urban freight tour models: state of the art and practice 1 josé holguín-veras, ellen thorson, qian...
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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 ConceptsUrban Freight Tours: Empirical EvidenceUrban Freight Tour Models
Simulation Based ModelsHybrid ModelsAnalytical Models
Analytical Tour ModelsConclusions
2
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
R2
R3
R5
R6
R4
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
Schiedam
Alphen Apeldoorn
Amsterdam
Denver
New York City
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 TruckCombination
TruckAverage 6.5 7.01 tour/day 7.2 7.72 tours/day 4.5 3.73 tours/day 2.8 3.3
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 stopsParcel deliveries: 50-100 stops/tour
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Urban Freight Tour Models
The UFT models could be subdivided into:Simulation modelsHybrid modelsAnalytical 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
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
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 Models
Based on entropy maximization theory (Wilson, 1969; Wilson, 1970; Wilson, 1970)
Computes most likely solution given constraintsKey concepts:
Tour sequence: An ordered listing of nodes visitedTour flow: The flow of vehicle-trips that follow a
sequence
The problem is decomposed in two processes: A tour choice generation processA 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
23/4/18
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 stateTotal 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.
The equivalent model of formulation 2:Entropy Maximization Tour Flow Model
Trip production constraints
Total travel time constraint
Total handling time constraint
22
M
mmmm tttZ
1
)ln(
i
M
mmim Ota
1
T
M
mmmT Ctc
1
H
M
mmmH Ctc
1
0mt
)( i
)( 1
)( 2
MIN
Subject to:
First-order conditions (tour distribution models)
Traditional gravity trip distribution model
Entropy Maximization Tour Flow Model
)exp()exp()exp( *
1
*
1
***m
N
iimi
N
imimim cacat
23
)exp( **ijjjiiij cDBOAt
)exp()exp( *2
*1
1
**HmTm
N
iimim ccat
Formulation 1:
Formulation 2:
Formulation 1 and the traditional GM model have exactly the same number of parameters
Entropy Maximization Tour Flow Model
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 usedMuch better than the traditional GM
24
25Case Study: Denver Metropolitan Area
Test network919 TAZs among which 182 TAZs contain home
bases of commercial vehicles613 tours, representing a total of 65,385 tour flows /
day Calibration done with 17,000 tours (from heuristics)
Estimation procedureSorting input data: aggregate the observed tour
flows to obtain trip productions and total impedanceEstimation: 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
TD-FTS Model
Bi-objective Program:
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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
mi
dmim
K
d
NiOxg11
(3)
M
mj
dmjm
K
d
NjDx11
(4)
M
m
dmm
K
d
Cxc11
(5)
KdMmx dm ,,0
(6)
Function to replicate TD traffic counts
Trip Production Constraint
I can drop Trip Attraction ConstraintImpedance 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 IndustryImpedance 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 EstimatesEarly 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
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 52
42
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
ijijT
ijmijm
mmT
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 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 Cost36
),( ctatta xxtc
),( ctacca xxtc
a
matam cC
ijaa
caij 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 flowsTt: 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
ijmijm
ct
Tt
TTW
,
)! !(
! !Max
m ijijijijmmm TTTtttz )ln ln(Min
Multiclass Equilibrium ODS FormulationsMin
Subject to
Subject to
38
m ij
ijijijmmm TTTtttz )ln ln(
0)(),()(),( ****** ij
ijijT
ijmijm
mmT
ijmm TTTtCttttC
NitOM
mimm
ti ,...,2,1
1
NitDM
mjmm
tj ,...,2,1
1
},...2,1{ 1
NiTON
jij
ci
},...2,1{ 1
NjTDN
iij
cj
a
matam cC
ijaa
caij cC
atXm
mamta
aTXij
ijaijca
},...2,1{ QaVVX ta
ta
ta
},...2,1{ QaVVX ca
ca
ca
0mt
0ijT 0 taX
0 caX
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
General principles
The models estimate commodity flows and vehicle trips that arise under competitive market equilibrium
Conceptual advantages:Account for toursProvide 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 companiesCarrier travels empty from its base to pick up cargo
at shipper’s location(s)Carrier delivers cargo to shipper’s customersCarrier travels empty back to its base
Integrated Shipper-Carrier Operations:Carrier and Shipper are part of the same companyCarrier is loaded at shipper’s location(s)Carrier delivers cargo to shipper’s customersCarrier 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)
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Legend:
(Contested nodes are shown as shaded circles)
Loaded trips made by suppliers
Empty tripsReceiver
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)
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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.
A Spatial Price Equilibrium UFT Model (P1)44
iE
ii dxxsES0
)()(
u j
uiji eE
0,
uij
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
uij
)1,0(uij
0uije 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
uij
uij
DN
ji
SN
ii eCESNSP
1 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
uiji eE
0,
uij
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(
0uije 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 problemA 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
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
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
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
ijtop
iijt ssD
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
(2.2
4, .7
3)
(2.84, .89)
C4
(1.61, .47)
(6.41, .10)
(3.76, .10)
(5.16, .10)
(2.39, .10)
(3.59, .97)
Two suppliers, four customers Vehicle-tours
Commodity flows and prices
Conclusions
There are reasons to be optimistic:The community is cognizant of the need to model toursCollecting data and developing tour modelsHowever:
The models developed are still in need of improvementsThe 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
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