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

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