a tour-based urban freight transportation model based on entropy maximization qian wang, assistant...
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A Tour-Based Urban Freight Transportation Model Based on Entropy Maximization
Qian Wang, Assistant ProfessorDepartment of Civil, Structural and Environmental EngineeringUniversity at Buffalo, the State University of New York
José Holguín-Veras, ProfessorDepartment of Civil and Environmental EngineeringRensselaer Polytechnic Institute
SHRP2 Innovations in Freight Demand Modeling and Data Symposium Sep 15, 2010
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Outline
•Background▫Motivations▫Objectives
•Methodology•Case study•Applications•Conclusions
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Motivations
•Complexity of freight activities▫Multiple measurement units used▫Multiple decision makers involved▫Diverse commodities shipped▫Trip chaining behavior
NYC: 5.5 stops/tour Denver: 3.2 stops/tour Passenger cars: 1 stop/tour
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Example of a tour
Base
Producer
Receiver 1
Receiver 2
Receiver 3Receiver 4
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Motivation (Cont.)
•How to model and forecast urban freight movements given the limited data sources
•How to use GPS data without infringing on privacy▫Aggregation takes care of that
Smaller zones could be used, providing better detail▫No need to model disaggregate flows
No data available for the foreseeable future Privacy issue will deter cooperation from carriers
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Objectives
•To develop a tour-based model given:▫Trip production and attraction from trip generation ▫Travel impedances (times, cost, distance)
•To assess the impact of different impedance variables, such as the travel time and the handling time, on the performance of tour estimation
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Modeling Framework
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Tour generation Tour flow distribution model
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Tour Flow Distribution Method
•Entropy maximization▫Formal procedure to find the most likely solutions
given a set of constraints▫Provides theoretical support to gravity models
• It provides the flexibility to incorporate secondary data (e.g., traffic counts) to demand forecasting▫e.g., entropy maximization can be used to reduce the
solution space for the ODS models
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Entropy of the System
•Three states of the urban freight system
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State State Variable
Micro state Individual tour starting and ending at a home base
Meso stateThe number of vehicle flows (called tour flows) following a node
sequence
Macro state Total number of trips generated by a node (production)
Total number of trips attracted to a node (attraction)
Formulation 1: C = Total time in the commercial network;
Formulation 2: CT = Total travel time in the commercial network;
CH = Total handling time in the commercial network.
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Entropy of the System (Cont.)
•Entropy: defined as the number of ways to generate the tour flow distribution solutions
•Entropy maximization: to find the most likely way to distribute tour flows given the constraints associated with the macro state
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Entropy Maximization Formulations•Formulation 1
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!
!...
1
)(2
2
1
m
M
m
ttT
tT
t
TCCWMax
(1)
Subject to:
},...,2,1{,1
NiOta i
M
mmim
(2)
},...,2,1{,1
NjDta j
M
mmjm
(3)
CtcM
mmm
1
(5)
},...,2,1{,0 Mmtm (6)
Trip production constraints
Trip attraction constraints
Entropy maximization
Cost constraint
Nonnegativity of tour flows
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Resulting Models
•First-order conditions (tour flow distribution models)▫Formulation 1:
•Traditional gravity trip distribution model
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)exp()exp( *
1
**m
N
iimim cat
)exp( **ijjjiiij cDBOAt
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Convexity of the Formulations•Second-order condition▫Objective function: Hessian is positive definite▫Constraints: linear▫Overall: convex program with one optimal solution
•Solution algorithm: primal-dual method for optimization with convex objectives (PDCO) (Saunders, 2005)
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Case Study: Denver Metropolitan Area• The Denver travel behavior inventory data (1998-1999)
(TBI) survey
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Case Study (Cont.)
•Test network▫919 TAZs among which 182 TAZs contain home
bases of commercial vehicles▫613 travel itineraries, representing a total of 65,385
tour flows per day
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Model Estimation Procedure
•Step 1: Obtain input data: Os, Ds•Step 2: Generate a set of candidate tours▫Using tour choice models▫Could be randomly and exhaustively generated too
•Step 3: Let the model find the optimal tours, i.e., the ones that match the trip generation constraints•Step 4: Compare the estimations with observations
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Performance of the Models
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Estimated Results Formulation 1
MAPE 6.71%
Tour-time-related Lagrange multiplier ( ) -0.000228
Tour-travel-time-related Lagrange multiplier ( ) /
Tour-handling-time-related Lagrange multiplier ( ) /
1
2
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Performance of the Models (Cont.)•Distribution of tour time (travel + handling time)
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Tour time (minutes)
Freq
uenc
y (%
)
9808407005604202801400
12
10
8
6
4
2
0
Histogram of tour time_observed
Tour time (minutes) (Formulation 1)
Freq
uenc
y (%
)
9808407005604202801400
12
10
8
6
4
2
0
Histogram of tour time_estimated
Observed
Estimated
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Performance of the Models (Cont.)•Distribution of tour travel time
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Tour travel time (minutes)
Freq
uenc
y (%
)
665570475380285190950
10
8
6
4
2
0
Histogram of tour travel time_observed
Tour travel time (minutes) (Formulation 2)
Freq
uenc
y (%
)
665570475380285190950
10
8
6
4
2
0
Histogram of tour travel time_estimatedObserved
Estimated
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Performance of the Models (Cont.)•Distribution of tour handling time
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Tour handling time (minutes)
Freq
uenc
y (%
)
7006005004003002001000
10
8
6
4
2
0
Histogram of tour handling time_observed
Tour handling time (minutes) (Formulation 2)
Freq
uenc
y (%
)
7006005004003002001000
10
8
6
4
2
0
Histogram of tour handling time_estimated
Observed
Estimated
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Potential Applications:
•Could be the engine of a freight origin-destination matrix estimation technique that explicitly considers delivery tours
•Could be used to construct commercial vehicle tours from commodity flow estimates, without ambiguity regarding the underlying rules
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Potential Applications (Cont.)•Given the base-year tours
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Input information: the base-year tours and the associated cost
Aggregate the base-year information to get the trip productions/attractionsand the total impedance
Estimate the parameters (Lagrange multipliers)in the tour distribution model
using the entropy maximization formulations:
)exp(1 1
N
i
Fm
K
a
ama
Fimi
Fm cpat
)exp(1
211
N
i
FHM
FTm
K
a
ama
Fimi
Fm ccpat
Estimate the future-yeartour flows usingthe tour distribution models
Cal
ibra
tio
n
Application
Formulation 1:
Formulation 2:
Predict future trip production and attraction
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Conclusions
•The model is a general form of the gravity model• It explicitly considers tour chains• It is the first freight demand model able to
represent tour behavior in a mathematical function• It is able to replicate calibration data quite well
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Future Work
•Consider more cost factors• Incorporate traffic counts (ODS)•Link commodity flows to vehicle flows
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Questions?
Qian WangDepartment of Civil, Structural and Environmental Engineering
University at Buffalo, the State University of New [email protected]
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