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towing is not specified in the model, so the controllers may decide when to tow
the flight and do so when the procedure causes fewest conflicts.
The conflict constraint is defined as follows:
X
f
1
,g
1
+
gGR
f∈C
d
(
f
1
)
X
f,g
≤
M,
∀
f
1
∈
F
(
g
1
)
,
∀
g
1
∈
GR,
(7)
∀
GR
∈{
GR
1
, ..., GR
5
}
where
GR
is one of the gate groups from Figure 1 and
C
d
(
f
1
) is a set of flights
which can be in conflict with the departure time of flight
f
1
if the flights are
allocated to the same group of gates. The flights which are in
C
d
(
f
1
) are identified
from the scheduled arrival and departure times. It is assumed that arrivals up to
10 minutes before/after the departure of
f
1
can conflict (i.e., a 10 minute long
time gap is required). Similarly if an aircraft departs close to the departure time
of
f
1
it would be in
C
d
(
f
1
), but the time margin can be shorter because it is a
lessconflictingsituationsoa3minutelongtimegaphasbeenused.The10and
3 minute gaps can be set to any values desired, but have here been set to be large
values to investigate and illustrate the effects of this constraint. An analogous set
exists for arrival time conflicts so there is a separate set
C
a
(
f
1
) which contains all
flights which are in conflict with the arrival of
f
1
and a constraint equivalent to
Constraint 7 is applied. This idea of finding conflicting flights can also be viewed
as a type of time window method. The conflicts within time windows would be
identified by two windows for the arrival conflicts (short and long gaps) and
two windows for the departure conflicts. But in a typical time window approach
an artificial step size would usually be used to move the windows along the
problem whereas here the positions of the windows are implicitly defined by the
flight times, with consequent benefits of avoiding time discretisation problems.
Our constraint allows up to
M
flight movements for each group of gates. The
number of allowed flights should be chosen depending on the particular airport
preference. For this paper, the concept is the important element, resulting in
lower congestion on the taxiways close to the gates.
Objective.
The objective function is a weighted sum of four elements. It aims
to ensure that time gaps between allocated flights are large enough to absorb
minor delays, that the gate spaces are used effectively, that the airline preferences
are maximised and that the number of remote ('dummy gate') allocations is
minimised. The following function is minimised:
n
n
m
n
m
n
m
n
p
f
1
,f
2
U
g,f
1
,f
2
+
r
f,g
X
f,g
+
a
f,g
X
f,g
+
dX
f,dummy
f
1
=1
f
2
=1
g
=1
f
=1
g
=1
f
=1
g
=1
f
=1
(8)
The first component of the function refers to the time gaps, the variable
U
g,f
1
,f
2
is forced to be one if two flights are at the same gate (see Inequality 5).
The gap cost function
p
f
1
,f
2
that is used in the objective function to penalise
such allocations is given by Equation 9:
p
f
1
,f
2
=
90
(
gap
f
1
,f
2
if
10
<gap
f
1
,f
2
<
90
)+1
(9)
1
if gap
f
1
,f
2
≥
90
,
