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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.

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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 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.