Davide Taviani
June 22, 2012
140

# Multifleet routing and multistop flight scheduling for schedule perturbation

What can we do in order to minimize the losses if some plane becomes suddenly unavailable?

June 22, 2012

## Transcript

1. 1
Multiﬂeet routing and multistop ﬂight
scheduling for schedule perturbation
Shangyao Yan, Yu-ping Tu (1995)
Davide Taviani
June 22, 2012

2. 2
Flight perturbation
Fact: Perturbations (some airplane suddenly unavailable) in ﬂight
schedules occur.
Causes:
meteorological conditions
congestion at the airport (also poor gate assignment schedule)
late or absent crew members
sudden war / terrorist threat

3. 3
Flight perturbation
How can we minimize our losses?
Some of the previous research:
local improvements of ﬂight scheduling
minimization of total passenger delay (nonlinear integer problem,
diﬃcult to solve for large instances)
development of greedy heuristics (minimizing ﬁrst the number of
cancelled ﬂights, then the overall passenger delay)
time-space framework and successive shortest path to cancel a
series of ﬂights (no indication on when we can resume normal
operations, only feasible solution for shortage of more than one
aircraft)

4. 4
Flight perturbation
Problem: all of these solutions consider only single ﬂeets (one type
of airplane).
In reality there are several types of airplanes which can support each
other (i.e. an idle large aircraft can serve ﬂights scheduled for a
missing small airplane, but not the other way around).
Our model: BMSPM (Basic Multiﬂeet Schedule Perturbation Model)

5. 5
The framework
Initial information:
Regular schedule
Incident informations
available aircraft

6. 5
The framework
Initial information:
Regular schedule
Incident informations
available aircraft
Allow ﬂight cancellation
and construct BMSPM

7. 5
The framework
Initial information:
Regular schedule
Incident informations
available aircraft
Allow ﬂight cancellation
and construct BMSPM
Solution of the problem (LRS)

8. 5
The framework
Initial information:
Regular schedule
Incident informations
available aircraft
Allow ﬂight cancellation
and construct BMSPM
Solution of the problem (LRS)
Is our
solution
satisfac-
tory?

9. 5
The framework
Initial information:
Regular schedule
Incident informations
available aircraft
Allow ﬂight cancellation
and construct BMSPM
Selection of
Delay of ﬂights
Ferry of spare
aircraft
Multistop ﬂight
modiﬁcation
Network modiﬁcation
Solution of the problem (LRS)
Is our
solution
satisfac-
tory?
no

10. 5
The framework
Initial information:
Regular schedule
Incident informations
available aircraft
Allow ﬂight cancellation
and construct BMSPM
Selection of
Delay of ﬂights
Ferry of spare
aircraft
Multistop ﬂight
modiﬁcation
Network modiﬁcation
Solution of the problem (LRS)
Is our
solution
satisfac-
tory?
Schedule ﬂights
no
yes

11. 6
Singleﬂeet time-space network
Assumption: for simplicity, only one plane is suddenly unavailable.
The pertubation is characterized by:
starting time: the airplane becomes unavailable
recovery time: the airplane is back (e.g. repaired)
ending time: the regular schedule resumes
We construct a multicommodity ﬂow network (multiple
supply/demand nodes).

12. 7
Singleﬂeet time-space network
Nodes:
(1) initial supply:
airplanes at airport
at starting time
(2) intermediate
supply: airplanes
ﬂying at starting
time, recovered
airplane
(3) intermediate
demand: airplane
ﬂying at ending time
(4) ﬁnal demand:
airplane at airport at
ending time

13. 8
Singleﬂeet time-space network
Arcs:
(5) position arc: the
planes travels empty
(6) ﬂight arc: normal
ﬂight with passenger
(7) ground arc: the
airplane remains on
ground
(8) overnight arc: the
airplane remains on
ground for the night

14. 9
Singleﬂeet time-space network
Given the arc (i, j) with cost Cij
, with the amount of ﬂow Xij
arc cost capacity
ﬂight Ccij
+ (Cij
− Ccij
)Xij
0 ≤ Xij
≤ 1
ground airport tax + various charges 0 ≤ Xij
≤ Xmax
overnight ground + overnight charge 0 ≤ Xij
≤ Xmax
position ﬂight - passenger revenue 0 ≤ Xij
≤ Xmax
Ccij
: cost of cancellation (e.g. passenger reimbursment)
Xmax
: maximum capacity of the destination airport

15. 10
Multiﬂeet time-space network
A singleﬂeet time-space net-
work for each type of aircraft.
More types of edges for big-
ger airplanes (they can per-
form also the duties of smaller
ones).
Example on the left: one
plane of type B is unavailable.

16. 11
Multiﬂeet time-space network
Remarks on BMSPM:
Using a large aircraft instead of a smaller one can have additional
costs (e.g. changing gate/crew) that should be taken into account
in the cost of the edge.
Once we have constructed our network, we proceed by computing
a minimum integer cost ﬂow: since we consider passenger
revenues to be negative, we are eﬀectively maximizing the total
proﬁt.

17. 12
Problem: Just cancelling the ﬂight of the unavailable airplane may
be too expensive.
We add ﬂexibility to our model by allowing:
delay of ﬂights
ferrying of idle aircrafts
modiﬁcation of multistop ﬂights

18. 13
We modify the network
(1) Alternative
delayed ﬂight arc
(2) Set of alternative
ﬂights

19. 14
Parameters of these arcs: similar to the original ones but with
additional delay costs and potential losses of passenger revenues.
Additional constraints: at most one ﬂight among these
alternatives can be chosen.
The number of sliding arcs can be set independently for each
ﬂight:
the problem
usually set according to carrier experience and needs

20. 15
Additional strategies: ferrying of idle airplanes
Whenever a plane is idle (i.e.
uses a ground arc) we might
consider to bring it some-
where else to load and trans-
port passengers.
We add more position arcs to
our networks: same costs and
capacities as before.
needed.

21. 16
Multistop ﬂights: obtained by
We go from i to j, stop brieﬂy,
and then go to k.
We can still perform independent
ﬂights, but if we join them in a
multistop ﬂight we obtain an ad-
ditional revenue (less charges).

22. 17
solve our problem, we add the
edge ik, allowing also a direct
ﬂight between the two airports
(i.e. we cancel the stop in j).
New constraints:
at most one choice between
ik and ijk
ﬂights from i to a and from
b to k must be served at
most once

23. 18
Solution of the problem: LRS
Issues of the problem:
integer multi-commodity network ﬂow problem is NP-hard
we want quickly a “good enough” solution to cope with
emergency of unavailable ﬂight(s)
Our strategy: Lagrangean Relaxation with Subgradient methods (LRS)
fast convergence
eﬃcient allocation of memory space

24. 19
The LRS works as follows:
1) Find a lower bound using Lagrangean relaxation;
2) Find an upper bound (feasible solution) starting from the lower
one;
3) Reduce the gap between the bounds by modifying the Lagrangean

25. 20
LRS: Lower bound
We use Lagrangean relaxation for the side constraints.
The Lagrangean problem can then be decomposed into several
independent network ﬂow subproblems, and solved with the eﬃcient
network simplex method.
The optimal objective of such subproblems is a lower bound on our
original one.

26. 21
LRS: Upper bound
We start from the lower bound, typically unfeasible, and use a
shortest path algorithm to ﬁnd a least cost ﬂow augmenting circuit
passing a speciﬁed arc.
If we have an unfeasible solution, then there must be (at least) a side
constraint (a bundle of arcs) violated, so more than one arc in this
bundle has 1 unit of ﬂow. For all those violated constraints:
1) we specify one arc between those with the largest cost (after being
modiﬁed by the Lagrangean relaxation) and reduce the ﬂow to 0.
2) to maintain ﬂow conservation, we ﬁnd a least cost ﬂow augmenting
path from the arc tail to the arc head and augment a unit of ﬂow
troughout the path.
3) If the side constraint is not yet satisﬁed, we repeat the procedure.
The networks are designed to have feasible solutions, so we can always
ﬁnd an initial upper bound.

27. 22
Obtaining the ﬁnal schedule
After we solved the problem we have
a good enough “ﬂeet ﬂow”.
We can use a ﬂow decomposition al-
gorithm to decompose the link ﬂows
into arc chains, each representing
the route of one airplane in the
perturbed period (routes are not
unique).
To reﬁne the solution we can employ
other choices (which depend on the
case considered) to get these routes:
e.g. “ﬁrst in, last out” for some
plane that may need extra mainte-
nance between ﬂights.

28. 23
Case study
Based on data from a major Taiwan airline’s international operations
(China Airlines) of 1993.
Data:
24 cities
weekly timetable
273 ﬂights ( 20% onestop)
Resulting problem size: 8635 nodes, 34067 arcs.
Results:
Most of the models converged to 1% gap in less than 30 minutes.
Four simpler models optimized within a minute