Qubits Europe 2019
DLR
Knowledge for Tomorrow
Flight Gate Assignment with a Quantum Annealer
Elisabeth Lobe, Tobias Stollenwerk
High Performance ComputingSimulation and Software TechnologyDLR German Aerospace Center
26th March 2019
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Algorithmic Quantum Computing Research at DLR
Quantum Optimization Algorithms
Quantum Compiling
Embedding strategies for Quantum Annealing
Complete graph in broken Chimera
Weight distribution problem
DLR
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Aerospace Applications at DLR
for Quantum Annealing
Air Traffic ManagementSatellite Telemetry VerificationEarth Observation Mission PlanningFlight Gate Assignment
for Gate-Based Quantum Computing
QAOA for scheduling problemsHHL for Radar Cross SectionQuantum Simulation for Battery Research
DLR
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Flight Gate Assignment
A day at Frankfurt Airport
about 1300 aircraft movements (arrival and departure)
more than 90% are passenger flights
more than 170000 passengers
about 60% transfer passengers
278 gates
DLR
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Passenger Flows
�i
gate γ
Tαγ
gate α
Tαβ
gate β
securitypoint
baggageclaim
�j �k
tdepα tarr
α
Nji Nik
tini touti
�`
tin`tbuff
narrindep
i
F , G sets of flights and gates
ndep/arri passengers which depart/ arrive with flight i
Nij transfer passengers from flight i to j
tin/outi arrival/departure time of flight i
Tαβ average time to get from gate α to β
tdep/arrα average time to arrive at/ leave from gate α
tbuff buffer time between two flights at the same gate
Which flight should be assigned towhich gate, such that the total transit
time of the passengers is minimal?
A : F → G
DLR
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FGA Binary Program �i
γ
Nik·Tαγ
α
Tαβ ·Nji
β
� �
�j �k
tdepα ·n
depi
tdepα ·n
depi
narri ·t
arrα
narri ·t
arrα
Variables x ∈ {0, 1}F×G with
xiα =
{1, if flight i takes gate α,
0, otherwise
Minimizing the total transfer time with objective function
T (x) = Tarr(x) + Tdep(x) + Ttransfer(x)
=∑iα
narri t
arrα xiα +
∑iα
ndepi tdep
α xiα +∑ijαβ
NijTαβ xiα xjβ
=∑iα
narri t
arrα xiα +
∑iα
ndepi tdep
α xiα︸ ︷︷ ︸linear
+∑ijαβ
NijTαβ xiα xjβ︸ ︷︷ ︸quadratic
⇒ Quadratic Assignment Problemfundamental problem in combinatorial optimization, NP-hard
seems to exploit possible advantages of the D-Wave machine
DLR
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Constraints and Penalty Terms
1. One gate per flight∑α
xiα = 1 ∀i ∈ F
2. Different gates if standing times of two flights overlap forbidden pairs
P ={
(i, j) ∈ F 2 : tini < tinj < touti + tbuff
}xiα + xjα ≤ 1 ⇔ xiα · xjα = 0 ∀(i, j) ∈ P ∀α ∈ G
⇒ Penalty terms Cone(x) =∑i
(∑α
xiα − 1
)2
,
Cnot(x) =∑α
∑(i,j)∈P
xiαxjα where Cone/not
{> 0, if constraint is violated
= 0, if constraint is fulfilled
DLR
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QUBO with Penalty Weights
Q(x) = T (x) + λoneCone(x) + λnotCnot(x)
Need to ensure that a solution always fulfills constraints, hence ∆C > ∆T
⇒ Comparing coefficients in worst cases for
not assigning a flight to any gate
λone > maxi,α
(n
depi tdep
α + narri t
arrα +max
βTαβ
∑j
Nij
)assigning a pair of forbidden flights to the same gate
λnot > maxi,α,γ
(n
depi tdep
α − ndepi tdep
γ + narri t
arrα − narr
i tarrγ +max
β
(Tαβ − Tγβ
)∑j
Nij
)
⇒ Refinement by bisection of weights yielding valid or invalid solutions
DLR
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Airport Data
M. Jung et al. (DLR-FW)
Flight schedule for one day from a mid-sized European airport
Passenger flow from agent-based simulation of Martin Jung
Extracted total instance: 293 flights and 97 gates
⇒ Over 28000 binary variables with about 400 Mio. couplings
DLR
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Instance Preprocessing
0 20000 40000 60000time in seconds
0
50
100
150
200
250
Flig
hts
flights with transfer passengers
Splitting too long on-block timesReducing to only flights with transfersExtracting connected subgraphsFurther slicing of largest
subgraph randomly
⇒163 instances:3 to 16 flights
2 to 16 gates
DLR
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Bin Packing
2 4 6 8 10N
1.0
1.5
2.0
2.5
3.0
3.5
R
50 %90 %99 %100 %
Maximum coefficient ratio of QUBO CQ =maxij |Qij |minij |Qij |
Reducing maximum coefficient ratio to overcome precisionissues
Tαβ , tarrα , t
depα → {0, 1, ...,T } with T ∈ {2, 3, 6, 10}
Nij , narri , n
depi → {0, 1, ...,N} with N ∈ {2, 3, 6, 10}
⇒ CQ � CQ
Approximation ratio (solved with SCIP)
R =Q(argminxQ(x))
minxQ(x)
⇒ Little effect on solution quality
DLR
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Annealing Setup
10 8 6 4 2 0JF
0.0
0.2
0.4
0.6
p
25 %50 %75 %
EmbeddingQuadratic overhead
Up to 84 logical qubits
(#Variables = #Flights ·#Gates)
Intra-logical coupling (JF)Influences success probability p
Best option by scanning: -1 in units of largest coefficient
(Standard) Run parametersAnnealing time 20µs with 1000 runs
Majority voting
DLR
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Annealing Results
0 200 400 600 800 1000Maximum Ising coefficient ratio CI
0.0
0.2
0.4
0.6
0.8
1.0
Succ
ess
Pro
babili
ty p
3 4 5 6 7Number of flights |F|
100
200
300
400
500
T99 (
)
s
25 %50 %75 %
3 4 5 6 7Number of flights |F|
50
100
150
200
CI
25 %50 %75 %
QUBO to Ising transformationincreases maximum coefficient ratio significantly
⇒ large Ising coefficients suppress success probability
Time to solution with 99% certainty T99 = log1−p(1− 0.99)Tanneal
grows with problem size→ because of larger coefficients?
due to small problem sizes asymptotic behaviour unclear
DLR
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Summary
Flight gate assignment is amenable to QA
Precision issues due to large coefficients
Mitigate limited precision by bin packing
Open questions:
How to recombine partial solutions?
How would larger instance perform?
Are these instances hard for classical solvers?
DLR
DLR
Knowledge for Tomorrow
Questions? Related Article:
Elisabeth Lobe Flight Gate Assignment with a Quantum AnnealerT. Stollenwerk, E. Lobe and M. Jung, QTOP, Springer, 2019High Performance Computing
Simulation and Software TechnologyDLR German Aerospace Center
