Passenger and freight train scheduling optimization
1. 1
PASSANGER AND FREIGHT TRAINS OPERATIONAL SHEDULE
PLANNING
Evgenii M. Kozhanov
e-mail: vinger4@gmail.com
There is a method which helps to solve the passenger and freight operational
timetable problem based on the MIP math model. The method includes accounting the
fixed timetabled trains (as a rule, passenger trains).
Keywords: operational planning, linear programming, passenger trains, freight
trains.
The method of the combined solution of passenger and freight operational
timetable problem is an efficient method of solving the train dispatching problem. This
method allows to delay a passenger train for passing other trains for the purpose of
decreasing costs.
This method is based on non-integer variables – values of departure time from each
station for every train. These variables are based variables. li
t representing the time of
departure train l from station i in minutes from starting point of planning period. Each
train has a route (sequence of stations), therefore set of value t represents the train
operational schedule.
There is a rail network only with single-line corridor-type tracks in the following
mathematical model.
The objective function minimizes the total amount of departure times from last
station of route of each train:
_
_
,
,
l la st
l la st
l s
l s
F t
_l la st
s - last station of route of train l.
(1)
2. 2
Subject to:
1) This constraint set ensures that departure time of each train from our
beginning route station is greater than the appearance time on this station
1
1 _
:li d ep l first
l i I
t t I i s
,
d ep
t - appearance time for train;
_l first
s - station of appearance train l.
2) Follow constraint set is a “continuous path of the each train” (the departure
time from next station is larger than the sum of departure time from the previous station
and the motion time between these stations)
, ( ( ), ) , , , ( ( ), ) , ( ( ), )
, ( ( ), ) , ( ( ), ) , , , ( ( ), )
2 2
, ; : ( ( ), ) ,
l n s m l i li х l i n s m l i р li з l n s m l i
l n s m l i l n s m l i li х l i n s m l i р li
lп р и б lп р и б
t t T t d t d
T d t t T t d
l i I I n s m l i s i s
, , ,х l i j
T - motion time of train l between station i and station j, in minutes.
li
d - sign that train l makes a stop at station i ( 1li
d - if train l makes a stop at
station i; 0li
d - if that isn`t so);
( , )n s m i - next station for station i in route number m;
( )m l - the number of route for train l;
р
t - acceleration time, in minutes;
з
t - braking time, in minutes;
T - value of planning period (in minutes).
T - величина периода планирования (в минутах).
3) Follow constraint set ensures that each train makes a stop at beginning and
ending stations
_
_
,
,
1,
1
l first
l la st
l s
l s
d
d l
(2)
(3)
(4)
(5)
(6)
3. 3
4) Follow constraint set ensures that the time between two trains, that arrival
from opposite directions at the same station, is larger than “non-synchronous interval“
, ( ( ), ) , , ( ( ), ), , ( ( ), )
, ( ( ), ) , , ( ( ), ), , ( ( ), )
, ( ( ), ) , , ( ( ), ), , ( ( ), )
, ( ( ), ) , ,
,
l p s m l i х l p s m l i i н р l p s m l i з li
k p s m k i х k p s m k i i р k p s m k i з ki lki
k p s m k i х k p s m k i i р k p s m k i з ki н
l p s m l i х l
t T t d t d
t T t d t d T A
t T t d t d
t T
( ( ), ), , ( ( ), )
(1 )
, , ; : ( ) ( ), ( ( ), ) 0 , ( ( ), ) 0
p s m l i i р l p s m l i з li lki
t d t d T A
l k i U U m l m k p s m l i p s m k i
lki
A - sign that train l will arrive at station i later than train k ( 1lki
A - if train l
arrives at station i later than train k; 0lki
A - if that isn`t so);
н
- “non-synchronous interval“, in minutes.
5) Follow constraint set ensures that time between the arrival of one train and
the departure of another train from the same track is larger than “crossing interval” (for
the single-line tracks)
, , , , ( ( ), ) , , ( ( ), )
, ( ( ), )
, ( ( ), ) , , ( ( ), ), , ( ( ), ) ,
,
,
(1 )
, , ; : , ( ) ( ), ( ( ), ) ( ( ), )
l i х l i n s m l i c р l i з l n s m l i
k p s m k i lki
k p s m k i х k p s m k i i c р k p s m l i з k i
l i lki
t T t d t d
t T A
t T t d t d
t T A
l k i U U l k m l m k p s m k i n s m l i
( , )p s m i - previous station for station i of route number m;
c
- “crossing interval” in minutes.
6) Follow constraint set ensures that the time interval between the departure of
two trains is larger than defined value
, ,
, ,
(1 )
, , ; : , ( ( ), ) ( ( ), )
lki k i l i
lki l i k i
T A t t I
T A t t I
l k i U U l k n s m k i n s m l i
I - defined value for interval between two trains in minutes.
There are points of arrival and departure at stations in route for the fixed
timetabling trains. The example of these points for passenger trains of single-track line
is represented on Table 1.
(8)
(9)
(10)
(11)
4. 4
The following variables are included for taking into account such points.
,н а гl i
t - the value of difference between minimum moving time between two
stations (station i and station ns(i,m(l))) and standard moving time between the same
stations for train l.
,о п о зд l i
t - delay for arriving train l as far as standard arriving time at station i (“value
of decrease moving time”).
Table 1. Points of departure and arrival for the fixed timetable trains
Train Station Arrival
point
Departure point
1 4 12 17
1 6 48 50
2 2 13 17
2 3 31 33
2 4 50 51
2 6 97 101
2 8 129 131
2 9 144 145
3 3 97 103
3 5 133 150
5 2 171 180
5 3 193 194
5 4 210 211
11 3 590 599
11 6 676 681
13 3 32 34
13 2 49 50
14 4 48 51
14 3 64 65
14 2 77 82
15 6 47 50
15 3 100 102
17 6 127 129
17 2 180 195
19 4 265 266
19 3 281 282
19 2 296 297
5. 5
Thereupon the following constraints were added (for taking into account fixed
timetable trains).
1) Follow constraint set ensures that the “value of decrease moving time” is
smaller than permissible value:
, m ax ,
, ,н агl i l i
t t l i ,
m ax ,l i
t - permissible value of the “value of decrease moving time” for train l while
moving from station i to station ns(i,m(l)).
2) Follow constraint set ensure that the delay is defined:
, , , ( , ( )) , ( , ( )) ,
, ( , ( )), , ( ) ,
,
о п о зд l i вхl i l p s i m l р l p s i m l з l i
х p s i m l i k l н а гl i
t T t t d t d
T t l i
,
,вхl i
T - time of array for train l to station i (“input point”).
3) Follow constraint set ensure that the departure time is larger than set value
(larger than “departure point”):
, ,
,l i вы хl i
t T l i ,
,вы хl i
T - time of “departure point” for train l which departures from station i.
Moving time is decreased by the “value of decrease moving time” ,н а гl i
t in
constraints (3), (6) – (9) (in constraint, which containes moving time). For example,
constraint (3) gets the following image:
, ( ( ), ) , , , ( ( ), ) , , ( ( ), )
, ( ( ), ) , ( ( ), ) , , , ( ( ), ) ,
2 2
, ; : ( ( ), ) ,
l n s m l i li х l i n s m l i н а гl i р li з l n s m l i
l n s m l i l n s m l i li х l i n s m l i н а гl i р li
lп р и б lп р и б
t t T t t d t d
T d t t T t t d
l i I I n s m l i s i s
New summand appears in the objective function:
, ,
100 10000н агl i опоздl i
F t t ,
Coefficients 100 and 1000 are conventional costs of one minute of “value of
decrease moving time” and one minute of delay concerning the “arrival point”.
Combined passenger and freight operational timetable was calculated for ”arrival
point” and “departure point” from Table 1. The result is presented on fig.1. Intervals for
boarding passengers and other operations with passenger trains are shown by using red
(12)
(13)
(14)
(15)
(16)
6. 6
horizontal lines. These intervals were defined by “array points” and “departure points”
from Table 1.
