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An algorithm for multi-class network equilibrium problem in PCE of trucks: Application to the SCAG travel demand model

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The analysis of mixed traffic, which includes both private and trucks has received more attention recently due to the rapid development of urban truck traffic in many cities and regions of the world. In this paper, we consider a multi-class network equilibrium model where several classes of traffic, with their own travel times, interact on the links of the network. The volume/delay functions depend on the mix of trucks and cars and other factors like the slope of the links. Hence, the resulting cost functions are nonlinear, non-smooth and asymmetric. The problem is formulated as a nonlinear, nonsmooth, variational inequality model. A linear approximation type algorithm, which uses step size based on the method of successive averages, is used to solve the problem. Numerical results are reported for a large scale problem (6 classes of traffic, 3217 zones and 99867 links).
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Transportmetrica
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AN ALGORITHM FOR MULTI-CLASS NETWORK
EQUILIBRIUM PROBLEM IN PCE OF TRUCKS:
APPLICATION TO THE SCAG TRAVEL DEMAND
MODEL
Jia Hao Wu , Michael Florian & Shuguang He
To cite this article: Jia Hao Wu , Michael Florian & Shuguang He (2006) AN ALGORITHM
FOR MULTI-CLASS NETWORK EQUILIBRIUM PROBLEM IN PCE OF TRUCKS:
APPLICATION TO THE SCAG TRAVEL DEMAND MODEL, Transportmetrica, 2:1, 1-9, DOI:
10.1080/18128600608685656
To link to this article: http://dx.doi.org/10.1080/18128600608685656
Published online: 07 Jan 2009.
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Transportmetrica, Vol. 2, No. 1 (2006), 1-9
1
AN ALGORITHM FOR MULTI-CLASS NETWORK EQUILIBRIUM
PROBLEM IN PCE OF TRUCKS: APPLICATION TO THE SCAG TRAVEL
DEMAND MODEL
JIA HAO WU1, MICHAEL FLORIAN2 AND SHUGUANG HE3
Received 28 January 2005; received in revised form 23 March 2005; accepted 29 March 2005
The analysis of mixed traffic, which includes both private and trucks has received more attention recently
due to the rapid development of urban truck traffic in many cities and regions of the world. In this paper, we
consider a multi-class network equilibrium model where several classes of traffic, with their own travel times,
interact on the links of the network. The volume/delay functions depend on the mix of trucks and cars and
other factors like the slope of the links. Hence, the resulting cost functions are nonlinear, non-smooth and
asymmetric. The problem is formulated as a nonlinear, nonsmooth, variational inequality model. A linear
approximation type algorithm, which uses step size based on the method of successive averages, is used to
solve the problem. Numerical results are reported for a large scale problem (6 classes of traffic, 3217 zones and
99867 links).
KEYWORDS: Network equilibrium, multi-class traffic assignment, variational inequality problem
1. INTRODUCTION
The modeling of urban freight movements has received more attention recently due to
the need of representing the increased truck traffic on urban road networks. The models
used for transportation planning of urban areas are enhanced with the flows of trucks in
the period of times studied. The traffic flow characteristics of mixed traffic of cars and
trucks are quite complex since the delay depends on the proportion of the different
vehicles on a given link, the grade of the link and the behavior of the drivers. The model
presented in this paper is motivated by a detailed study carried out in the area covered by
the jurisdiction of the Southern California Association of Governments (SCAG) by a
consulting firm (Meyer, Mohaddes Associates Inc., 1999). The model that is developed
in this contribution is a multi-class network equilibrium model with asymmetric costs,
which is formulated as a variational inequality. It is solved by a linear approximation
algorithm, which uses step sizes computed with the method of successive averages. The
algorithm was implemented in a widely available transportation planning software
package and was applied to the network used by SCAG for transportation planning
purposes. This is a very large network consisting of 3,217 centroids, 25,430 nodes and
99,687 links.
The most difficult part of the model is that, as mentioned above the link travel times
depend on the mix of trucks and cars on the link and on the slope of the link as well. The
resulting link travel time functions are nonlinear and asymmetric. The volumes of trucks
of different classes (that is size) are converted into passenger car equivalents, or PCE’s
(we will use this notation even though the reference to PCU’s, passenger car units is
1 INRO Solutions Inc., 5160 Décarie Boulevard, Suite 610, Montreal, Quebec H3X 2H9, Canada, and Centre
for Research on Transportation, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, QC H3C
3J7 Canada. Present address: TJKM Transportation Consultants, 5960 Inglewood Dr. Suite 100, Pleasanton,
CA, USA 94588.
2 INRO Solutions Inc., 5160 Décarie Boulevard, Suite 610, Montreal, Quebec H3X 2H9, Canada, and Centre
for Research on Transportation, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, QC H3C
3J7 Canada. Corresponding author (E-mail: mike@inro.ca).
3 INRO Solutions Inc., 5160 Décarie Boulevard, Suite 610, Montreal, Quebec H3X 2H9, Canada.
2
common as well). The conversion factors depend on the percentage of each class of
traffic on the link since the interaction between the various vehicle types is complex and
quite variable.
An example of PCE conversions used may be found in the report of Meyer, Mohaddes
Associates Inc. (1999) where the total PCE’s on a link are a nonlinear function of three
classes of trucks and three classes of private cars. The difficulty as well as the novelty of
this problem arises from the fact that such a network equilibrium model was not
previously studied in the literature. A new mathematical formulation as well as a
solution algorithm must be developed for this model.
In this paper we focus on the model formulation, the solution algorithm and the
presentation of the computational results obtained with the SCAG network. The only
theoretical contributions in the literature that are relevant to the algorithm used in this
model are asymmetric cost single class models. Marcotte and Zhu (1996) and Magnanti
and Perakis (1994, 1997) use an LP based operator and a projection operator respectively
and prove the convergence of these algorithms under certain conditions. We extend these
types of algorithms for this multi-class of network equilibrium problem. The theoretical
study of the convergence of the algorithm that is presented in the following is beyond the
scope of this paper.
This paper is organized as follows. In Section 2, the mathematical model for the
problem is defined. In Section 3, a solution method is developed for the mathematical
model, while Section 4 is devoted into the application of the method in a real network.
Section 5 concludes the paper.
2. MODEL FORMULATION
In this section, the notation used is introduced in order to state the mathematical
formulation for the problem.
2.1 Notations
In this paper, the following notations are used. The links of the road network are
designated by Aa
, where A is the set of links. The demand for travel by user class
Mm for the origin-destination pair ),( ji is denoted as m
ij
T where M is the set of
classes. This demands may use paths RRm
ij where R is the set of all routes
m
ij
mji RR ),,( U=. Table 1 summarizes the definition of sets and indices used in this paper.
TABLE 1: Sets and indices used
A : link set of base network in period t a : link index of base network
W : total OD pairs w : OD pair index
m
ij
R : set of routes for pair (i,j), class m r : path index
M : class set m : class index
Given the sets and indices above, the following variables and given data are defined:
The variables include:
m
r
h : The path flow of class m on the route r,
3
m
a
f : The total link flow of class m on link a,
m
r
C : The path travel time of class m on route r,
a
v : The total link flow in PCE (passenger car equivalence) that include flows of
all classes on link a (which will be discussed in next section),
m
a
v : Link flow in PCE of class m on link a,
(
)
=mm
aa vv , and
)( aa vc : The travel time on link a for total link flow a
v in PCE.
The given data consists of:
m
ij
T : The travel demand from origin i to destination j of class m.
2.2 Total link flow in PCE
The total link flow in PCE, that is,
(
)
Mmffv m
aaa = | can expressed as a very
general function of link flows (in vehicles) of all classes Mmf m
a,, and the
parameters associated with link a. It is not just simply be a linear combination of the link
flows of all classes such as
β= mm
a
m
afv where m
β is a conversion factor into PCE
from the total link flow of class m. In particular, the following nonlinear function is
considered:
(
)
Mmgladjkvcpffv aa
m
aaa
m
a
m
a
m
a= |,,,,,, , (1)
where
m
a
p is the percentage of the link flow of class m on link a,
=Mm m
a
m
a
m
affp , (2)
a
vc is the link flow in PCE, a
v, over the capacity of link a, a
k, given as,
aaa kvvc /=, (3)
a
k is the link capacity in PCE on link a,
()
a
m
avcadj is an adjustment factor of class m on link a based on a
vc ,
a
l is the length on link a, and
a
g is the grade on link a.
This type of nonlinear function has been calibrated and validated with observed data
and can be formulated in the form of a look-up table in practice. Consider the following
example with a
v expressed as
(
)
×=
Mm
m
aaaa
m
aa
m
a
m
aa fglvcadjkpPCEv ,),(,, ,
where
(
)
aaa
m
aa
m
a
m
aglvcadjkpPCE ,),(,, can be presented in the comprehensive look-up
table (See Section 4 for an example).
2.3 Mathematical model
The feasible region of the problem is defined as follows:
4
MmijTh m
ij
r
m
r=
),(, , (4)
m
ij
m
rRrMmh ,,0 , (5)
where
MmAahf Rr ar
m
r
m
aδ=
,, , (6)
(
)
m
ij
Aa ara
m
a
m
rRrMmvcC δ=
,, , (7)
where (4) are the equations of conservation of flow, (5) is the nonnegativity of the path
flows, ar
δ is 1 if link a is on route r and is zero otherwise and (7) is the definition of
path travel time of class m on route r, m
r
C.
The multi-class network equilibrium problem can be formulated as a variational
inequality problem. Find
*
h such that
(
)
(
)
∈∈
hhhhC
WwMmRrrr
m
r
w
,0
** (8)
It is well known that while the total link flow v in PCE is unique if
(
)
)(vcm
a is strictly
monotone, the composed volumes
(
)
Mmf m, may not be unique with )()( vcvc a
m
a=
for all m. However the strict monotonicity conditions are difficult to verify for this
model. It is clear that the solution of the problem satisfies the following equilibrium
conditions:
w
m
r
m
w
m
r
m
w
m
rRrMm
hu
hu
C
=
>= ,,
0 if
0 if , (9)
where
}
m
rRr
m
wCu w
=min is the minimum travel time for pair w of class m, which are
the well known Wardrop (1952) user’s optimal conditions. The derivation of the
variational inequality formulation (8) from Wardrop’s user optimal principle (9) is well
known and may be referenced in the seminal work of Smith (1979). We may reference
also the survey chapter of Florian and Hearn (1995).
3. SOLUTION ALGORITHM
In the literature, there are many recursive averaging schemes. Some were studied by
Marcotte and Zhu (1996) and Magnanti and Perakis (1997). These can be used to solve
the variational inequality problem with only a single class and a mapping of either LP-
based operator or projection operator. The evaluation of the projection operator is
equivalent to a minimization of a quadratic convex optimization problem, while the LP-
based (linear approximation) operator can be used for multi-classes of demand, which
can be solved by solving a shortest path problem with a network loading (a
decomposable linear program problem by class) instead of the quadratic optimization
problem. We choose the LP-based operator for the development of the solution
algorithm, which is given as follows.
5
3.1 Multi-class mixed flow network assignment algorithm
Step 0 : Initialization. Start with 0,0 ,== lm
a
fl .
Step 1: Compute percentage of link flow, v/c ratio and link flow in PCE.
MmAaffp Mm lm
a
lm
a
lm
a=
,,
,,, ,
Aakvcvc a
l
a
l
a= +,
1,
(
)
lm
aaa
l
a
m
aa
lm
a
m
a
lm
afglvcadjkpPCEv ,,, ,),(,, ×= .
Step 2: Computation of link cost.
(
)
Aavc aa
,.
Step 3: Computation of the shortest path flow problem for each class m.
∈∈WwRrm
r
ml
r
m
whC ,
min ,
s.t. MmWwTh m
w
Rr m
r
m
w=
,, ,
WwRrMmh m
w
m
r,,,0 .
Step 4: Computation of link flow MmAahf Rr ar
m
r
m
aδ= ,, .
Step 5: Computation of step size (MSA)
(
)
MmAalffff lm
a
lm
a
lm
a
lm
a++=
+,,)1(
,,,1, .
Step 6: Apply a stopping criterion.
Step 7: 1
+
=ll . Go to Step 1.
4. APPLICATION
This algorithm was applied to solve the particular multi-class network equilibrium
problem in PCE in the network used by SCAG for transportation planning, which was
mentioned in Section 1. The algorithm was implemented in the EMME/2 (INRO, 1998)
transportation planning software by using its macro language and the judicious use of the
multi-class assignment module. Step 2, Step 3 and Step 4 of the algorithm use the first
iteration of a symmetric cost multi-class assignment module to compute shortest paths
and load the demand for different classes, while Step 1 and Step 5 are implemented with
network calculations. Step 6 is coded directly with a macro language procedure.
There are three classes of passenger vehicles and three classes of trucks. The well
known BPR volume delay function was used in its original forms:
(
)
(
)
(
)
4
/15.01 aaaaa kvvc ×+β= ,
where a
β
and a
k (practical capacity) are both parameters. The six classes of demand of
traffic are listed in Table 2.
TABLE 2: Six classes of demands
Class ID Descriptions
1 Passenger cars of one person demand that can not access HOV links
2 Passenger car of two person demand that can access whole network
3 Passenger cars of three person or more demand that can access whole network
4 Light-heavy duty trucks that can not access HOV links
5 Medium-heavy duty trucks that can not access HOV links
6 Heavy-heavy duty trucks that can not access HOV links
6
The trip demands for the six classes of traffic considered in the model are as follows:
Class 1 : 5,119,365 Passenger cars
Class 2 : 2,070,009 Passenger cars
Class 3 : 384,592 Passenger cars
Class 4 : 79,583 Light-Heavy trucks
Class 5 : 51,794 Medium-Heavy trucks
Class 6 : 34,809 Heavy-Heavy trucks
This is probably one of the largest multi-class assignment problems used in the practice
of transportation planning. The three classes of trucks are classified as
1. Light-Heavy : 8,500 to 14,000 GVW
2. Medium-Heavy : 14,000 to 30,000 GVW
3. Heavy-Heavy : over 30,000 GVW
In the computation, the various variables and data information for each link are stored, as
shown in Table 3.
TABLE 3: Partial link attributes
Notation Description
a
v Current total equivalent car flow
m
a
f Successive average flow on link a of class m
a
g Attribute as the grade for the link
Thus total link flow in PCE a
v on link a is a nonlinear function of three heavy duty
trucks and three auto vehicles which is defined as:
(
)
(
)
(
)
m
aa
m
aaa
m
a
m
aaaa
m
aa
m
a
m
apvcadjglpPCEglvcadjkpPCE ,,,,),(,, ×= ,
where
(
)
m
aa
m
apvcadj , and
(
)
aa
m
a
m
aglpPCE ,,
(
)
(
)
3,2,1for ,1,, == mglpPCE aa
m
a
m
a are
defined in Table 4 and Table 5. Thus the nonlinear functions of (1) are defined in the
form of look-up tables instead of the continuous functions. These functions have been
validated with observed data, as reported in Meyer, Mohaddes Associates Inc. (1999).
TABLE 4: m
a
adj values m = 4, 5, 6
4
a
p 5
a
p
6
a
p
a
vc
0-5 5-10 >10 0-5 5-10 >10 0-5 5-10 >10
<0.5 0.60 0.66 0.90
0.66 0.77 0.93 0.90 0.77 0.93
0.5-1.0 0.77 0.89 1.15 0.89 1.01 1.20 1.15 1.01 1.20
1.0-1.5 1.10 1.20 1.30 1.20 1.25 1.34 1.30 1.25 1.34
1.5-2.0 1.00 1.05 1.22 1.05 1.22 1.25 1.22 1.22 1.25
>2.0 1.19 0.66 1.26
1.05 1.24 1.29 1.26 1.24 1.29
Two measures of the convergence of the algorithm are provided. The first measure
(M1) is the relative difference l
d at iteration l between the volume at iteration l and
successive average volume at iteration l:
∑∑∑∑ = am lm
a
am lm
a
lm
a
lfffd ,,,
7
TABLE 5: Heavy duty truck PCE values
(
)
aa
m
a
m
aglpPCE ,,
m = 4 m = 5 m = 6
a
g a
g
a
g
m
a
p a
l
0-2 3-4 >4 0-2 3-4 >4 0-2 3-4 >4
1 2.0 4.2 6.4 3.4 6.9 8.8 4.3 8.0 11.3
0-5
>1 2.0 5.5 7.5 5.2 8.4 10.7 6.7 10.5 13.5
1 2.0 3.4 4.8 3.1 5.0 6.4 3.5 5.8 8.8 5-10
>1 2.0 4.2 5.3 3.9 5.9 7.8 4.8 7.8 13.5
1 2.0 3.3 4.1 2.8 4.5 6.0 3.2 5.1 8.3
>10 >1 2.0 3.5 5.0 3.7 5.3 7.5 4.0 7.5 12.5
The second measure (M2) is the relative gap l
rgap at iteration l computed with the
flow lm
a
f, which is the “all-or-nothing” assignment on shortest paths and the last flow
1, lm
a
f weighted by the current travel time
(
)
l
aa vc :
(
)
(
)
∑∑∑∑ = am lm
a
l
aa
am lm
a
lm
a
l
aa
lfvcffvcrgap 1,,1,
It is well known that if 0
l
d or 0
l
rgap as
l, then one can obtain an
equilibrium solution. This solution procedure was implemented in the EMME/2 software
package as a macro. The algorithm converges on this network, as shown in Figure 1,
where M1 and M2 are the two convergence measures mentioned above.
Convergence Curve
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1 2 3 4 5 6 7 8 9 101112131415
iterations
relative value
M1
M2
FIGURE 1: Convergence of the solution procedure
The computations require about 40 min per iteration on a SUN SPARC ULTRA 5
workstation and about 7 minutes on an IBM Thinkpad T40 based on an Intel Centrino
8
1.6 Mhz. If the algorithm is used in a travel demand forecasting model, one may need
only 4-6 iterations for a reasonable convergence of an inner loop. The assignment results
are shown in Figures 2 and 3 where private vehicles (that is, m = 1, 2, 3) are not shown.
FIGURE 2: Three classes of trucks – general view
FIGURE 3: Three classes of trucks – a window
9
5. CONCLUSION
In this paper, the issue of a multi-class mixed car and truck traffic network equilibrium
problem was considered. Based on the analysis of the problem, a variational inequality
problem was formulated and an adapted successive averaging method was developed for
solving a six-class variational inequality problem. Then the linear approximation based
method was applied to a real network with good numerical results. This model is
actually used in practice.
ACKNOWLEDGEMENT
This project is supported in part by NSERC individual operation grants OGP0157735
and OGP0007406.
REFERENCES
Florian, M. and Hearn, D. (1995) Network equilibrium methods. Chapter in Handbook
on Operations Research – Network Routing, 8, pp. 485-550.
INRO Consultants Inc. (1998) EMME/2 User’s Manual, Montreal.
Magnanti, T.L. and Perakis, G. (1994) Averaging schemes for variational inequalities
and systems of equations. Mathematics of Operations Research, 22, 568-587.
Magnanti, T.L. and Perakis, G. (1997) Solving variational inequality and fixed point
problems by averaging and optimizing potentials, OR 324-97, MIT.
Marcotte, P. and Zhu, D. (1996) Convergence properties of feasible descent algorithms
for solving variational inequalities in banach spaces. CRT-96-12, University of
Montreal.
Meyer, Mohaddes Associates Inc. (1999) Heavy Duty truck Model and VMT Estimation.
Report to Southern California Association of Governments.
Smith, M.J. (1979) The existence, uniqueness and stability of traffic equilibria.
Transportation Research Part B, 13, 295-304.
Wardrop, J.G. (1952) Some theoretical aspects of road traffic research. Proceedings of
the Institution of Civil Engineers, Part II, 1, 325-378.
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... Assets condition is obtained using deterioration models such us HDM-4 [15]. Level of Service is estimated using travel time models such us the Bureau of Public Roads (BPR) model [36]. The model of road performance variation over time includes 4 stages [23]: stage 1 is the normal operation; stage 2, in which the performance drops and the operation is limited; stage 3 represents the recovery of the infrastructure performance, and stage 4 is the normal operation until the occurrence of a new natural event. ...
... • Road segment performance (D ij , dimensionless): is the normal travel time to the travel time under deficient operating conditions ratio, multiplied by the traffic volume under deficient operating conditions and the normal traffic volume ratio. The travel time is estimated with Eq. (6), assuming there is no traffic congestion [36]. ...
Article
Natural events affect road performance, causing traffic disruptions and increased travel time. Road agencies face the dilemma of selecting more cost-effective alternatives for recovering the road performance, minimizing the costs, recovery time, and the loss of serviceability. The paper aims to develop a model integrating the cost-saving and the resilience evaluation to prioritize recovery strategies. Indexes are proposed to estimate the relative change in resilience, road agency and user costs produced by various recovery strategies from a base strategy. Both indices are integrated into a priority index that combines resilience, cost savings and redundancy. The procedure was applied to a case study in which earthquakes were simulated using recurrence models previously calibrated for Chile. Its effect on a simple road network composed of road platforms and bridges was estimated. The priority index was sensitized concerning traffic, damage of the road platform, and re-routing length. It was concluded that the variation in resilience and costs associated with recovery strategies cannot be considered as separate prioritization criteria, especially when changes in one or other decision variable are small.
... The initial structural and non-structural estimated costs, based on the Iranian building price list [36], are 100,000$ and 28,000$, respectively. In the life-cycle cost analysis, it is assumed that the monetary discount rate is 10% [37], the bridge life span is 100 years [14], and 2 years are required for downtime bridge renovation [36]. ...
... One of the major relations for the travel timevolume function employed for each arc length, which is valid for this site, is Eq. (31) [37]: ...
Article
While rubber isolators are widely used for the seismic protection of bridges, their influence on the total life-cycle costs (TLCC) is generally neglected. This study aims to provide a framework to obtain the most cost-effective design solutions of Lead-Rubber Bearing (LRB) isolators for minimum objective functions based on the life-cycle cost of bridges. Also, a comprehensive analytical study is conducted to assess the effects of LRB characteristics on the total life-cycle costs (TLCC) consisting of the initial costs of LRB isolators and structural elements, traffic damage costs, and structural and non-structural damage costs due to possible earthquake events during the bridge life-span. While changing the lead core diameter, confined diameter, and the total thickness of the rubber layers could influence the TLCC of the bridges, the effect of lead core diameter is shown to be more significant. By using lead core diameter as the main design parameter, increasing the initial cost of LRBs up to a certain level can significantly reduce (up to 38%) TLCC of the bridges. However, beyond this limit, the TLCC may increase. Based on the results of over 13,000 Reinforced Concrete (RC) bridges under different earthquake hazard levels, the best design solutions are identified for different design objectives. The results of this study should prove the benefit of the proposed method for a more efficient design of bridges with lower TLCC.
... (10). In conventional multiclass user equilibrium with multiple types of vehicles (e.g., trucks and cars), different types of vehicle flows are usually converted into equivalent passenger car flows using a metric called passenger car equivalent (Wu et al., 2006;Hajibabai et al., 2014;de Andrade et al., 2017). Inspired by the concept of passenger car equivalent, Liu and Song (2019) ...
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Although autonomous vehicle (AV) technology is expected to bring dramatic societal, environmental, and economic benefits, the high vehicle cost might slow the adoption of AVs. This paper explores an infrastructure-enabled autonomous driving system, which is a promising remedy to the high cost of AVs. Specifically, the system combines vehicles and infrastructure in the realization of autonomous driving. Equipped with roadside sensing, computing, and communicating devices, an ordinary road can be upgraded into an “automated road” enabling autonomous driving service for vehicles with the minimum required on-board devices. The vehicle costs can thus be significantly reduced. We envision that automated roads will be deployed in transportation networks to serve a new type of vehicle called infrastructure-enabled autonomous vehicles (IEAVs), which can be driven autonomously only on automated roads but manually on ordinary roads. Therefore, IEAV users may experience inconvenience costs due to transitions between autonomous driving and manual driving, and the frequent switch will inevitably yield high inconvenience costs. Therefore, to minimize their individual travel cost, they have to decide whether to switch to the autonomous driving mode when heading to an automated road. Considering such a unique feature of IEAVs, we proposed a group of driving-mode-choice equilibrium conditions to describe IEAV drivers’ driving mode choice behaviors, in which we considered drivers’ travel time costs, service charges of autonomous driving, and inconvenience costs due to driving mode change. Combining traditional route-choice equilibrium conditions with the proposed driving-mode-choice equilibrium conditions, we developed a new user equilibrium (UE) model to describe the equilibrium flow distributions in a road network with automated roads and mixed-autonomy traffic. The UE model is formulated as a novel non-linear complementarity problem, and its solution existence is discussed. To solve the UE model, a network expansion method is proposed to reformulate the UE model as a standard path-based UE model. A route-swapping-based solution algorithm is then used to solve the reformulated UE model. Numerical studies are presented to demonstrate the proposed models and algorithms.
... For a review, the reader is referred to [34,37]. In this study, we use an efficient sequential linear approximation algorithm with step sizes according to the method of successive averages (MSA) introduced in [38] to solve the lower level equilibrium problem. ...
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This study focuses on network configurations to accommodate automated vehicles (AVs) on road networks during the transition period to full automation. The literature suggests that dedicated infrastructure for AVs and enhanced infrastructure for mixed traffic (i.e., AVs on the same lanes with conventional vehicles) are the main alternatives so far. We utilize both alternatives and propose a unified mathematical framework for optimizing road networks for AVs by simultaneous deployment of AV-ready subnetworks for mixed traffic, dedicated AV links, and dedicated AV lanes. We model the problem as a bilevel network design problem where the upper level represents road infrastructure adjustment decisions to deploy these concepts and the lower level includes a network equilibrium model representing the flows as a result of the travelers’ response to new network topologies. An efficient heuristic solution method is introduced to solve the formulated problem and find coherent network topologies. Applicability of the model on real road networks is demonstrated using a large-scale case study of the Amsterdam metropolitan region. Our results indicate that for low AV market penetration rates (MPRs), AV-ready subnetworks, which accommodate AVs in mixed traffic, are the most efficient configuration. However, after 30% MPR, dedicated AV lanes prove to be more beneficial. Additionally, road types can dictate the viable deployment plan for certain parts of road networks. These insights can be used to guide planners in developing their strategies regarding road network infrastructure during the transition period to full automation.
... Dedicated AV lanes in the network produce a net benefit (e.g., reduced travel cost). Their study merges a multi-class user-equilibrium model (Wu et al., 2006) and a diffusion model describing the evolution of AV market penetration in a time-dependent deployment model, which is solved by a heuristic algorithm as a bi-level model. They applied to the south Florida case study, and the results showed that AV lanes should be deployed progressively when the AV market penetration rate reaches above 20 percent. ...
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Traffic flow measurement is very important for traffic management systems. However, the existing traditional measurement approaches are highly time-consuming and expensive to continuously gather the required data and to maintain the corresponding equipment, such as loop detectors and video cameras. On the other hand, many services on the web propose to estimate automobile travel time taking into account traffic conditions thanks to crowd sourced data (Floating Car Data). This work proposes to reconstruct, from estimated travel time, traffic flows using machine learning method. In particular, we evaluate the capacity of Gaussian Process Regressor (GPR) to address this issue. After obtaining estimated travel time on a given route, a clustering process shows that travel duration profiles in each day can be associated to different “types of day”. Then, different regressors are trained in order to estimate traffic flows from travel duration. In the “multi-model” variant, we trained a Regressor for each type of day. Conversely, in the “single model” variant, only one Regressor is trained (the type of day is not taken into account). This is an innovative work to estimate and reconstruct the traffic flow in transportation networks with machine learning method from aggregated Floating Car Data (FCD). A series of experiments are conducted to compare the estimated traffic flows, obtained by the proposed single model and multi-model, and the real ones from actual sensors. The obtained results show that both single model and multi-models can capture the tendency of real traffic flows. Furthermore, the performance can be improved by regulating parameters in GPR machine learning model, such as half width of sample window and sample size (a whole week or only weekdays), and multi-models can highly increase the performance compared with the single model. Therefore, the proposed GPR machine learning and FCD based new method can replace those traditional loop detectors for the measurement of traffic flow.
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Road networks, especially those built in an extreme environment, are at risk of service failure after long-term exposure to various types of disruptive events. With growing costs of service failure, it has become increasingly important to evaluate the dynamic operability characteristics, i.e., the dynamic resilience, of road networks. However, existing studies in this field generally considered a one-time disaster event during a relatively short period, and thus became insufficient for assessing the resilience of road networks in an extreme environment with frequent disaster events. This paper aims to propose a Time-dependent Resilience Analysis Framework (TIDRAF) by modeling the dynamic resilience of a road network in an extreme environment (where there are frequent disruptive events) and sequential repair actions. The original resilience triangle diagram is extended into a long-term paradigm to fit the road network with its frequent-disaster environment. The road network in the Tibetan Plateau, which has for years severely suffered from frequent disaster events, was selected as the case to validate the proposed TIDRAF. Based on the results of the analysis, three optimization strategies are presented to improve the resilience of the network against failures.
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"December 1997." Includes bibliographical references (p. 41-46). Supported in part by NSF Grant. 9634736-DMI T.L. Magnanti, G. Perakis.
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We study averaging methods for solving variational inequalities whose underlying maps are nonexpansive and for solving systems of (asymmetric) equations. Our goal is to establish global convergence results using weaker assumptions than are traditional in the literature. We examine averaging schemes for relaxation algorithms and for their specialization as projection and linearization methods and as Cohen’s auxiliary framework. For solving systems of equations, we consider averaging for a general class of methods that includes, as a special case, a generalized steepest descent method. We also develop a new interpretation of a norm condition typically used for establishing convergence of relaxation schemes, by associating it with a strong-f-monotonicity condition.
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The paper formulates link-flow definitions of equilibrium and stability, and gives conditions which guarantee the existence, uniqueness and stability of traffic equilibria. Junction delays in towns usually depend on the traffic flow along intersecting links; the theory presented here is designed to be applicable when there are such junction interactions.
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This work is concerned with the analysis of convergence properties of feasible descent methods for solving monotone variational inequalities in Banach spaces. Keywords: Banach spaces, variational inequalities, Frank-Wolfe algorithm, global convergence, projection. 1 Introduction Let X be a reflexive Banach space, X the topological dual space of continuous linear functionals defined on X , C a closed, convex and nonempty subset of X and F a continuously differentiable (in the Gateaux sense) operator from C into X . In this paper, we analyze the convergence behaviour of descent algorithms for finding points of C that satisfy the variational inequality: hF (x ); x Gamma x i 0 8x 2 C: (1) We assume that the solution set of (1) is nonempty. To find a solution, we introduce a nonnegative and continuously differentiable functional OE : C ! R + such that OE(x) = 0 if and only if x is a solution of the variational inequality. Such merit functions and the associated descent alg...
Heavy Duty truck Model and VMT Estimation
  • Meyer
Meyer, Mohaddes Associates Inc. (1999) Heavy Duty truck Model and VMT Estimation. Report to Southern California Association of Governments.
Network equilibrium methods Chapter in Handbook on Operations Research – Network Routing Averaging schemes for variational inequalities and systems of equations
  • References Florian
  • M Hearn
  • D Manual
  • Montreal
  • T L Magnanti
  • G Perakis
Downloaded by [University of Hong Kong Libraries] at 08:28 11 November 2014 REFERENCES Florian, M. and Hearn, D. (1995) Network equilibrium methods. Chapter in Handbook on Operations Research – Network Routing, 8, pp. 485-550. INRO Consultants Inc. (1998) EMME/2 User's Manual, Montreal. Magnanti, T.L. and Perakis, G. (1994) Averaging schemes for variational inequalities and systems of equations. Mathematics of Operations Research, 22, 568-587.
If the algorithm is used in a travel demand forecasting model, one may need only 4-6 iterations for a reasonable convergence of an inner loop. The assignment results are shown in Figures 2 and 3 where private vehicles (that is, m = 1, 2, 3) are not shown
  • Mhz
Mhz. If the algorithm is used in a travel demand forecasting model, one may need only 4-6 iterations for a reasonable convergence of an inner loop. The assignment results are shown in Figures 2 and 3 where private vehicles (that is, m = 1, 2, 3) are not shown. FIGURE 2: Three classes of trucks – general view FIGURE 3: Three classes of trucks – a window
Network equilibrium methods
  • M Florian
  • D Hearn
Florian, M. and Hearn, D. (1995) Network equilibrium methods. Chapter in Handbook on Operations Research -Network Routing, 8, pp. 485-550.