Information Technology Reference
In-Depth Information
function. The following model QPDF (quadratic problem with direct flights),
which is established in practice, is an operational implementation of this idea.
It incorporates that the reference units are not single days but a set of time
intervals
I
=
, which are also used by the
forecasting module to estimate the market response. The market response during
time interval
i
{
365
−
181
,
180
−
121
,...,
7
−
0
}
I
for a price
p
is denoted as
MR
i
(
p
)
1
.
∈
Model 1: QPDF
max
i∈I
q
i
p
i
(1.1)
subject to
i∈I
q
i
≤ Q
(1.2)
q
i
≤ MR
i
(
p
i
)
(1.3)
p
365
−
181
≤ p
180
−
121
≤ ... ≤ p
7
−
0
(1.4)
i∈I
q
i
Q
≥ minLF
(1.5)
p
i
≥
0
∀i ∈ I
(1.6)
q
i
≥
0
∀
i
∈
I
(1.7)
The objective function (1.1) expresses the revenue calculated as price per
seat times the number of seats made available for that price. (1.2) ensures that
total aircraft capacity is not exceeded and (1.3) limits the number of seats for
each interval to the market response. (1.4) models that prices increase over the
booking period and with (1.5) we require a minimum loadfactor. Note that this
constraint is neglected in some optimization runs.
This model is rather straightforward and easy to understand. Yet, unfortu-
nately, it is computationally rather complex and impractical. First, the objective
function is quadratic and non-convex. Thus specific solvers are required and the
solution time is significantly larger than solution times for equal sized linear
programs. Even more critical is the fact that a solution will most likely be non-
integer and not be compatible with the prespecified price schema. Thus, it will
for instance recommend to sell 34.45 tickets for a price of 76.13 Euro. With a
solution like that, two problems arise. First, since it is not possible to sell 0.45
tickets the solution has either to be rounded, which may cause feasibility prob-
lems or the model has to be modified to an integer program. Defining
q
i
as an
integer variable turns the model into a quadratic mixed integer problem and
the solution becomes even more complex and time-consuming. Secondly, and
more importantly, an LCC in general does not want to sell tickets at arbitrarily
patched prices like 76.13 Euro. Germanwings, for instance, uses a predefined
schema of potential prices for all flights starting from 19.99 Euro and increasing
in specific steps including 69.99 Euro and 79.99 Euro, for instance.
1
For the notation we refer to [9].
