Content uploaded by Michael Florian
Author content
All content in this area was uploaded by Michael Florian on Oct 15, 2021
Content may be subject to copyright.
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ttra20
Download by: [Bibliothèques de l'Université de Montréal] Date: 23 March 2017, At: 06:44
Transportmetrica
ISSN: 1812-8602 (Print) 1944-0987 (Online) Journal homepage: http://www.tandfonline.com/loi/ttra20
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.
Submit your article to this journal
Article views: 59
View related articles
Citing articles: 13 View citing articles
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 Ω 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
a∈∈∀δ=
∑
∈
,, , (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 Ω∈
*
h such that
(
)
(
)
Ω∈∀≥−
∑
∑
∑
∈∈∈
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
a∈∈∀δ= ∑∈,, .
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.