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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. l it 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 s t

l la s t

l s

l s

F t

_l la s ts - last station of route of train l.

(1)

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 _:

l i d e p l f ir s t

l i I

t t I i s

,

d e pt - appearance time for train;

_l fir s ts - 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 l i 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 l i 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 jT - motion time of train l between station i and station j, in minutes.

l id - sign that train l makes a stop at station i ( 1

lid - 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;

рt - acceleration time, in minutes;

зt - 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 f ir s t

l la s t

l s

l s

d

d l

(2)

(3)

(4)

(5)

(6)

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 l i

k p s m k i х k p s m k i i р k p s m k i з k i lk i

k p s m k i х k p s m k i i р k p s m k i з k i 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 l i lk it d t d T A

l k i U U m l m k p s m l i p s m k i

lk iA - sign that train l will arrive at station i later than train k ( 1

lk iA - if train l

arrives at station i later than train k; 0lk i

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 lk i

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 lk i

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 )

, , ; : , ( ( ) , ) ( ( ) , )

lk i k i l i

lk i 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

The following variables are included for taking into account such points.

,н а г l it - 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 it - 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

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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 ,, ,

на гl i l it t l i ,

m ax ,l it - 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 в х l 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 г l i

t T t t d t d

T t l i

,

,вх l iT - 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 it T l i ,

,вы х l iT - 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 г l it 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 l i х l i n s m l i н а г l i р li з l n s m l i

l n s m l i l n s m l i l i х l i n s m l i н а г l 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:

, ,1 0 0 1 0 0 0 0

на гl i о по зд l iF 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)

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horizontal lines. These intervals were defined by “array points” and “departure points”

from Table 1.

7

67

6307

6311

6309

2533

2535

2537

6308

6312

502

504

86

376

5410 6

912

2526

2528

2530

Figure 1. Example of combined passenger and freight operational timetable