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A Hierarchical Approach to Integrated
Transit

Derek Edwards

Georgia Institute of Technology


Co
-
Authors:
Aarjav

Trivedi
,
Arun

Kumar
Elangovan
, and Steve Dickerson

IEEE Intelligent Transportation Systems Conference: October 6, 2011

•
Why is Atlanta’s mass transportation not as efficient and widely
used as those in New York City

and Washington DC?


Crowded Manhattan and Washington
Transit Stations Subway Station
1

Empty Midtown Atlanta Bus Stop

1
http://gothamist.com/2008/05/13/confirmed_nyc_s.php

2

New York,

NY

Washington,
DC

Atlanta, GA

Population

Density

(people/mi
2
)


27,532

9,800

4,018

Average
Weekday
Unlinked Transit
Trips

10,303,095

1,460,125

504,420

Typical

Headway
Between Buses

5
-
15 minutes

8
-
20 minutes

20
-
45

minutes

U.S. Census Bureau,
U.S. Census Bureau
, County and City Data Book: 2000.

U.S. Census Bureau, Annual Estimates of the Resident Population for Incorporated Places of 100,000: 2009.

Rogoff
, P.M. “Transit Profiles: The Top 50 Agencies national transit database 2009 report year”: 2010.

Metropolitan Transportation Authority,
MTA System Schedules
, March 2011.

Metropolitan Atlanta Rapid Transit Authority,
Bus Routes and Schedules
, March 2011.

WMATA.com Bus Routes and Scheduled, 2011.


3

Enabling Technologies:

Ubiquitous mobile networks,
smart phones, GPS.

Remove inefficiencies from
transportation

•
Optimize bus routes in real
time.

•
Automate the car
-
pooling
process

•
Leverage existing
infrastructure

4

5

The dial
-
a
-
ride problem (DARP), is the problem of creating
M

dynamic vehicle routes to optimally service a set of N passengers
curb
-
to
-
curb with
a priori
information of the passenger’s origins and
destinations.

CORDEAU, J.
-
F. and LAPORTE, G., “The dial
-
a
-
ride problem: models and algorithms,”

Annals of Operations Research, vol. 153, no. 1, pp. 29
–
46, 2007.

6

http://www.gebweb.net/optimap/

What is the best way for a salesman to visit N
cities or locations?

•
For N passengers there are
N! permutations.

•
NP
-
Hard

•
Solved heuristically for
large numbers of cities.



7

http://www.gebweb.net/optimap/

•
For N passengers there are
N! permutations.

•
NP
-
Hard

•
Solved heuristically for
large numbers of cities.


•
Solution found using Ant
Colony Optimization:

•
Distance 14km

•
Travel Time 31:27



What is the best way for a salesman to visit N
cities or locations?

8

1

2

3

7

5

6

4

8

1

2

3

4

5

6

7

8

9

9

What is the best way for one or more vehicles to
service N pickup and delivery requests?

•
For N passengers there are
2N locations that must be
visited.

•
Additional Constraint: A
passenger drop
-
off location
cannot be visited before the
pick
-
up location.

•
2

!
2
𝑁

possible permutations.

•
NP
-
Hard

•
Solved heuristically for large
numbers of passengers.



9

What is the best way for one or more vehicles to
service N pickup and delivery requests?

•
For N passengers there are
2N locations that must be
visited.

•
Additional Constraint: A
passenger drop
-
off location
cannot be visited before the
pick
-
up location.

•
2

!
2
𝑁

possible permutations.

•
NP
-
Hard

•
Solved heuristically for large
numbers of passengers.


•
Solution
found using Ant
Colony Optimization:

•
Distance
16km

•
Travel Time
38:35




10

11

High Speed Data Trunk

Local Data Connection

Router/Gateway

Local Data Subnet

On
-
Demand
Transportation Subnet

Transit Station

Intra
-
City Transit

High Speed Commuter Rail

12

•
Provides solution to the last mile problem.


•
Outperforms static transit options in low
density areas.


•
Breaks up large DAR network into many
small semi
-
independent networks.



13

The Network
-
Inspired Transportation System

•
Subnets

•
Static Transit System

•
Metro
-
Wide Transit System

𝜙

=
{
𝜎
𝜙
𝑖
,

𝜙
𝑖
}

𝑇
=
{
𝑣
,


}

Ψ
=
{
Φ
,
𝑇
,

}

𝜙
1

𝜙
2

𝜙
3

𝜙
4

𝑣
1

𝑣
2

𝑣
3

𝑣
4

A

B

C

A

B

C

Where,
Φ

is the set of all subnets, and
D
is the set of all on
-
demand vehicles
in
Ψ
.

14

Defining the Optimization Problem

•
Global Objective Function:

•
Operator’s Objective Function:

•
Passenger’s Objective Function:

𝐽

𝑜𝑡𝑎𝑙
=

𝐽

+


𝐽


𝐽

=

𝐷
𝐽
𝐷
+


𝐽


𝐽

=

𝑑



=
1

𝐽

=

𝑝






=
1


𝑱
𝑫
:


Total cost of operating the dynamic vehicles


𝑱
𝑺
:

Total cost of operating the static vehicles



𝑱


:


Total cost of routing the passengers

𝒑


: Cost of routing passenger
j

N

: Total number of passengers


𝑱

:


Total cost incurred by the operator

𝑱
𝑺
:

T
otal cost incurred by the passenger

𝒅

: Cost of operating dynamic vehicle
i

M

:
Total
number of dynamic vehicles

15

Street Network: Node 2 is a Transit Station.

EDWARDS, D., et.
a
l.,“The

Network
-
Inspired Transportation System:
A
Hierarchical Approach to Bi
-
Modal Transit”, 14
th

International IEEE Conference on Intelligent Transportation Systems, October, 2011.

Route of Static Bus.

On
-
demand transit out performs static transit for solving the
last mile problem.

16

EDWARDS, D., et.
a
l.,“The

Network
-
Inspired Transportation System:
A
Hierarchical Approach to Bi
-
Modal Transit”, 14
th

International IEEE Conference on Intelligent Transportation Systems, October, 2011.

Route of Static Bus.

𝐽
=


𝑙

+
2

+
1

=
1



(
𝑝
𝑤
,


+
𝑝
𝑟
,


)




=
1

N

= Number of Passengers

l
i

is the length of route segment
i

𝑝
𝑤
,



is the length of time passenger

j

waited
for the bus.

𝑝
𝑤
,



is the length of time passenger
j

rode
the bus


17

EDWARDS, D., et.
a
l.,“The

Network
-
Inspired Transportation System:
A
Hierarchical Approach to Bi
-
Modal Transit”, 14
th

International IEEE Conference on Intelligent Transportation Systems, October, 2011.

Route of Static Bus.

Results:

Objective: Minimize VMT

Objective: Minimize Passenger Wait
and Ride Time

18

19

Subnets
–

The on
-
demand regions where entire passenger trips
can be served by a single vehicle.

•
Size, Shape, Allocation (geographic versus functional)

20

•
The NITS should accommodate the ride
-
share option.


•
The ride
-
share option introduces semi
-
static routes. A driver
with a car has a known origin and destination, but is willing
to alter his trip to accommodate others.


•
How should these trips be integrated with static transit?



21

22

Derek Edwards

School of Electrical and Computer Engineering

Georgia Institute of Technology

dedwards@gatech.edu

Steve Dickerson

School of Mechanical Engineering

Georgia Institute of Technology

s
teve.dickerson@me.gatech.edu

Arun

Kumar
Elangovan

RideCell
, LLC

arunmib@ridecell.com

Aarjav

Trivedi

RideCell
, LLC

aarjav@ridecell.com

23

1.
E
ncode neighborhood as a graph. Using distances
between intersections as weights.

2.
Preprocessing: Using
Dijkstra’s

Algorithm, create a
complete distance graph of the neighborhood.


24

3.
Identify location of passengers and destinations of
passengers.

4.
Use a Genetic Algorithm to determine the optimal
order in which to visit the passengers.

25

Proof of Concept Objective Function

𝐽

𝑜𝑡𝑎𝑙
=


𝑙

+
2

−
1

=
1




[
𝜆
1
,


𝑝
1
,


+




=
1
𝜆
2
,


𝑝
2
,



+
𝜆
3
,


𝑝
3
,


]

𝜆
1
,


=

1
if

passenger

j

wishes

to

minimize

wait

time
0
else

𝜆
2
,


=

1
if

passenger

j

wishes

to

minimize

ride

time
0
else

𝜆
3
,


=

1
if

passenger

j

wishes

to

minimize

total

trip

time
0
else

p
1,j

the
wait

time for passenger

j


𝒍

:

length of the
i
th

segment traversed by the vehicle.

p
2
,j

the
ride

time for passenger

j


p
3
,j

the
total

trip time for passenger

j


26

Total Vehicle Mile
Traveled:


11.59


Minimize Wait (Green)

Minimize Ride (Blue)

Minimize Total

27

Total Vehicle Mile
Traveled:


4.25


Minimize Wait (Green)

Minimize Ride (Blue)

Minimize Total

28

Total Vehicle Mile
Traveled:


5.55


Minimize Wait (Green)

Minimize Ride (Blue)

Minimize Total

29