Geoscience Reference
In-Depth Information
X
x ik ¼
1
8 i 2 U
ð 10 Þ
k 2 K ð i Þ
u i ; v i
0
8 i 2 U
ð 11 Þ
x ik 2f
0
1
g8 i 2 U 8 k 2 K ð i Þ
ð 12 Þ
;
y aa
ij 2f
0
1
g8 i ; j 2 U ; i \ j
ð 13 Þ
;
y ad
g8 i ; j Þ2 V ad
ij 2f
0
1
ð 14 Þ
;
y da
g8 i ; j Þ2 V da
ij 2f
0
1
ð 15 Þ
;
y dd
g8 i ; j Þ2 V dd
ij 2f
0
1
ð 16 Þ
;
Model Description
Objective function ( 1 ) minimises the weighted deviations of the arrival and
departure times proposed by the model from the times speci
ed by the timetable.
The weights cause that long-distance/high-speed trains will respect planned times
and regional trains will be postponed if necessary. The second criterion maximises
the desirability of the platform tracks to be assigned to the trains.
Constraint ( 3 ) ensures that a minimum dwell time needed for boarding and
alighting must be kept.
Constraint ( 4 ) states that connecting train i with real departure vi i þ t Pd
has to
i
wait in station to time at least u j þ t P j þ t Cn .
Constraints ( 5 )
(12) ensure that a minimum headway will be kept between
-
con
fl
icting trains. More precisely, constraint ( 5 ) states that if trains
i and i
have
max and train i arrives at platform track k before train i
planned arrival times within t
arrives at track k
, i.e.
; y aa
x ik ¼
1
; x i 0 k 0 ¼
1
ii 0 ¼
1
ð 17 Þ
and trains are on con
fl
icting routes (i.e. a(Ii, i , k, I i , k
)istrue), then train i
is allowed
) aa minutes later than train i. If at least one of the
conditions ( 23 ) is not met (e.g. train i is not assigned to track k), then constraint ( 5 )
becomes irrelevant as the right-hand side is negative (M is a suitably picked high
positive number). If train i
to arrive at least h(i,
k,
i
,
k
is followed by train i ð y aa
, then i is allowed to
arrive at least h(i , k , i, k) aa minutes later than i , which is ensured by constraint ( 6 ).
Constraints (7)
ii 0 ¼
0
Þ
(12) have a similar meaning for the other combinations of arrival
-
departure.
Constraints ( 13 )
( 14 ) ensure that a train will not be dispatched to an occupied
-
is followed by train i ð y aa
track. If train i
ii 0 ¼
0
Þ
and both trains arrive at the same
i, k) da minutes after train i
is allowed to arrive at least h(i
track k, then i
, k,
leaves
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