block viii-j: freight demand modeling -tour models-

Urban Freight Tour Models: State of the Art and Practice

1

José Holguín-Veras, Ellen Thorson, Qian Wang,

Ning Xu, Carlos González-Calderón, Iván

Sánchez-Díaz, John Mitchell

Center for Infrastructure, Transportation,

and the Environment (CITE)

Outline

Introduction, Basic Concepts

Urban Freight Tours: Empirical Evidence

Urban Freight Tour Models

Simulation Based Models

Hybrid Models

Analytical Models

Analytical Tour Models

Conclusions

2

Introductionand Basic Concepts

3

Basic Concepts

A supplier sending shipments from its home base (HB) to six receivers

The goal is to capture both the underlying economics of production and consumption, and realistic delivery tours

4

HB

S-2

Loaded vehicle-trip

Commodity flow

Notation:

Consumer (receiver)

Empty vehicle-trip

S-1S-3

R1

R2R3

R5

R6

R4

Empirical Evidence onUrban Freight Tours

5

Characterization of Urban Freight Tours (UFT)

Number of stops per tour depends on: Country, city, type of truck, the number of trip chains, type of carrier, service time, and commodity transported

6

0

1

2

3

4

5

6

7

8

9

10,000 100,000 1,000,000 10,000,000

Ave

rag

e n

um

be

r o

f sto

ps/to

ur

Population

Schiedam

Alphen Apeldoorn

Amsterdam

Denver

New York City


Denver, Colorado (Holguín-Veras and Patil,2005):

Port Authority of NY and NJ (HV et al., 2006):

By type of company:

Common carriers: 15.7 stops/tour

Private carriers: 7.1 stops/tour

By origin of tour:

New Jersey: 13.7 stops/tour

New York: 6.0 stops/tour

7

Stops/Tour Single TruckCombination

Truck

Average 6.5 7.0

1 tour/day 7.2 7.7

2 tours/day 4.5 3.7

3 tours/day 2.8 3.3


NYC and NJ (Holguin-Veras et al. 2012):

Average: 8.0 stops/tour 12.6%: 1 stop/tour

54.9%: < 6 stops/tour 8.7% do > 20 stops

Parcel deliveries: 50-100 stops/tour

8


9


The UFT models could be subdivided into:

Simulation models

Hybrid models

Analytical models

10

Simulation Models

11

Simulation Models

Simulation models attempt to create the needed isomorphic relation between model and reality by imitating observed behaviors in a computer program

Examples include:

Tavasszy et al. (1998) (SMILE)

Boerkamps and van Binsbergen (1999) (GoodTrip)

Ambrosini et al., (2004) (FRETURB)

Liedtke (2006) and Liedtke (2009) (INTERLOG)

12

Hybrid Models

13

Hybrid Models

Hybrid models incorporate features of both simulation and analytical models (e.g., using a gravity model to estimate commodity flows, and a simulation model to estimate the UFTs)

Examples include:

van Duin et al. (2007)

Wisetjindawat et al. (2007)

Donnelly (2007)

14

Analytical Models

15

Analytical Models

Analytical models attend to achieve isomorphism using formal mathematic representations based on behavioral, economic, or statistical axioms

Two main branches:

Spatial Price equilibrium models (disaggregate)

Entropy Maximization models (aggregate)

Examples include:

Holguín-Veras (2000), Thorson (2005)

Xu (2008), Xu and Holguín-Veras (2008)

Holguín-Veras et al. (2012)

Wang and Holguín-Veras (2009), Sanchez and Holguín-Veras (2012)

16

Entropy Maximization Tour Flow Model

17

Entropy Maximization Tour Flow Models

Based on entropy maximization theory (Wilson, 1969; Wilson, 1970; Wilson, 1970)

Computes most likely solution given constraints

Key concepts:

Tour sequence: An ordered listing of nodes visited

Tour flow: The flow of vehicle-trips that follow a sequence

The problem is decomposed in two processes:

A tour choice generation process

A tour flow model

18


Tour choice: To estimate sensible node sequences

Tour flow: To estimate the number of trips traveling along a particular node sequence

2014/4/15

19

Tour choice generation Tour flows

Entropy Maximization Tour Flow Model20

Definition of states in the system:

State State Variable

Micro stateIndividual commercial vehicle journey starting and ending at a home base

(tour flow) by following a tour ;

Meso stateThe number of commercial vehicle journeys (tour flows) following

a tour sequence.

Macro state Total number of trips produced 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.

Static version of EM Tour Flow Model

21

The equivalent model of formulation 2:Entropy Maximization Tour Flow Model

Trip production

constraints

Total travel time

constraint

Total handling time

constraint

22

M

m

mmm tttZ1

)ln(

i

M

m

mim Ota 1

T

M

m

mmT Ctc 1

H

M

m

mmH Ctc 1

0mt

)( i

)( 1

)( 2

MIN

Subject to:

First-order conditions (tour distribution models)

Traditional gravity trip distribution model


)exp()exp()exp( *

1

*

1

***

m

N

i

imi

N

i

mimim cacat

23

)exp( **

ijjjiiij cDBOAt

)exp()exp(*

2

*

1

1

**

HmTm

N

i

imim ccat

Formulation 1:

Formulation 2:

Formulation 1 and the traditional GM model

have exactly the same number of parameters


The optimal tour flows are found under the objective of maximizing the entropy for the system

The tour flows are a function of tour impedance and Lagrange multipliers associated with the trip productions and attractions along that tour

Successfully tested with Denver, Colorado, data:

The MAPE of the estimated tour flows is less than 6.7% given the observed tours are used

Much better than the traditional GM

24

25

Case Study: Denver Metropolitan Area

Test network

919 TAZs among which 182 TAZs contain home bases of commercial vehicles

613 tours, representing a total of 65,385 tour flows / day

Calibration done with 17,000 tours (from heuristics)

Estimation procedure

Sorting input data: aggregate the observed tour flows to obtain trip productions and total impedance

Estimation: estimate the tour flows distributed on these tours using the entropy maximization formulations

Assessing performance: compare the estimated tour flows with the observed tour flows

Estimated vs. observed tour flowsPerformance of the Models

Time-Dependent Freight Tour Synthesis Model

27

TD-FTS Model

Bi-objective Program:

28

PROGRAM TD-ODS 1

Minimize

M

m

dm

dm

dm

K

d

xxxxz11

)ln()( (1)

Minimize

2

1 1

'

1 12

1)(

A

a

K

k

K

d

M

m

dm

kdam

ka xvxe (2)

Subject to:

M

m

idmim

K

d

NiOxg11

(3)

M

m

jdmjm

K

d

NjDx11

(4)

M

m

dmm

K

d

Cxc11

(5)

KdMmxdm ,,0

(6)

Function to replicate TD traffic counts

Trip Production Constraint

I can drop Trip Attraction Constraint

Impedance Constraint

Possible combinations of flow

TD-FTS Model

Multi-attribute Value formulation:

29

PROGRAM TD-FTS3

Minimize )()()(),( 2211 xxxx zvevzeU

Subject to:

M

m

yj

dymjm

K

d

YyNjDx1

,

1

,

)( ,yj

K

d

M

m

Y

y

dymm Cxc

1 1 1

,

)(

YyKdMmxdym ,,,0,

Trip Production Constraint per Industry

Impedance Constraint

Utility function from DM

Application to the Denver Region30

RMSE MAPE RMSE MAPE RMSE MAPE RMSE MAPE RMSE MAPE RMSE MAPE

S/TD-EM 39.8 31.2% 58.5 274.7% 45.6 42.3% 55.5 46.4% 39.5 106.1% 50.3 117.0%

S/TD-FTS-A 17.2 16.6% 14.5 231.0% 14.7 29.2% 14.8 31.0% 9.5 68.7% 13.6 76.1%

S/TD FTS-B 8.0 0.8% 9.5 46.3% 9.1 3.8% 6.4 4.9% 5.9 12.3% 9.1 22.9%

DCGM 116.0 79.5% 39.6 364.8% 42.0 94.5% 47.2 78.2% 25.8 148.1% 60.4 170.4%

S/TD-EM 32.1 21.0% 58.3 257.0% 44.1 36.9% 56.9 42.7% 41.6 100.4% 50.7 109.0%

S/TD-FTS-A 13.8 11.3% 12.6 153.4% 12.9 23.6% 14.6 25.8% 12.6 61.4% 13.2 66.1%

S/TD FTS-B 5.7 5.2% 7.5 42.9% 8.9 9.0% 9.3 6.3% 10.5 21.6% 9.1 19.8%

Tour Flows

OD Flows

Modeling

Approach

Daily Flows

Estimates

Time Intervals Flows Estimates

Early Morning Morning After Noon Evening Overall*

TD-FTS MAPE’s: 0.8%-76.1%

Static Entropy Maximization (S-EM): MAPE’s 31.2%-117%

Gravity Model (DCGM): MAPE 79.5%

Temporal aspect better captured using TD-FTS

TD-FTS Performance per Time Interval31

y = 0.755xR² = 0.913

0

50

100

150

200

250

300

350

400

450

0 200 400 600

Est

ima

ted

T

D T

ou

r F

low

s

Actual TD Tour Flows

TD-ODS AM OH

y = 1.049xR² = 0.996

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

Est

ima

ted

T

D T

ou

r F

low

s


TD-ODS AM PH

y = 1.032xR² = 0.998

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

Est

ima

ted

T

D T

ou

r F

low

s


TD-ODS PM PH

y = 0.928xR² = 0.983

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000

Est

ima

ted

T

D T

ou

r F

low

s


TD-ODS PM OH

Multiclass Equilibrium Demand Synthesis

32

Multiclass: two or more classes of travelers with different behavioral or choice characteristics

Vehicle classes are related under the same objective function

Multiclass equilibrium demand synthesis (MEDS)

Multiclass traffic33

Travel time function depends on the vector of traffic flows for passenger cars and trucks

The travel cost is affected by the value of time of each one of the classes

The formula is a second order Taylor expansion of a general link-performance function

Where:

Xt: Vector of truck traffic flow

Xc: Vector of passenger cars traffic flow

Link Travel Cost34

tctctccta XXXXXXXXt 5

2

4

2

3210),(

In a multiclass equilibrium, the cost functions of the modes are asymmetric, traffic flows interact

The user optimal assignment cannot be written as an optimization problem

The User Equilibrium (UE) problem could be addressed using a Variational Inequality (VI) Problem

Multiclass Equilibrium35

0)(),()(),( ****** ij

ijij

T

ijmij

m

mm

T

ijmm TTTtCttttC00 * mmm tCC

00 * ijijij TCC

Although the vehicles share the transportation network and the travel time is the same (in equilibrium), the travel cost will be affected by the value of time of each one of the partiesvalue of time

Considering paths, the cost functions for truck tours and passenger cars will be given by:

In a multiclass equilibrium, the cost functions of modes are asymmetric, traffic flows interact. UE Variational Inequality

Where: :A binary variable indicating whether tour m uses link a

:A binary variable indicating whether trip from i to j uses link a

Link Travel Cost36

),( ctat

t

a xxtc

),( ctac

c

a xxtc

a

ma

t

am cC

ija

a

c

aij cC

ma

ija

Multiclass EM Formulations

The multiclass EM formulations is given by:

Where:W: System entropy that represents the number of ways of distributing commercial vehicles tour flows and passenger cars flows

Tt: Total number of commercial vehicle tour flows in the network;

Tc: Total number of passenger cars flows in the network;

tm : Number of commercial vehicle journeys (tour flows) following tour m;

Tij : Number of car trips between i and j

37

ijm

ijm

ct

Tt

TTW

,

)! !(

! !Max

m ij

ijijijmmm TTTtttz )ln ln(Min

Multiclass Equilibrium ODS Formulations

Min

Subject to

Subject to

38

m ij

ijijijmmm TTTtttz )ln ln(

0)(),()(),( ****** ij

ijij

T

ijmij

m

mm

T

ijmm TTTtCttttC

NitOM

m

imm

t

i ,...,2,1 1

NitDM

m

jmm

t

j ,...,2,1 1

},...2,1{ 1

NiTON

j

ij

c

i

},...2,1{ 1

NjTDN

i

ij

c

j

a

ma

t

am cC

ija

a

c

aij cC

atXm

mam

t

a

aTXij

ijaij

c

a

},...2,1{ QaVVX t

a

t

a

t

a

},...2,1{ QaVVX c

a

c

a

c

a

0mt

0ijT 0 t

aX

0 c

aX

VI problem to

obtain a UE

condition for cars

and trucks

The objective is to

find the most likely

ways to distribute

tours considering

congestion

Spatial Price Equilibrium Tour Models

39

General principles

The models estimate commodity flows and vehicle trips that arise under competitive market equilibrium

Conceptual advantages:

Account for tours

Provide a coherent framework to jointly model the joint formation of commodity flows and vehicle trips

Based on the seminal work of Samuelson (1952), as it seeks to maximize the economic welfare associated with the consumption and transportation of the cargo, taking into account the formation of UFTs

40

Two flavors

Independent Shipper-Carrier Operations:

Carrier and Shipper are independent companies

Carrier travels empty from its base to pick up cargo at shipper’s location(s)

Carrier delivers cargo to shipper’s customers

Carrier travels empty back to its base

Integrated Shipper-Carrier Operations:

Carrier and Shipper are part of the same company

Carrier is loaded at shipper’s location(s)

Carrier delivers cargo to shipper’s customers

Carrier travels empty back to its base

41

Integrated Shipper-Carrier Operations

Five suppliers deploy tours from their bases (rhomboids) to distribute the cargo they produce to various consumer (demand) nodes (circles)

42

Legend:

(Contested nodes are shown as

shaded circles)

Loaded trips made

by suppliers

Empty trips

Receiver

Supplier

A Spatial Price Equilibrium UFT Model

Samuelson’s model is reformulated to consider freight tours. A supplier i sends a cargo (eip1, eip2, eip3, and eip4) to different customers (p1, p2, p3, and p4)

43

Supplier

p1

p2 p3

p4

i

eip1

eip3

eip2

eip4Commodity flow

Notation:

Loaded Vehicle-trip

Empty Vehicle-trip

The cost of delivering to p3 is not the

(path) cost along i-p1-p2-p3.

It it is the (incremental) cost from p2

to p3, plus part of the empty trip cost.

A Spatial Price Equilibrium UFT Model (P1)44

iE

ii dxxsES0

)()(

u j

u

iji eE

0,

u

ij

u

jpiQwe u

l

Ttttt u

u

ul

ul

u L

L

lpppii

)(0

1

1,,0

11

Qwe

u u

uj

ui

L

i

L

ij

u

pp

1

1 1,

1,

uj

u

ij

)1,0(u

ij

0u

ije 0,, HTD ccc 0, ,, jiji td

(Social Welfare)

(Area under excess supply function)

(Excess supply)

(Linking flows to vehicle-trips)

(Tour length constraint)

(Capacity constraint)

(Conservation of flow)

(Integrality)

(Non-negativity)

u

SN

i

u

ij

u

ij

DN

j

i

SN

i

i eCESNSP1 11

)()(

ul

ul

ul

ul

ul

ul

ul

ul piHpEppTppDpi

u

piecctcdceC

,,,,, 11

)(

(Delivery cost to demand node)

MAX

Subject to:

This term is equal to the

summation of tour costs.

Thus, it could be replaced by

the summation of tour

costs, which allows to

eliminate the delivery cost

constraint

A Spatial Price Equilibrium UFT Model (P2)

MAX

Subject to:

45

u

uVi

SN

ii CESNSP )(

1

iE

ii dxxsES0

)()(

u j

u

iji eE

0,

u

ij

u

jpiQwe u

l

Ttttt u

u

ul

ul

u L

L

lpppii

)(0

1

1,,0

11

Qwe

u u

uj

ui

L

i

L

ij

u

pp

1

1 1,

juj

uij 1

,

ujiuij ,,)1,0(

0u

ije 0,, HTD ccc 0, ,, jiji td

(Net Social Payoff)

(Area under excess supply function)

(Excess supply)

(Linking flows to vehicle-trips)

(Tour length constraint)

(Capacity constraint)

(Conservation of flow)

(Integrality)

(Non-negativity)

However…

P2 is a nasty combinatorial and non-linear problem that is notoriously difficult to solve

To solve it, frame it as:

A dispersed SPE problem

A problem of profit maximization subject to competition (which is equivalent to the NSP formulation produced by Samuelson)

A dynamic problem in which competitors adjust decisions based on the market competition results

Use heuristics

46

Heuristic Solution Approach (Dispersed SPE)47

Pro

fit-

maxim

izin

g

rou

tin

g:

First-stage: Production

level, prices, profit margins, net

profits

Second-stage pricing: Compute

optimal prices, profit margins

Phase 1: Initialization of

TS, Generate initial solutions

Phase 2: Initial Improvement:

Perform neighborhood search

procedurePhase 3: Second Improvement, 2-

Opt procedure and repeat 2

Phase 4: Intensification, Perform

neighbor search on solutions from 3

Equilibrium?Production dynamics:

Update production level

Initialization: Assume

prices

Convergence?No

Market competition: Compute

purchases from suppliers

No

Yes

STOP Equilibrium? Production dynamics:

Update production level

NoYes

Equilibrium Results48

0

20

40

60

80

100

0 20 40 60 80 100

Y (m

iles

)

X (miles)

Consumers

SuppliersC2, D=8

C4, D=10

C3, D=4

C1, D=6

S1

S2

0

20

40

60

80

100

0 20 40 60 80 100

Y (m

iles

)

X (miles)

Consumers

SuppliersC2, D=8

C4, D=10

C3, D=4

C1, D=6

S1

S2

)(

)()1(

)1(

12

,,

iij

hujITijuhtQijuht

ijt

op

i

ijtssD

mmP

sP

0

20

40

60

80

100

0 20 40 60 80 100

Y (m

iles

)

X (miles)

Consumers

SuppliersC2, D=8

C4, D=10

C3, D=4

C1, D=6

S1

S2

C4

C4

(3.59, .97)

Two suppliers, four customers Vehicle-tours

Commodity flows and prices

Concluding Remarks

49

Conclusions

There are reasons to be optimistic:

The community is cognizant of the need to model tours

Collecting data and developing tour models

However:

The models developed are still in need of improvements

The data collected are small and not comprehensive

Simulations and hybrid models require better behavioral foundations that are not always validated

The most theoretically appealing models present significant computational challenges to be overcome

Entropy Maximization models offer an interesting avenue, though disregarding commodity flows

50

References

51

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block viii-j: freight demand modeling -tour models-

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