By
Pranamesh Chakraborty
Sharath M N
Department of Civil Engineering
Indian Institute of Technology Kanpur
India
1st of March, 2013
Schedule Determination of a
Multiple- Route Transit System
Introduction
Transit system plays a very important role in handling
the high travel demand and simultaneously reducing
the problem of congestion and pollution in any urban
area. A public transport system can be made efficient
by scheduling it in such a manner so that the waiting
time for passengers as well as transfer time from one
route to another is minimized.
Problem Statement
Route 2
Route 1
Route 3
Route 6
Route 5
Route 4
Fig 1. Typical Transit Network
The solid lines represent the routes and the circled
intersections represent transfer stations
The problem is to determine the schedule such that waiting
time for the passengers is minimized which includes Initial
Waiting Time (IWT) as well as Transfer Time (TT).
Mathematical Formulation
The objective function is1
, ,
, 1
, , , , , , , , , ,
1 1 0
( ) ( )( )
k ki m i mji
a annk l l k k k k
i j m j m i m i j m i k m i m i m
k lm i j m li
d a v t a a t dt
First term represents the total transfer time while the
second term represents the total waiting time. th th th
,
th th th
,
: Arrival time of k transit unit of i route at m transfer stop
: Departure time of k transit unit of the i route from m transfer stop
k
i m
k
i m
a
dth th th
, ,
, th th
, ,
: Arrival pattern of pasengers for k transit unit of i route at m transfer stop
: A binary variable that takes the value of 1 if transfer from k transit unit of i route
i k m
k l
i j m
v
th th th
th th th th
, ,
to l transit unit of j route at m transfer point is ideal to passengers; 0 otherwise
: Number of passengers transferring from k transit unit of i route to j route at m
tr
k
i j m
ansfer point
Constraints (Chakroborty, P., Das, A .,2003)The transit unit should stop for a certain minimum period as physical transfer of
passengers takes some time and at the same time, the transit unit cannot stop at a stop
for a very long time. The constraints g1 and g2 will take care of these aspects.
Constraint g3 states that transferring of passengers cannot happen to transit units that
have departed the transfer stations before the arrival of passenger at that transfer point.
A person can transfer to only one transit unit and it is dictated by the constraint g4.
. The time headway between any two successive vehicles must be positive and must
be less than the policy headway. It is governed by constraints g5 and g6.
No transferring passenger shall wait for more than a certain period of time and it is
being dictated by g7.
The arrival time of a transit unit is dependent on arrival time of a transit unit at a stop
previous to transfer stop and also on the travel time between those two stops;
constraints g8 and g9 state this.
The constraint g10 states the minimum fleet size for each route while g11 states that
the total fleet size of all the routes is to be equal to N.
Constraintsmax
1 , ,
min
2 , ,
,
3 , , , ,
,
4 , ,
, 1, 2,.......,
, 1, 2,.......,
(1 ) 0 , , 1, 2,......., 1, 2,.......,
1 , ,
k k
i m i m i i
k k
i m i m i i
k k k l
j m i m i j m i j
k l
i j m
l
g d a S i m and k n
g d a S i m and k n
g d a M i m j i and k n and l n
g i m j i
1
5 , ,
1
6 , ,
,
7 , , , ,
1, 2,......., 1, 2,.......,
, 1, 2,.......,
0 , 1, 2,.......,
( ) , , 1, 2,......., 1, 2,
i j
k k
i m i m i i
k k
i m i m i
l k k l
j m i m i j m i
and k n and l n
g a a H i m and k n
g a a i m and k n
g d a T i m j i and k n and l
max
8 , , 1 , 1,
min
9 , , 1 , 1,
min
10
11
.......,
, 1, 2,.......,
, 1, 2,.......,
n
j
k k
i m i m i m m i
k k
i m i m i m m i
i i
i
i
n
g a d t i m and k n
g a d t i m and k n
g n i
g n N
th
min th
max
: Policy headway of i route
: An arbitrary large number
: The minimum period for which the transit unit must stop at any stop on i route
: The maximum period for which the transit unit can
i
i
i
H
M
S
S
1 2
1 2
th
min th
, 1 2
max th
, 1 2
stop at any stop on i route
: Minimum travel time of transit unit of i route from stop m to stop m
: Maximum travel time of transit unit of i route from stop m to stop m
: Max
i m m
i m m
t
t
T imum transfer time for any transferring passenger
N : Total fleet size
n : Fleet size of i routeth
i
where
Solution Technique
The scheduling problem formulated is a mixed integer
nonlinear programming problem. The objective
function and the constraint g7 are nonlinear whereas
the variable is a 0-1 binary variable and the other
variables are real. It is difficult to solve such a
formulation problem using traditional optimization
techniques. Hence Genetic Algorithms (GA) are used
to solve such problems.
Genetic Algorithms
Genetic algorithms are optimization algorithms based
on the principles of natural genetics and natural
selection.
GAs are generally used to solve maximization
problems. For minimization problems, the objective function (referred as Fitness function Ƒ(x) in GA
problems) can be transformed into an equivalent
function to be maximized.
.
The design variables are first coded randomly in string structures.
The length of the string depends on desired level of accuracy.
Then the fitness value of each string is evaluated.
The population is then operated by three main operators- reproduction, crossover
and mutation thus forming a new population set.
Genetic Algorithms
Genetic Algorithms
Reproduction
The good strings are selected and reproduced
with a probability (pi) proportional to its fitness
value.
Crossover
Two strings are chosen at random from the
population and some portion of the strings is
exchanged between the pair. The crossover is
done with a small probability of pc (i.e. 100pc % of
the strings are used in crossover).
Mutation
The mutation operator changes randomly a
particular bit of a string from 1 to 0 or vice versa
with a small mutation probability pm.
Revised Formulation
(considering only one transfer stop)
The new decision variables , and ni are introduced
where,
= Headway between the kth and the (k—l)th bus of the ith route;
= Stopping time of the kth bus of the ith route;
The bounds for the variable is provided by constraint g1 and g2.
The upper bound for is provided by g5, lower bound by g6.
The variables is also not required because the schedule
encoded in a string is known and thus it is simple to find which
transfers were made. Thus constraints g3 and g4 are also eliminated.
The constraint g11 is used to eliminate one of the ni variables.
Thus we are left with only g7.
This revised formulation can be used to determine the optimal
solution using Penalty methods in satisfying the constraints
(Chakroborty, Deb, & Sharma, 2001).
k
ih
k
ih
k
is
k
is
k
isk
ih,
,
k l
i j
String
A string is concatenation of several substrings
There will be as many substrings as number of
variables
Number of bits in a string is governed by level of
accuracy required
The upper bound and lower bound of substrings
can be defined
String
n1 is the first substring
s1 is the second substring
The third substring is
si = for all k
Number of variables is
r is the number of routes
No substring for is required
1
khk
is
1
khmax
1 1n
rn
ni , si and strings are decoded
ni is checked for its range
Only first ni number of are decoded and rest are ignored
Last head way for each route is computed based on sum of all headway which should be equal to scheduling period and checked if it is in permissible range
Thus the entire schedule can be determined
The objective function and rest of constraints can be calculated
If a constraint is not satisfied, a heavy penalty is assigned to the string
Evaluation of Stringsk
ih
k
ih
Example (Chakroborty, Deb, & Sharma, 2001)
Scheduling is done for a single transfer station
Number of routes, r=3
Total fleet size, N=30
Scheduling time period, H=240 minutes
Minimum fleet size=5
Policy headway is assumed to be proportional to
demandPercentage of total demand Policy headway (mins)
5-15 56
15-25 47
25-45 40
45-60 35
>60 31
Example (contd.)
Minimum stopping time=2mins
Maximum stopping time=5mins
Headway range=32mins
Maximum transfer time=30mins
=300, for all routes
Arrival pattern is triangular with =0.8
,
k
i j
GA parameters
Population size =800
Maximum number of generations =800
Cross over probability = 0.95
Mutation probability = 0.005
Example (contd.)
Variable Number of bits
Fleet size 4
Headway 5
Stopping time 2
Example (contd.)
Objective function with only IWT term
Case 1
Equal Demand on all routes
Theoretical optimum fleet size for each route is 10
Theoretical global optimum objective function
value=3456mins
Source: (Chakroborty, Deb, &
Sharma, 2001)
Example (contd.)
Case 2
Demand ratio among routes=1:4:9
Theoretical fleet size allocation should be in the
ratio 1:2:3
Optimum objective function value=13825mins
Source: (Chakroborty, Deb, &
Sharma, 2001)
Objective function with both IWT and TT
This problem can not be solved by using
traditional optimisation techniques
Case 1
Demand ratio is 1:4:4
Example (contd.)
Source: (Chakroborty, Deb, &
Sharma, 2001)
Case 2
Demand ratio 1:4:9
Example (contd.)
Source: (Chakroborty,
Deb, & Sharma, 2001)
Example (contd.)
Variables IWT+TT IWT only
Fleet size distribution 8:11:11 7:12:11
Number of matches 8 1
Comparison for Demand ratio 1:4:4
Variables IWT+TT IWT only
Fleet size distribution 7:11:12 6:10:14
Number of matches 7 2
Comparison for Demand ratio 1:4:9
Chakroborty P, Das. A. (2003). Principles of Transportation Engineering. New
Delhi: Prentice Hall of India.
Chakroborty, P., Deb, K., & Sharma, R. K. (2001). Optimal fleet size distribution
and scheduling of transit systems using genetic algorithms. Transportation Planning
and Technology , 24.3, 209-225.
Chakroborty, P., Deb, K., & Subrahmanyam, P. S. (1995). Optimal Scheduling of
Urban Transit Systems using Genetic Algoritms. Journal of Transportation
Engineering , 544-553.
Deb, K. (2006). Optimization for Engineering Design. New Delhi, India: Prentice
Hall of India Private Limited.
References
Thank You