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valid prices used by the LCC then the model has to specify capacities
q
f,i,p
for
each price, flight and time interval. We introduce a binary parameter
u
f,l
with
u
f,l
= 1 if flight
f
uses leg
l
and
u
f,l
=0otherwise.
Model 3: LPCF
max
f∈F
p · q
f,i,p
(3.1)
i∈I
p∈P
subject to
f∈F
u
f,l
· q
f,i,p
≤ Q
l
∀l ∈ L
(3.2)
i∈I
p∈P
q
f,i,p
≤ d
f,i,p
· b
f,i,p
∀p ∈ P,f ∈ F,i ∈ I
(3.3)
b
f,i,p
=1
∀f ∈ F,i ∈ I
(3.4)
p∈P
b
f,i,p
≤ B
f,p
∀f ∈ F,i ∈ I,p ∈ P
(3.5)
B
f,p
≤ n
∀f ∈ F
(3.6)
p∈P
q
f,i,p
≥ sold
f,i,p
∀p ∈ P,f ∈ F,i ∈ I
(3.7)
b
f,i,p
≤
p≤p
0
b
f,i−
1
,p
∀f ∈ F,p
0
∈ P,i ∈ I\{
1
}
(3.8)
p≤p
0
B
f,p
∈{
0
,
1
}
∀p ∈ P,f ∈ F
(3.9)
b
f,i,p
∈{
0
,
1
}
∀p ∈ P,f ∈ F,i ∈ I
(3.10)
q
f,i,p
≥
0
,
integer
∀p ∈ P,f ∈ F
(3.11)
In the objective function we sum revenue over all flights. While class limits
are on flights, seat capacity constraints are on legs. Here we allow each leg to
have a different capacity
Q
l
which gives flexibility for use of different aircraft
types. As for the first model no rounding is necessary, the model can be solved
by standard MIP-solvers, yet, the dimension of an instance is now depending on
the number of prices in schema
P
, the number of flights
F
and the number of
time intervals to be optimized. We will report on computational results in the
next section.
4 Application - A Simulation Study
We have compared the models introduced in the previous section using real data
provided by Germanwings. This data covers a subset of the flight network with
six direct flights and five connecting flights between seven European airports.
The starlike network is shown in Figure 4. All flights are scheduled on a weekly
basis and operated up to 3 times a day which results in a total of 118 direct
flights per week.
