[PPT] - Urban Freight Tour Models: State of the Art and Practice Jos PowerPoint Presentation

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Urban Freight Tour Models: State of the Art and Practice

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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)

SLIDE 2

Outline

Introduction, Basic Concepts Urban Freight Tours: Empirical Evidence Urban Freight Tour Models

Simulation Based Models Hybrid Models Analytical Models

Analytical Tour Models Conclusions

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Introduction and Basic Concepts

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SLIDE 4

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

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HB

S-2

Loaded vehicle-trip Commodity flow Notation: Consumer (receiver) Empty vehicle-trip

S-1 S-3

R1 R2 R3 R5 R6 R4

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Empirical Evidence on Urban Freight Tours

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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

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1 2 3 4 5 6 7 8 9

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

Average number of stops/tour Population

Schiedam Alphen Apeldoorn Amsterdam Denver New York City

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Characterization of Urban Freight Tours (UFT)

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

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Stops/Tour Single Truck Combination 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

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Characterization of Urban Freight Tours (UFT)

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

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Urban Freight Tour Models

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Urban Freight Tour Models

The UFT models could be subdivided into:

Simulation models Hybrid models Analytical models

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Simulation Models

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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)

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Hybrid Models

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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)

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Analytical Models

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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)

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Entropy Maximization Tour Flow Model

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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

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Entropy Maximization Tour Flow Model

 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

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Entropy Maximization Tour Flow Model

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Definition of states in the system:

State State Variable Micro state

Individual commercial vehicle journey starting and ending at a home base (tour flow) by following a tour ;

Meso state The 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.

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Static version of EM Tour Flow Model

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The equivalent model of formulation 2:

Entropy Maximization Tour Flow Model

Trip production constraints Total travel time constraint Total handling time constraint

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



 

M m m m m

t t t Z

1

) ln (

i M m m im

O t a 



1 T M m m m T

C t c 



1 H M m m m H

C t c 



1



m

t

) (

i

 ) (

1

 ) (

2



MIN Subject to:

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First-order conditions (tour distribution models) Traditional gravity trip distribution model

Entropy Maximization Tour Flow Model

) exp( ) exp( ) exp(

* 1 * 1 * * * m N i im i N i m im i m

c a c a t    

 

 

  

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) exp(

* * ij j j i i ij

c D B O A t  

) exp( ) exp(

* 2 * 1 1 * * Hm Tm N i im i m

c c a t     





Formulation 1: Formulation 2:

Formulation 1 and the traditional GM model have exactly the same number of parameters

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Entropy Maximization Tour Flow Model

The optimal tour flows are found under the objective

f 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

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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

btain 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

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Estimated vs. observed tour flows

Performance of the Models

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Time-Dependent Freight Tour Synthesis Model

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TD-FTS Model

 Bi-objective Program:

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PROGRAM TD-ODS 1 Minimize

 

 

 

 

M m d m d m d m K d

x x x x z

1 1

) ln( ) ( Minimize

2 1 1 ' 1 1

2 1 ) (

 

   

         

A a K k K d M m d m kd am k a

x v x e  Subject to:

 

 

  

M m i d m im K d

N i O x g

1 1

 

 

  

M m j d m jm K d

N j D x

1 1



 

 



M m d m m K d

C x c

1 1

K d M m xd

m

    , ,

Function to replicate TD traffic counts Trip Production Constraint I can drop Trip Attraction Constraint Impedance Constraint Possible combinations

f flow

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TD-FTS Model

Multi-attribute Value formulation:

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PROGRAM TD-FTS3 Minimize

     

) ( ) ( ) ( ), (

2 2 1 1

x x x x z v e v z e U       Subject to:

 

 

   

M m y j d y m jm K d

Y y N j D x

1 , 1

, 

) (

,y j





  



K d M m Y y d y m m

C x c

1 1 1 ,

) (

Y y K d M m xd

y m

     , , ,

,

Trip Production Constraint per Industry Impedance Constraint Utility function from DM

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Application to the Denver Region

30 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

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TD-FTS Performance per Time Interval

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y = 0.755x R² = 0.913 50 100 150 200 250 300 350 400 450 200 400 600 Estimated TD Tour Flows Actual TD Tour Flows

TD-ODS AM OH

y = 1.049x R² = 0.996 500 1000 1500 2000 2500 500 1000 1500 2000 2500 Estimated TD Tour Flows Actual TD Tour Flows

TD-ODS AM PH

y = 1.032x R² = 0.998 500 1000 1500 2000 2500 500 1000 1500 2000 2500 Estimated TD Tour Flows Actual TD Tour Flows

TD-ODS PM PH

y = 0.928x R² = 0.983 100 200 300 400 500 600 700 800 200 400 600 800 1000 Estimated TD Tour Flows Actual TD Tour Flows

TD-ODS PM OH

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Multiclass Equilibrium Demand Synthesis

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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 traffic

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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

ne 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 Cost

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t c t c t c c t a

X X X X X X X X t

5 2 4 2 3 2 1

) , (            

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In a multiclass equilibrium, the cost functions of the modes are asymmetric, traffic flows interact The user optimal assignment cannot be written as an

ptimization problem

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

Multiclass Equilibrium

35

) ( ) , ( ) ( ) , (

* * * * * *

     

 

ij ij ij T ij m ij m m m T ij m m

T T T t C t t t t C

*

   

m m m

t C C

*

   

ij ij ij

T C C

SLIDE 36

 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 parties value 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 Cost

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) , (

c t a t t a

x x t c    ) , (

c t a c c a

x x t c   



 

a ma t a m

c C 

ija a c a ij

c C   

ma



ija



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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

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

  

ij m ij m c t

T t T T W

,

) ! ! ( ! ! Max



     

m ij ij ij ij m m m

T T T t t t z ) ln ln ( Min

SLIDE 38

Multiclass Equilibrium ODS Formulations

Min

Subject to Subject to

38



     

m ij ij ij ij m m m

T T T t t t z ) ln ln (

) ( ) , ( ) ( ) , (

* * * * * *

     

 

ij ij ij T ij m ij m m m T ij m m

T T T t C t t t t C

 

N i t O

M m im m t i

,..., 2 , 1

1

  





 

N i t D

M m jm m t j

,..., 2 , 1

1

  





} ,... 2 , 1 {

1

N i T O

N j ij c i

  



} ,... 2 , 1 {

1

N j T D

N i ij c j

  





 

a ma t a m

c C 

ija a c a ij

c C   

a t X

m ma m t a

   a T X

ij ija ij c a

  

} ,... 2 , 1 { Q a V V X

t a t a t a

      } ,... 2 , 1 { Q a V V X

c a c a c a

     



m

t



ij

T



t a

X



c a

X

VI problem to

btain a UE

condition for cars and trucks The objective is to find the most likely ways to distribute tours considering congestion

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Spatial Price Equilibrium Tour Models

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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

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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

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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

SLIDE 43

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)

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Supplier

p1 p2 p3 p4 i eip1 eip3 eip2 eip4

Commodity 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.

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A Spatial Price Equilibrium UFT Model (P1)

44



 

i

E i i

dx x s E S ) ( ) (





u j u ij i

e E

,

 

 u ij u j p i

Q we

u l

 T t t t t

u u u l u l u

L L l p p p i i

   



 



) (

1 1 , ,

1 1

Q we

u u u j u i

L i L i j u p p



 

    1 1 1 ,

1

,





u j u ij

 ) 1 , ( 

u ij

 

u ij

e

, , 

H T D

c c c ,

, ,



j i j i

t d

(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

e C E S NSP

1 1 1

) ( ) (

u l u l u l u l u l u l u l u l

p i H p E p p T p p D p i u p i

e c c t c d c e C

, , , , ,

1 1

) (    

 

(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

SLIDE 45

A Spatial Price Equilibrium UFT Model (P2)

MAX Subject to:

45

   

 u u V i SN i i

C E S NSP ) (

1



 

i

E i i

dx x s E S ) ( ) (





u j u ij i

e E

,

 

 u ij u j p i

Q we

u l

 T t t t t

u u u l u l u

L L l p p p i i

   



 



) (

1 1 , ,

1 1

Q we

u u u j u i

L i L i j u p p



 

    1 1 1 ,

j

u j u ij

   1

, 

u j i

u ij

, , ) 1 , (   



u ij

e

, , 

H T D

c c c ,

, ,



j i j i

t d

(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)

SLIDE 46

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

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Heuristic Solution Approach (Dispersed SPE)

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Profit-maximizing routing: First-stage: Production level, prices, profit margins, net profits Second-stage pricing: Compute

ptimal prices, profit margins

Phase 1: Initialization of TS, Generate initial solutions Phase 2: Initial Improvement: Perform neighborhood search procedure Phase 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 No Yes

SLIDE 48

Equilibrium Results

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20 40 60 80 100 20 40 60 80 100 Y (miles) X (miles) Consumers Suppliers C2, D=8 C4, D=10 C3, D=4 C1, D=6 S1

S2

20 40 60 80 100 20 40 60 80 100 Y (miles) X (miles) Consumers Suppliers C2, D=8 C4, D=10 C3, D=4 C1, D=6 S1

S2

) ( ) ( ) 1 ( ) 1 ( 1

2 , , i i j h u j ITijuht Qijuht ijt

p

i ijt

s s D m m P s P        



  

20 40 60 80 100 20 40 60 80 100 Y (miles) X (miles) Consumers Suppliers C2, 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

SLIDE 49

Concluding Remarks

49

SLIDE 50

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

SLIDE 51

References

51

SLIDE 52

References

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