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

Outline

•Background▫Motivations▫Objectives

•Methodology•Case study•Applications•Conclusions

2

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

3

Example of a tour

Base

Producer

Receiver 1

Receiver 2

Receiver 3Receiver 4

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

4

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

5

Modeling Framework

6

Tour generation Tour flow distribution model

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

7

Entropy of the System

•Three states of the urban freight system

8

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.

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

9

Entropy Maximization Formulations•Formulation 1

10

!

!...

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

Resulting Models

•First-order conditions (tour flow distribution models)▫Formulation 1:

•Traditional gravity trip distribution model

11

)exp()exp( *

1

**m

N

iimim cat

)exp( **ijjjiiij cDBOAt

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)

12

Case Study: Denver Metropolitan Area• The Denver travel behavior inventory data (1998-1999)

(TBI) survey

13

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

14

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

15

Performance of the Models

16

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

Performance of the Models (Cont.)•Distribution of tour time (travel + handling time)

17

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

Performance of the Models (Cont.)•Distribution of tour travel time

18

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

Performance of the Models (Cont.)•Distribution of tour handling time

19

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

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

20

Potential Applications (Cont.)•Given the base-year tours

21

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

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

22

Future Work

•Consider more cost factors• Incorporate traffic counts (ODS)•Link commodity flows to vehicle flows

23

Questions?

Qian WangDepartment of Civil, Structural and Environmental Engineering

University at Buffalo, the State University of New Yorkqw6@buffalo.edu

24