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
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