were also true for bigger
datasets. As presented in Figure 3, for SASampler and
Kerberos, the obtained solutions are highly concen-
trated in terms of OF costs indicating that their OF
costs are similar. This may have happened because
both of these solvers were only able to find a few valid
solutions. Furthermore, for SASampler, the solutions
found had lower OF costs when compared with the
solutions found by Kerberos. Regarding HSS, it was
able to find valid solutions for all the 10 shots. Fol-
lowing the same trend verified for the other datasets,
HSS found solutions with a big range of OF costs. Al-
though the most expensive solution found for dataset
B was obtained by this solver, 50% of the solutions
found had a lower OF cost when compared with any
of the solutions obtained by Kerberos. Comparing
HSS with SASampler, both of them were able to find
the minimum cost solution. However, as previously
analysed, SASampler was able to find a valid solution
only 20% of the times. Thus, it may indicate that us-
ing SASampler for big datasets can lead to multiple
unfeasible solutions.
Figure 3: Box plot of the OF costs regarding the obtained
solutions from the three solvers for dataset B.
Summing up, the results shown that when comparing
the classical solver SASampler with the hybrid solvers
Kerberos and HSS none of the solvers revealed be-
ing perfect for finding the best solutions. Nonethe-
less, they diverge on the performance when solving
the problem considering different datasets. In fact, al-
though all the solvers were able to find feasible solu-
tions for all the datasets, none of them was able to find
the solution with minimum total cost in the majority
of the shots. Kerberos was the one that performed
the worst, as the distribution of the results obtained
were highly expensive for the majority of the runs
and therefore were not satisfactory. SASampler re-
vealed better results, being able to find the minimum
energy solution at least once for the datasets. For big-
ger datasets, this solver revealed not to be so useful
to solve this QUBO model as the majority of the so-
lutions found were either unfeasible or considerably
more expensive than the cheapest solution found. Fi-
nally, regarding HSS, it was the solver that performed
the best overall. However, it is relevant to note that
some of the solutions found by this solver were highly
expensive which represent a drawback on the confi-
dence on the usage of this solver to this problem.
A QUBO Model to the Tail Assignment Problem
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6 CONCLUSIONS AND FUTURE
WORK
We have presented a formulation of the Tail Assign-
ment Problem as a QUBO model to be solved using
QA based on the current commercial solution devel-
oped by D-Wave Systems. Our results demonstrate
that using an hybrid solver for the proposed QUBO
model may represent a considerable advantage in the
probability of finding valid non-expensive solutions
when compared with a classical solver. Nonetheless,
it is relevant to note that, even though it may show
encouraging results, quantum computation is in an
early stage and, therefore, the current limitations do
not allow scaling it to complete real-world datasets.
Throughout this study, we opted to implement mul-
tiple simplifications to narrow the scope of the prob-
lem in analysis. A non-consideration of minor main-
tenance tasks is not realistic as they have to occur in
a real scenario. Furthermore, a robust approach may
be a pivotal achievement as flight delays are frequent
and tight schedules can be significantly affected by
that. Additionally, to understand the effectiveness in
a deeper level of the proposed modelling technique,
when applied to multiple solvers using different sce-
narios, it would be important to run more tests using
different datasets. Finally, as HSS revealed to perform
better for solving this problem than using only a clas-
sical algorithm such as SA, further studies on hybrid
solvers could be relevant for a better understanding
on the real advantage of using such technique to solve
complex optimisation problems.
REFERENCES
Albash, T. and Lidar, D. A. (2018). Adiabatic quantum
computation. Reviews of Modern Physics, 90(1).
Chapuis, G., Djidjev, H., Hahn, G., and Rizk, G. (2019).
Finding maximum cliques on the d-wave quantum
annealer. Journal of Signal Processing Systems,
91(3):363–377.
Cruz-Santos, W., Venegas-Andraca, S. E., and Lanzagorta,
M. (2019). A QUBO Formulation of Minimum Multi-
cut Problem Instances in Trees for D-Wave Quantum
Annealers. Scientific Reports, 9(1):1–12.
Froyland, G., Maher, S. J., and Wu, C. L. (2013). The recov-
erable robust tail assignment problem. Transportation
Science, 48(3):351–372.
Gr
¨
onkvist, M. (2005). The tail assignment problem.
PhD thesis, Chalmers University of Technology and
G
¨
oteborg University.
Ikeda, K., Nakamura, Y., and Humble, T. S. (2019). Appli-
cation of quantum annealing to nurse scheduling prob-
lem. Scientific Reports, 9(1):12837.
Kadowaki, T. and Nishimori, H. (1998). Quantum an-
nealing in the transverse ising model. Phys. Rev. E,
58:5355–5363.
Lucas, A. (2014). Ising formulations of many NP problems.
Frontiers in Physics, 2(February):1–14.
Montoito, F. (2016). Application of the Simulated Anneal-
ing with Adaptive Local Neighborhood Search to the
Tail Assignment Problem The Case Study of TAP.
Master’s thesis, Instituto Superior T
´
ecnico, Universi-
dade de Lisboa.
Neukart, F., Compostella, G., Seidel, C., von Dollen, D.,
Yarkoni, S., and Parney, B. (2017). Traffic flow opti-
mization using a quantum annealer. Frontiers in ICT,
4:29.
Papalitsas, C., Andronikos, T., Giannakis, K.,
Theocharopoulou, G., and Fanarioti, S. (2019).
A QUBO model for the traveling salesman problem
with time windows. Algorithms, 12(11):1–18.
Ray, P., Chakrabarti, B. K., and Chakrabarti, A. (1989).
Sherrington-kirkpatrick model in a transverse field:
Absence of replica symmetry breaking due to quan-
tum fluctuations. Phys. Rev. B, 39:11828–11832.
Ruther, S., Boland, N., Engineer, F. G., and Evans, I.
(2017). Integrated aircraft routing, crew pairing, and
tail assignment: Branch-and-price with many pricing
problems. Transportation Science, 51(1):177–195.
Stollenwerk, T., Lobe, E., and Jung, M. (2019). Flight
Gate Assignment with a Quantum Annealer. Lecture
Notes in Computer Science (including subseries Lec-
ture Notes in Artificial Intelligence and Lecture Notes
in Bioinformatics), 11413 LNCS:99–110.
Tran, T. T., Do, M., Rieffel, E. G., Frank, J., Wang,
Z., O’Gorman, B., Venturelli, D., and Beck, J. C.
(2016). A hybrid quantum-classical approach to solv-
ing scheduling problems. In SOCS.
Ushijima-Mwesigwa, H., Negre, C. F. A., and Mniszewski,
S. M. (2017). Graph partitioning using quantum an-
nealing on the d-wave system. In Proceedings of the
2nd International Workshop on Post Moores Era Su-
percomputing, page 22–29, New York, USA. Associ-
ation for Computing Machinery.
Venturelli, D., Marchand, D. J. J., and Rojo, G. (2016). Job
Shop Scheduling Solver based on Quantum Anneal-
ing. arXiv:1506.08479 [quant-ph].
Vikst
˚
al, P., Gr
¨
onkvist, M., Svensson, M., Andersson, M.,
Johansson, G., and Ferrini, G. (2019). Applying the
Quantum Approximate Optimization Algorithm to the
Tail Assignment Problem. arXiv:1912.10499 [quant-
ph].
Yadav, K. K. (2017). An Effective and Efficient Solution
to Tail Assignment Problem. International Journal
of Advanced Research in Computer Science and Soft-
ware Engineering, 7(4):378–386.
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