road pricing for congestion control with unknown

Transportation Research Part C 18 (2010) 157–175

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Transportation Research Part C

journal homepage: www.elsevier .com/locate / t rc

Road pricing for congestion control with unknowndemand and cost functions

Hai Yang a,*, Wei Xu b, Bing-sheng He c, Qiang Meng d

a Department of Civil Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR Chinab School of Management and Engineering, Nanjing University, Nanjing, PR Chinac Department of Mathematics, Nanjing University, Nanjing, PR Chinad Department of Civil Engineering, The National University of Singapore, Singapore

a r t i c l e i n f o

Article history:Received 29 January 2008Received in revised form 14 May 2009Accepted 17 May 2009

Keywords:Road pricingCongestion controlTraffic equilibriumUnknown demand

0968-090X/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.trc.2009.05.009

* Corresponding author. Tel.: +852 2358 7178; faE-mail address: [email protected] (H. Yang).

a b s t r a c t

It is widely recognized that precise estimation of road tolls for various pricing schemesrequires a few pieces of information such as origin–destination demand functions, link tra-vel time functions and users’ valuations of travel time savings, which are, however, not allreadily available in practice. To circumvent this difficulty, we develop a convergent trial-and-error implementation method for a particular pricing scheme for effective congestioncontrol when both the link travel time functions and demand functions are unknown. Thecongestion control problem of interest is also known as the traffic restraint and road pric-ing problem, which aims at finding a set of effective link toll patterns to reduce link flowsto below a desirable target level. For the generalized traffic equilibrium problem formu-lated as variational inequalities, we propose an iterative two-stage approach with a self-adaptive step size to update the link toll pattern based on the observed link flows and givenflow restraint levels. Link travel time and demand functions and users’ value of time are notneeded. The convergence of the iterative toll adjustment algorithm is established theoret-ically and demonstrated on a set of numerical examples.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Road pricing has been recognized as an effective approach to traffic demand management and control. By charging a suit-able toll, traffic demand and flow distribution on the road network can be influenced to achieve more efficient utilization ofexisting limited road capacity and generation of toll revenues for road maintenance and construction. From a theoretical per-spective, road pricing has been a subject of substantial research for a few decades by transportation economists and scien-tists (Lindsey, 2006). The initial idea of road pricing was suggested by Pigou (1920), who used the example of a congestedroad to make points about externalities and optimal congestion charges. Seminal works on both intellectual and practicaldevelopments after Pigou’s idea include Walters (1961), Beckmann (1965) and Vickrey (1969). A notable application of eco-nomic theory to road network pricing is the well-known first-best or marginal-cost pricing. According to this theory, theadditional cost or congestion externality that a road user imposes on other users can be internalized by charging tolls, there-by driving a user equilibrium (UE) flow pattern toward the system optimum (SO). In a congested network, the optimal toll tobe levied on each link is equal to the difference between the marginal social cost and the marginal private cost and it can bedetermined by solving a socially optimal traffic equilibrium problem. Various extensions of marginal-cost pricing can bemade to networks with link flow interactions, multiple vehicle types, such as trucks and cars, and with flow capacity

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158 H. Yang et al. / Transportation Research Part C 18 (2010) 157–175

constraints and queues (Dafermos and Sparrow, 1971; Dafermos, 1973; Smith, 1979a; Yang and Huang, 1998). In spite of itsperfect theoretical basis, the first-best pricing scheme is of little practical interest. The second-best charging schemes aremore practically relevant and indeed have received ample attention recently. A wide variety of second-best pricing schemesin general networks have been developed to determine the optimal tolls for system performance optimization under givenphysical and economic pricing constraints. Readers may refer to Yang and Huang (2005) and Lawphongpanich et al. (2006)for recent comprehensive treatments of the first-best and second-best pricing problems in general networks.

A congestion pricing scheme is often introduced in practice to avoid traffic jams by eliminating queuing delays duringpeak periods and to ensure that all the link flows do not exceed their physical capacities. Nevertheless, with the emergingproblem of sustainable development and transportation faced by most major cities, it is often necessary to go beyond simplephysical capacity and queuing control by achieving more strict flow restrictions on certain road sections. For instance, whenthe link flow reaches its saturation level, the road network reaches the maximum acceptable risk; the excessive noise gen-erated and the waste of fuel indicate the many serious environmental problems caused by congested roads. It is necessary toseek a set of link tolls to maintain traffic demand within a desirable level such as the ‘‘environmental” capacity, which can beregarded as the maximum traffic volume that can be accommodated along the streets without exceeding a certain thresholdof environmental damage. Indeed, for this type of traffic restraint and road pricing problem, Ferrari (1995) proposed a heu-ristic method to determine tolls through the modification of link cost functions. Larsson and Patriksson (1995) discussed linkcapacity-constrained traffic equilibrium models using an augmented Lagrange dual scheme. Yang and Bell (1997) formulatedthe problem as an elastic-demand network equilibrium model with ‘‘queuing” and then solved it by an inner penalty func-tion method (Inouye, 1987). All these methods are intimately associated with the extension of the notion of the Wardropiantraffic equilibrium (Wardrop, 1952) to networks with capacity constraints, which was in fact considered in the 1960s and1970s to improve the modeling of congestion effects (Charnes and Cooper, 1961; Jorgensen, 1963; Thompson and Payne,1975; Daganzo, 1977a,b). Apart from deriving link tolls for traffic restraints in congested networks, capacity constraintsare also used in many traffic control and management schemes (Yang and Yagar, 1994; Smith and van Vuren, 1993) and sys-tem-optimal route guidance (Correa et al., 2004; Jahn et al., 2005).

A fundamental question in the actual design and implementation of road pricing schemes is how to choose the optimalcharge level of congestion tolls in a simple yet practical manner. Because the demand curve is usually unknown and difficultto estimate in practice, Vickrey (1993) and Downs (1993) first argued that congestion pricing could be determined on a trial-and-error basis without demand functions, but neither proposed an actual mechanism to implement such a novel idea. Onlyrecently did Li (2002) propose an iterative toll adjustment procedure based on the observed link flow on a single road link,which was soon extended to a general road network by Yang et al. (2004) for the perfect marginal-cost pricing problem. Intheir developed iterative procedures, the marginal-cost-based link tolls are adjusted with changes in the observed link flowpatterns, which are in reaction to different trial tolls by individual road users. Although the proposed iterative trial-and-errorprocedure does not require knowledge of demand functions, the link travel time functions and the users’ value of time (VOT)are required for each trial, and they are not readily available in actual implementations.

The trial-and-error toll adjustment procedures developed by Li (2002) and Yang et al. (2004) are restricted to the perfectmarginal-cost pricing problem. One naturally wonders whether an efficient toll adjustment procedure can be developed forthe practical traffic restraint and road pricing problem mentioned above, i.e., to seek a set of link tolls to reduce traffic demandto a desirable threshold level with unknown demand and link travel time functions. In this inquiry, we have to remark that astraightforward way to remove traffic queues is to set a toll equivalent to the observable queuing delay at each bottleneck link,given that the value of time of all users is identical and known. By doing so, we can obtain an non-queued equilibrium networkflow pattern that satisfies the physical capacity constraint of the network. The reason is that the toll charge produces no lossesfor road users, as long as it is not beyond queuing delay, because it simply substitutes a charge for wasted time, betweenwhich road users are indifferent (Yang and Bell, 1997). Nonetheless, in most traffic congestion management problems withlink toll charges, link flows need to be restrained to satisfy the environmental capacity constraints that are more severe thanthe physical capacity, as mentioned earlier. There should be no queue visible at all at the stationary point of the traffic equi-librium, and hence the previous simple method of substituting queuing delay with a toll charge is not available. In this situ-ation, Meng et al. (2005) developed a practical trial-and-error method for seeking the desirable link tolls for effectivecongestion management without demand and link travel time functions. The basic method is based on the dual theory of non-linear programming (Bazaraa et al., 1993) and the convergence result of the Lagrangian dual algorithm established by Larssonet al. (1996). Nonetheless, the analyses and convergence proofs in Meng et al. (2005) are based on an optimization model thatrequires that the travel time functions are separable and thus the flow interactions among different links are ignored.

In this study, we develop an iterative toll adjustment method for the traffic restraint and road pricing problem with un-known demand and link travel time functions. The method is developed within the framework of the generalized trafficequilibrium problem with elastic demand and asymmetric link flow interactions. Variational inequality (VI) formulation(Smith, 1979b; Dafermos, 1980) is adapted for the generalized elastic-demand traffic equilibrium problem with capacityconstraints. By introducing the Lagrange multipliers associated with capacity constraints, a kind of alternating directionmethod can be applied for solving the resulting VI formulation with linear capacity constraints, which in turn motivatesus to develop a novel two-stage updating scheme for the iterative trial-and-error adjustment of link toll charges. Unlikethe trial-and-error method for the marginal-cost pricing problem, the whole procedure proposed here updates link tollsbased on the observed link flows after each trial and the given flow restraint levels only; link travel time and demand func-tions and users’ value of time are not needed.

H. Yang et al. / Transportation Research Part C 18 (2010) 157–175 159

This paper is organized as follows: The next section presents the VI formulation for the generalized, capacity-constrainedtraffic equilibrium problem. In Section 3, we develop a two-stage iterative pricing scheme with the general VI based trafficequilibrium model and prove its convergence. A set of numerical examples are provided in Section 4 to illustrate the poten-tial application of the proposed method. A general discussion of the deficiency and potential further extensions of the meth-od are provided in Section 5 and conclusions are given in Section 6.

2. Traffic equilibrium problem with capacity constraints

Consider a general road network with a set of directed links denoted by A, the flow on link a is denoted by va andv ¼ fva; a 2 AgT is a column vector of all link flows. Each link, a 2 A, is associated with a travel time, ta, and lett ¼ fta; a 2 AgT . Suppose that the travel time on a link is in general dependent upon the flow on every link in the network;that is, ta ¼ taðvÞ. This function contains the separable link travel time function (travel time on a link as a function of the flowon that link only) as a special case. Let W denote the set of origin–destination (O–D) pairs in the network and Rw the set of allsimple routes between O–D pair w 2W . The demand associated with traveling between each O–D pair, w 2W , is denoted asdw and all the O–D demands are grouped into a vector, d ¼ fdw; w 2WgT . We consider the general non-separable travel de-mand function defined in terms of the travel costs for that O–D pair as well as the other O–D pairs; that is, dw ¼ DwðcÞ, wherec ¼ fcw; w 2WgT is a column vector of all O–D travel costs and cw is the generalized travel cost between O–D pair w 2W .The separable demand function is included as a special case.

The following standard assumptions are required in the model formulation:

Assumption 1. The route choice behavior of users on the network under a given link toll pattern follows the deterministicUE principle.

Assumption 2. All network users have a uniform VOT denoted by s (homogeneous users).

Assumption 3. The link travel time function, tðvÞ ¼ ftaðvÞ; a 2 AgT , is non-negative and strongly monotone with respect tolink flow vector v.

Assumption 4. The O–D demand function, DðcÞ ¼ fDwðcÞ; w 2WgT , is non-negative and �DðcÞ is monotone with respect toO–D travel cost vector c; furthermore, DðcÞ is bounded and invertible. The inverse demand function is denoted byD�1ðdÞ ¼ D�1

w ðdÞ; w 2Wn oT

.

Let Ca be the desirable upper bound or acceptable threshold level of link flow va; a 2 A. As mentioned before, ourobjective is to determine a set of link tolls, denoted by u�a; a 2 A

� �, to satisfy the following environmental capacity

constraints:

va 6 Ca; a 2 A: ð1Þ

Let u� ¼ u�a; a 2 A� �T be the column vector of all these valid link tolls. After implementing such a given set of toll charges, by

Assumption 1 we have an elastic-demand network equilibrium flow pattern v�; d�ð Þ, where v� ¼ fvaðu�Þ; a 2 AgT is the UElink flow vector and d� ¼ fdwðu�Þ; w 2WgT is the O–D demand vector. The following equilibrium conditions are thus satis-fied when UE is achieved:
P
a2Ataðv�Þdw

ar þPa2A

u�as

� �dw

ar ¼ c�w; if f wr > 0;

Pa2A

taðv�Þdwar þ

Pa2A

u�as

� �dw

ar P c�w; if f wr ¼ 0;

8>><>>: r 2 Rw; w 2W; ð2Þ

D�1w ðd

�Þ ¼ c�w; if d�w > 0;

D�1w ðd

�ÞP c�w; if d�w ¼ 0;

(w 2W ; ð3Þ

where c�w is the shortest generalized travel cost between O–D pair w 2W and f wr is the flow on route r 2 Rw between O–D pair

w 2W; dwar is 1 if route r between O–D pair w 2W uses link a, and 0 otherwise.

Without reducing traffic demand excessively by the considered traffic restraint and road pricing scheme, we require thatthe chosen link toll pattern should be meaningful in the sense that a link is free of charge if traffic flow on this link is less thanits environmental capacity; otherwise, it would be subject to a toll charge to restrain its flow actively to its prescribed re-straint level. That is,

u�a ¼ 0; if v�a < Ca;

u�a P 0; if v�a ¼ Ca;

�a 2 A: ð4Þ

Clearly, when Ca; a 2 A represent physical link capacities, the above conditions ensure that the traffic queue is eliminatedcompletely by the link toll pattern at links with binding capacity constraints (Hearn, 1980; Larsson and Patriksson, 1994;Yang and Bell, 1997; Yildirim and Hearn, 2005).


Without difficulty it can be shown that the equilibrium demand and link flow vector satisfying conditions (2)–(4) is thesolution of the VI given in (5) below, where S is the feasible demand and the link flow set satisfying link capacity constraintsas defined by (6), and X represents the feasible demand and the link flow set defined by (7).

Find ðv�; d�Þ 2 S such that

tðv�ÞTðv � v�Þ � D�1ðd�ÞTðd� d�ÞP 0; 8ðv; dÞ 2 S; ð5Þ

where

S ¼ fðv; dÞ 2 Xjva 6 Ca; a 2 Ag; ð6Þ

and X represents the feasible set defined by

X ¼ ðv ;dÞjva ¼Xw2W

Xr2Rw

f wr dw

ar; a 2 A;Xr2Rw

f wr ¼ dw; f w

r P 0; r 2 Rw

( ): ð7Þ

Note that the link toll set u�a; a 2 A� �

in (4), after being divided by the uniform VOT, s, is implicitly given by the non-neg-ative Lagrange multipliers associated with the capacity constraints in (6). For a comprehensive exposition of VI, readers mayconsult the monographs by Nagurney (1993) and Facchinei and Pang (2003).

Alternatively, by explicitly introducing the non-negative Lagrange multiplier vector, k, to the capacity constraints set, S,the problem in (5)–(7) can be reformulated as the following equivalent VI problem:

ðtðv�Þ þ k�ÞTðv � v�Þ � D�1ðd�ÞTðd� d�ÞP 0;

ðk� k�ÞTðC � v�ÞP 0;

(8ðv ;d; kÞ 2 X�RjAjþ ; ð8Þ

where C ¼ fCa; a 2 AgT is the vector of all link capacity constraints, RjAjþ denotes the non-negative orthant of RjAj and jAj is thecardinality of link set A. Moreover, since k ¼ u=s with s > 0, we can substitute it into (8) and obtain the following equivalentproblem:

tðv�Þ þ u�s

� �Tðv � v�Þ � D�1ðd�ÞTðd� d�ÞP 0;

ðu� u�ÞTðC � v�ÞP 0;

(8ðv ;d; uÞ 2 X�RjAjþ : ð9Þ

The VI problem (9) can be solved by the alternating direction method that seeks the solution by replacing the originalproblem with a sequence of sub-VI problems. The alternating direction method, referred to as the augmented Lagrangianmethod in the literature, can be found in He and Yang (1998) and He et al. (2004).

It is well-known that, in the special case where the link travel time and O–D demand functions are separable, the UE flowpattern ðv�; d�Þ in the above VI formulation can be reduced to the following constrained convex optimization problem:

minðv ;dÞ2S

Xa2A

Z va

0taðxÞdx�

Xw2W

Z dw

0D�1

w ðxÞdx: ð10Þ

Various solution methods for this type of traffic equilibrium problem with link capacity constraints have been proposedby, for example, Hearn (1980), Hearn and Lawphongpanich (1990), Larsson and Patriksson (1994, 1995, 1999) and Nie et al.(2004). A comprehensive review of the model formulations and algorithmic developments for the traffic equilibrium prob-lem with general side constraints can be found in Patriksson (1994).

3. The trial-and-error pricing scheme and its convergence

Given the general capacitated, non-separable traffic equilibrium model characterized by the original VI formulation, (5)–(7), or the alternative VI formulation, (8) or (9), the alternating direction method or the augmented Lagrangian methodsdeveloped in He and Yang (1998) and He et al. (2004) work only when full network demand and supply information is avail-able. For the implementation of the trial-and-error pricing scheme in the absence of both the link travel time and the inversedemand functions, we have to explore a new method tailored for the general VI-based formulation to find the desired linktoll solution in terms of the Lagrange multipliers associated with the link capacity constraints using observed link flows.

3.1. The trial-and-error pricing scheme

For the general VI model, we propose a new, two-stage, iterative trial-and-error scheme to perform toll-updating. Thescheme is first given below:

3.1.1. The iterative trial-and-error scheme

Step 0. (Initialization) Let uð0Þa ; a 2 An o

be an initial link toll pattern. Set k ¼ 0.


Step 1. (Observe link flow) Observe the revealed link flow denoted by v ðkÞa ; a 2 An o

after imposition of the obtained tollsuðkÞa ; a 2 An o

.

Step 2. (Update toll charge) Set an intermediate toll charge for each link by:

�uðkÞa ¼max uðkÞa þ bk v ðkÞa � Ca� �

; 0� �

; a 2 A; ð11Þ

and again observe the revealed link flow denoted by �v ðkÞa ; a 2 An o

after levying link tolls �uðkÞa ; a 2 An o

. Then, thenewly updated toll for the next iteration is given by

uðkþ1Þa ¼max uðkÞa þ akbk �v ðkÞa � Ca

� �; 0

� �; a 2 A: ð12Þ

Step 3. (Check convergence) If

kuðkþ1Þ � uðkÞkkuðkÞk 6 e; ð13Þ

then stop; otherwise set k :¼ kþ 1 and go to Step 1.

In the above procedure, uðkÞ ¼ uðkÞa ; a 2 An oT

and ‘k � k’ denote the Euclidean norm; e is a small positive number of conver-gence tolerance; ak; ak > 0, is a step size parameter and fbkg; bk > 0, is a sequence of parameters that must satisfy the fol-lowing condition to guarantee the convergence of the proposed iterative scheme:

bkkv ðkÞ � �v ðkÞk 6 ckuðkÞ � �uðkÞk; c 2 ð0;1Þ; ð14Þ

where c 2 ð0;1Þ is a fixed constant.To execute this scheme, we can start with an appropriate initial (such as zero) link toll pattern. The whole iterative

scheme is terminated after the difference between two successive trial tolls is within a predetermined tolerance. At the con-vergence point, the observed flows are all below or equal to the preset levels of the environmental capacity constraint. Thebasic consideration of the scheme is that, after imposition of any link toll pattern, we can observe, rather than calculatenumerically, the revealed link load pattern on the network that actually occurs under the current price. The observed linkload pattern is exactly the solution of the first inequality in (9) with given uðkÞ, but the second inequality in (8) may notbe satisfied with uðkÞ. Therefore, we consider (11) and (12), a two-stage scheme for updating link toll charges based onthe observed link flows. This scheme provides us with a practicable toll-updating scheme, which only needs the revealedequilibrium link flow pattern that can be very easily available via observations of the road in real applications. Hence, knowl-edge of the travel time functions, the users’ VOT or the demand functions is not required. It should be pointed out that thelink toll and the resulting flow patterns that satisfy the capacity constraints are not unique and the toll pattern identified bythe above trial-and-error scheme is effective for traffic restraints but it is not necessarily optimal. One may introduce a sec-ondary objective function to select the best toll pattern among the feasible solutions based on pre-specified criteria using abi-level programming approach (Yang and Bell, 1997; Lawphongpanich and Hearn, 2004) but it requires the full informationon the demand and travel time function as well as users’ VOT.

To facilitate the proof of convergence of the newly proposed trial-and-error implementation scheme, we now elucidateand summarize the scheme in a more compact vector form. Firstly, given uðkÞ 2 RjAjþ , the revealed link flow, v ðkÞ, is in fact apart of ðv ðkÞ; dðkÞÞ 2 X, which is the solution of the following VI problem:

ðtðv ðkÞÞ þ uðkÞ

s ÞTðv � v ðkÞÞ � D�1ðdðkÞÞTðd� dðkÞÞP 0; 8ðv ;dÞ 2 X; ð15Þ

and then the updating scheme for u kþ1ð Þ is divided into two-stages:

(1) Prediction:

�uðkÞ ¼maxfuðkÞ þ bkðv ðkÞ � CÞ;0g; ð16Þ

(2) Correction:

uðkþ1Þ ¼ maxfuðkÞ þ akbkð�v ðkÞ � CÞ;0g; ð17Þ

where �v ðkÞ, the revealed link flow after imposing toll charges, �uðkÞ, again corresponds to the partial solution of the following VIproblem:

tð�v ðkÞÞ þ�uðkÞ

s

T

ðv � �v ðkÞÞ � D�1ð�dðkÞÞTðd� �dðkÞÞP 0; 8ðv; dÞ 2 X; ð18Þ

and parameter bk satisfies condition (14).

It is worth noticing once again that the revealed link flow solution v ðkÞa ; a 2 An o

and �v ðkÞa ; a 2 An o

of VI (15) and VI (18)can be observed or measured at ease in practice using advanced traffic detectors. In other words, finding the link flow


solutions for given link toll charges does not require solving VI (15) and VI (18) numerically, and hence it does not requireexact knowledge of the link travel time and travel demand functions. Furthermore, the toll prediction (16) and correction(17) are made from iteration to iteration based on the observed link flow and prescribed flow restraint level; the users’ VOTsare not involved at all. Therefore, the implementation scheme does not require the VOT information either.

Now, we look at the choice of parameter bk, which is required to satisfy condition (14) for ensuring the convergence of thealgorithm. The following proposition indicates that such a parameter, bk, certainly exists.

Proposition 1. Suppose that the link travel cost function, tðvÞ, is strongly monotone with modulus l and the negative of thedemand function, �D�1ðdÞ, is also monotone. Then,

ðuðkÞ � �uðkÞÞTð�v ðkÞ � v ðkÞÞP slk�v ðkÞ � v ðkÞk2: ð19Þ

Proof. Since ð�v ðkÞ; �dðkÞÞ 2 X and ðv ðkÞ; dðkÞÞ 2 X, substituting them into VI (15) and VI (18), respectively, leads to

tðv ðkÞÞ þ uðkÞ

s

T

ð�v ðkÞ � v ðkÞÞ � D�1ðdðkÞÞTð�dðkÞ � dðkÞÞP 0; ð20Þ

tð�v ðkÞÞ þ�uðkÞ

s

T

ðv ðkÞ � �v ðkÞÞ � D�1ð�dðkÞÞTðdðkÞ � �dðkÞÞP 0: ð21Þ

Adding (20) and (21) leads to the following inequality:

1sðuðkÞ � �uðkÞÞTð�v ðkÞ � v ðkÞÞP ðtð�v ðkÞÞ � tðv ðkÞÞÞTð�v ðkÞ � v ðkÞÞ � ðD�1ð�dðkÞÞ � D�1ðdðkÞÞÞTð�dðkÞ � dðkÞÞP lk�v ðkÞ � v ðkÞk2

;

where the second inequality follows from the assumptions on the strong monotonicity of tðvÞ and the monotonicity of�D�1ð�Þ. Hence, Eq. (19) holds obviously for s > 0. h

Note that Eq. (19) can be further relaxed to the following inequality:

kuðkÞ � �uðkÞkP slk�v ðkÞ � v ðkÞk; ð22Þ

which shows that condition (14) is satisfied automatically provided that bk 6 csl, i.e., such a parameter, bk, does exist. Be-cause it is generally difficult to know the modulus, l, and the VOT, s, may not be available as well, we suggest a self-adaptiveprocedure to find such a suitable small bk in practical implementations. For a given uðkÞ > 0 and a trial bk > 0, note that v ðkÞ isautomatically produced according to uðkÞ. We set the trial �uðkÞ according to the proposed updating scheme and then calculate

rk :¼ bkkv ðkÞ � �v ðkÞkkuðkÞ � �uðkÞk ; ð23Þ

where �v ðkÞ is likewise automatically produced according to �uðkÞ. If rk 6 c ð0 < c < 1Þ, then the trial �uðkÞ is accepted as the pre-dictor; otherwise, reduce bk by bk :¼ bk � c� 0:9=rk and repeat the procedure until a suitable one is found. The choice of stepsize ak is discussed in convergence proof.

3.2. Some properties of projection and variational inequality

Now, we summarize some important properties of projection and VI, which will be used in the subsequent convergenceanalysis. Let U � Rn be a nonempty, closed convex set. For any given x 2 Rn, the projection mapping under the Euclideannorm, denoted by PUðxÞ, is defined as follows:

PUðxÞ ¼ arg minfkx� zk jz 2 Ug:

It is easy to see that the operator, maxf�;0g, amounts to the projection operation, PRþ ð�Þ, where Rþ represents the non-negative orthant. A basic property of the projection mapping on a closed convex set is:

ðx� PUðxÞÞTðPUðxÞ � zÞP 0; 8 x 2 Rn; z 2 U: ð24Þ

Following from (24), we readily have

kPUðxÞ � PUðzÞk 6 kx� zk; 8 x; z 2 Rn ð25Þ

and

kPUðxÞ � zk26 kx� zk2 � kx� PUðzÞk2; 8 x 2 Rn; 8 z 2 U: ð26Þ

For a given continuous monotone function, FðxÞ, denoting the residue function

eðx;bÞ :¼ x� PU½x� bFðxÞ�; b > 0; ð27Þ

and then its zero point is equivalent to the solution of the following VI problem (Bertsekas and Tsitsiklis, 1989):


ðx� x�ÞT Fðx�ÞP 0; 8x 2 U: ð28Þ

Therefore, in the problem of solving VI, keðx; bÞk is often regarded as an error bound, which quantitatively measures howmuch x fails to be in the solution set of VI (28).

3.3. The convergence proof

After introducing the above equivalence and notation, the whole trial-and-error scheme is equivalent to the followingsequential expressions: Firstly, given a trial, u 2 RjAjþ ,

vd

¼ PX

vd

�

tðvÞ þ us

D�1ðdÞ

!( ); ð29Þ

and then a predicted interim pattern, ð�v; �d; �uÞ, is obtained by:

�u ¼ PRjAjþ½uþ bðv � CÞ�; ð30Þ

�v�d

¼ PX

�v�d

�

tð�vÞ þ �us

D�1ð�dÞ

!( ): ð31Þ

Finally, the truly updated Lagrange multiplier vector is given by:

uðaÞ ¼ PRjAjþ½uþ abð�v � CÞ�: ð32Þ

Here, v ðkÞ; dðkÞ; uðkÞ; �uðkÞ; uðkþ1Þ; ak and bk are, respectively, replaced by v ; d; u; �u; uðaÞ; a and b for convenience.

Proposition 2. If �u ¼ u, then ðv; d;uÞ is a solution of VI (9).

Proof. Since �u ¼ u, by Eq. (30), we have

u ¼ PRjAjþ½uþ bðv � CÞ�:

According to the equivalence property of solving VI problems stated in (27) and (28), the proposition holds. h

This proposition indicates that the distance, k�u� uk, can be regarded as a convergence criterion of the iterative process.Note that in the trial-and-error pricing scheme presented in Section 3.1, the relative error measure of link toll charges be-tween two trial experiments as given in (13) is used as the convergence criterion. This measure is valid because one can eas-ily show that as kuðkþ1Þ � uðkÞk converges to zero, the residual keðuðkÞ; bkÞk in (27) or equivalently, k�uðkÞ � uðkÞk converges tozero as well.

Theorem 1. For any y and a solution y�, we have

HðaÞP 2aðu� �uÞT gðv; uÞ � a2kgðv ;uÞk2; ð33Þ

where

HðaÞ :¼ ku� u�k2 � kuðaÞ � u�k2 ð34Þ

and

gðv ;uÞ :¼ ðu� �uÞ � bð�v � vÞ: ð35Þ

The proof of Theorem 1 is given in Appendix A. This theorem is the foundation for choosing the best value of a in (32). Sincethe right-hand-side of (33) is a quadratic function of a, it reaches its maximum at

a� ¼ ðu��uÞT gðv ;uÞ

kgðv ;uÞk2 : ð36Þ

From the definition of gðv ;uÞ in (35) and inequality (14), we have

ðu� �uÞT gðv ;uÞP ð1� cÞku� �uk2: ð37Þ

Therefore, whenever ku� �uk– 0, this step size, a�, is always positive. Now, let g 2 ð0;2Þ be a relaxation factor and substitutea ¼ ga� in (33). By simple manipulation, we can get

HðaÞP gð2� gÞa�ðu� �uÞT gðv; uÞP gð2� gÞð1� cÞa�ku� �uk2: ð38Þ

Finally, the convergence of the proposed method is proved by the following Theorem 2.


Theorem 2. Let fuðkÞg be the sequence generated by the iterative scheme proposed in Section 3.1. Then, fuðkÞg converges to somesolution, u�.

The proof of Theorem 2 is given in Appendix A.

4. Numerical experiments

In this section, we present a set of numerical experiments to demonstrate the proposed iterative trial-and-error pricingscheme with the generalized traffic equilibrium model. For comparative analysis, we also consider the separable traffic equi-librium model presented in (10) and apply both the VI-based pricing scheme proposed in this study and the MathematicalProgramming (MP) based trial-and-error scheme developed by Meng et al. (2005). The trial-and-error scheme that appliesonly to the MP-based traffic equilibrium problem (10) uses the following simple toll-updating equation:

uðkþ1Þa ¼maxfuðkÞa þ bkðv ðkÞa � CaÞ;0g; a 2 A; ð39Þ

where the step size sequence, fbkg, should satisfy the following conditions for convergence:

bk > 0; 8 k;X1k¼1

bk ¼ 1;X1k¼1

b2k <1 ð40Þ

and a typical step size sequence, fbk ¼ 1=kg, is used unless otherwise mentioned. The procedure is repeated until two suc-cessive trial toll patterns are close enough to each other.

Four examples are presented. The first one is taken from Yang and Bell (1997) and the second one uses the Sioux FallsNetwork. Both examples involve separable link travel time functions and are solved by both the MP- and the VI-based meth-ods for comparison. The third and the forth examples are taken from Nagurney (1984). They involve general asymmetric,non-separable link travel time functions and are solved by the VI-based method only. It should be mentioned here thatthe link travel time and demand functions for all examples are purely used for generating the observed link flows; noneof them is needed for updating link tolls in the implementation of the trial-and-error schemes.

In all four examples, un-tolled link flows are used to obtain an initial set of link tolls. For given toll charge fuðkÞa ; a 2 Ag, theobserved link flows at each trial are simulated by accurately solving the elastic-demand traffic assignment problem with thelink travel time functions:

tðkÞa ðvÞ ¼ taðvÞ þuðkÞa

s; ð41Þ

where s, as defined before, is the users’ VOTs that are inherent in their route choice decisions under pricing, and s is assumedto be 60 (HK$/h). Namely, given the toll pattern fuðkÞa ; a 2 Ag at iteration k, the observed link flow pattern, fv ðkÞa ; a 2 Ag, isgenerated by solving VI (15). For parameter bk, we set initial b0 ¼ 1 and c ¼ 0:95 for the self-adaptive algorithm. For bothMP- and VI-based methods, the relative error measure, kuðkþ1Þ � uðkÞk=kuðkÞk, is adopted to check the rate of convergence,and a convergence tolerance, e ¼ 10�3, is adopted for all four examples.

4.1. Numerical experiments with separable link travel time functions

Example 1. The road network, as shown in Fig. 1, consists of 11 links, 7 nodes and 4 O–D pairs ð1! 7; 2! 7; 3! 7 and6! 7Þ. The true, but unknown, demand functions are given as:

1

4

2

7

6

5

3

1

2

3

4

5

6

7

8

9

10

11

Fig. 1. The network used for Example 1.

Table 1Input d

Link no

1234567891011

Table 2Estimat

Link no

1234567891011


D1!7ðc1!7Þ ¼ 600 expð�0:04c1!7Þ; D2!7ðc2!7Þ ¼ 500 expð�0:03c2!7Þ;D3!7ðc3!7Þ ¼ 500 expð�0:05c3!7Þ; D6!7ðc6!7Þ ¼ 400 expð�0:05c6!7Þ:

The following Bureau of Public Roads (BPR) link travel time function is used together with the link free-flow travel time,t0

a , link physical capacity, CPa , and link environmental capacity, CE

a, given in Table 1.

taðvaÞ ¼ t0a 1þ 0:15

va

CPa

!40@

1A; a 2 A: ð42Þ

The final results (the toll and the ratio of flow to environmental capacity for each link) obtained from the implementationof both the MP- and the VI-based trial-and-error schemes are given in Table 2. Fig. 2 depicts the change in the selected fourlink tolls with the number of iterations of the trial experiments and Fig. 3 displays the convergence of the two methods.

Example 2. This example uses a larger network, the Sioux Falls Network, for testing of both methods. The network topology,as shown in Fig. 4, consists of 24 nodes and 76 links. Again, the standard BPR link travel time function (42) is used, with thelink input data (free-flow travel time, t0

a , and physical capacity, CPa) given in Table 3. The environmental capacity, CE

a, is simplyset to be 0:8� CP

a for each link a 2 A. On the demand side, the following true but unknown negative exponential demandfunction is assumed:

dw ¼ d0w exp �nw

lw

l0w� 1

� �; w 2W ; ð43Þ

where nw > 0 is a cost sensitivity parameter, l0w is the minimum free-flow travel time, and d0

w is the maximum potential de-mand when lw ¼ l0

w. Here, a uniform value of nw ¼ 0:04 is taken for all O–D pairs and the value of d0w for each O–D pair,

w 2W , is presented in Table 4.

The final results (the toll and the ratio of flow to environmental capacity for each link) obtained from the MP- and VI-based trial-and-error schemes are given in Table 5, and the rate of convergence by both methods is displayed in Fig. 5.

4.1.1. Discussion of the resultsIt can be seen from Table 2 for Example 1 and Table 5 for Example 2 that the ratio of flow to environmental capacity for

each link is less than or equal to one; a link is subject to a toll charge only when its flow reaches its environmental capacity,

ata for the network in Example 1.

. a Free-flow travel time t0a (min) Physical capacity CP

a (veh/min) Environmental capacity CEa (veh/min)

6 200 1605 200 1506 200 2007 200 1506 100 1001 100 1005 150 15010 150 10011 200 16011 200 16015 200 150

ed link flow ratios and link tolls for network in Example 1.

. a Method based on MP Method based on VI

Ratio, va=CEa Link toll, ka Ratio, va=CE

a Link toll, ka

1.00 1.19 1.00 1.190.34 0.00 0.33 0.001.00 6.38 1.00 6.401.00 8.50 1.00 8.531.00 2.74 1.00 2.831.00 1.37 1.00 1.330.27 0.00 0.27 0.001.00 1.83 1.00 1.951.00 6.68 1.00 6.661.00 6.61 1.00 6.631.00 5.13 1.00 5.15

0.0

10.0

20.0

30.0

40.0

50.0

60.0

1 11 21 31 41 51 61

Number of Iterations Number of Iterations

Link

Tol

l Cha

rge

Toll on Link 4

Toll on Link 5

Toll on Link 9

Toll on Link 11

(a) Method based on MP

0.0

1.5

3.0

4.5

6.0

7.5

9.0

1 4 7 10 13 16 19

Link

Tol

l Cha

rge

Toll on Link 4 Toll on Link 5


(b) Method based on VI

Fig. 2. Change in the link tolls with iteration for Example 1.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

1 7 13 19 25 31 37 43 49 55 61

Number of Iterations

Con

verg

ence

Err

or

Method based on MP

Method based on VI

Fig. 3. Change in the convergence error with iteration for Example 1.


otherwise it is free of charge. The objective of environmental capacity constraints is achieved perfectly through the effectivelink toll charge identified from the iterative trial-and-error scheme. It is noted that the resulting flow to environmentalcapacity ratios identified by the MP- and VI-based methods is almost identical for each link but there are certain differencesin the link tolls, which are largely attributed to the non-uniqueness of those effective link toll charges.

From the changes in the toll charges with iteration of trials shown in Fig. 2 for the four tolled links selected for Example 1,we can see that it takes more than 60 iterations for the MP-based method to reach the convergence criterion and the linktolls exhibit significant fluctuations especially in several initial trials. This is certainly not welcome in real applications asa dramatic upward and downward adjustment of toll levels will increase public unacceptability of the trial-and-error pricingscheme. Fortunately, in Fig. 2b, the VI-based method exhibits a gentle change tendency and quick convergence and thus ismuch more preferred in practice. Corresponding to the change of individual link tolls is the aggregate rate of convergencedepicted in Fig. 3, which further confirms the superiority of the new VI-based method. In the pricing experiment on the lar-ger Sioux Falls Network in Example 2, we found that the simple step-length sequence, f1=kg, where k is the number of iter-ations, does not perform well with the MP-based method. After a few trials, the sequence, f0:01=kg, is found to be moresatisfactory and adopted to produce the convergence result depicted in Fig. 5. Even with this preselection of the step-lengthsequence for the MP-based method, the new VI-based method converges more sharply as shown in Fig. 5.

We can, without difficulty, understand why the VI-based method performs much better than the MP-based method. TheMP-based method, the step-length sequence fbkg is predetermined independently of the specific question, which is some-what arbitrary and thus greatly affects the rate of convergence. In contrast, with the VI-based method, the sequence, fbkg, isnot given a priori but is chosen during the process of implementation according to the self-adaptive or self-optimizing strat-egy described in Section 3.1, which is of course more reasonable.

1

8

4 5 63

2

15 19

17

18

7

12 11 10 16

9

20

23 22

14

13 24 21

3

12

6

8

9

11

5

15

122313

2116 19

17

2018 54

55

5048

2951 49 52

58

24

27

32

33

36

7 35

4034

41

44

57

45

72

70

46 67

69 65

25

28 43

53

59 61

56 60

66 62

6863

7673

30

7142

647539

74

37 38

26

4 14

22 47

10 31

Fig. 4. The Sioux Falls Network used for Example 2.


4.2. Numerical experiments with asymmetric link travel time functions

Here, we further consider two examples taken from Nagurney (1984) with asymmetric link travel time functions. In bothexamples, the following form of nonlinear and asymmetric link travel time function is used:

taðvÞ ¼ gaav4a þ

Xb2A

gabvb þ ha; 8 a 2 A: ð44Þ

The specific functions can be found in Nagurney (1984). The only difference here is that elastic travel demands are consid-ered and appropriate linear inverse demand functions are assumed.

Example 3. This example uses the network shown in Fig. 6, consisting of 20 nodes, 28 links and the following eight O–Dpairs: w1 ¼ ð1;20Þ, w2 ¼ ð1;19Þ, w3 ¼ ð2;17Þ, w4 ¼ ð4;20Þ, w5 ¼ ð6;19Þ, w6 ¼ ð2;20Þ, w7 ¼ ð2;13Þ and w8 ¼ ð3;14Þ. Theasymmetric link travel time functions in Table 6 are used. The following inverse demand functions are assumed and usedonly for simulating and generating observed link flows for given toll charges rather than for the actual implementation of thetrial-and-error pricing scheme.

kw1 ðdw1 Þ ¼ �0:05dw1 þ 20; kw2 ðdw2 Þ ¼ �0:06dw2 þ 40;



kw7 ðdw7 Þ ¼ �0:05dw7 þ 20; kw8 ðdw8 Þ ¼ �0:04dw8 þ 40:

The environmental capacity, CEa , of each link imposed for the capacity constraint is given in Table 7. After implementing

the VI-based trial-and-error scheme, the convergent link toll pattern and the ratio of the equilibrium link flow to the linkenvironmental capacity are summarized in Table 8.

Example 4. This example uses the network shown in Fig. 7, consisting of 25 nodes, 37 links and the following six O–D pairs:w1 ¼ ð1;20Þ, w2 ¼ ð1;25Þ, w3 ¼ ð2;20Þ, w4 ¼ ð3;25Þ, w5 ¼ ð1;24Þ and w6 ¼ ð11;25Þ. A complete set of asymmetric link traveltime functions is given in Table 9. Like in Example 3, the following set of true but unknown linear inverse demand functionsis used for link flow simulation.

Table 3Input data for the Sioux Falls Network in Example 2.

Link no. a t0a (min) CP

að103 veh/h) Link no. a t0a (min) CP

að103 veh/h)

1 3.60 25.90 39 2.40 5.092 2.40 23.40 40 2.40 4.883 3.60 25.90 41 3.00 5.134 3.00 4.96 42 2.40 4.925 2.40 23.40 43 3.60 13.516 2.40 17.11 44 3.00 5.137 2.40 23.40 45 2.40 15.658 2.40 17.11 46 2.40 10.329 1.20 17.78 47 3.00 5.0510 3.60 4.91 48 3.00 5.1311 1.20 17.78 49 1.20 5.2312 2.40 4.95 50 1.80 19.6813 3.00 10.00 51 4.20 4.9914 3.00 4.96 52 1.20 5.2315 2.40 4.95 53 1.20 4.8216 1.20 4.90 54 1.20 23.4017 1.80 7.84 55 1.80 19.6818 1.20 23.40 56 2.40 23.4019 1.20 4.90 57 2.40 15.6520 1.80 7.84 58 1.20 4.8221 2.00 5.05 59 2.40 5.0022 3.00 5.05 60 2.40 23.4023 3.00 10.00 61 2.40 5.0024 2.00 5.05 62 3.60 5.0625 1.80 13.92 63 3.00 5.0826 1.80 13.92 64 3.60 5.0627 3.00 10.00 65 1.20 5.2328 3.60 13.51 66 1.80 4.8929 3.00 5.13 67 2.40 10.3230 4.20 4.99 68 3.00 5.0831 3.60 4.91 69 1.20 5.2332 3.00 10.00 70 2.40 5.0033 3.60 4.91 71 2.40 4.9234 2.40 4.88 72 2.40 5.0035 2.40 23.40 73 1.20 5.0836 3.60 4.91 74 2.40 5.0837 1.80 25.90 75 1.80 4.8938 1.80 25.90 76 1.20 5.09

Table 4Potential O–D demands (�103veh/h) for the Sioux Falls Network in Example 2.

O/D 1 2 4 5 10 11 13 14 15 19 20 21 22 24

1 1.20 1.20 1.20 0.98 1.00 1.14 0.90 0.86 0.82 0.54 0.54 0.70 0.672 1.20 1.14 1.18 1.00 1.02 0.82 0.86 0.85 1.18 0.54 0.62 0.61 0.544 1.20 1.14 1.20 0.98 0.97 0.86 0.82 0.76 0.73 0.56 0.58 0.54 0.735 1.20 1.18 1.20 1.03 0.88 0.83 0.80 0.74 0.66 0.73 0.74 0.85 0.5410 0.98 1.00 0.98 1.03 1.21 0.82 0.90 1.20 1.06 0.86 0.82 0.88 0.5411 1.00 1.02 0.97 0.88 1.21 0.85 1.20 1.01 0.86 0.67 0.55 1.00 0.9513 1.14 0.82 0.86 0.83 0.82 0.85 0.79 0.78 0.62 0.54 0.56 0.61 1.2014 0.90 0.86 0.82 0.80 0.90 1.20 0.79 1.20 1.03 0.86 0.79 0.82 1.0315 0.86 0.85 0.76 0.74 1.20 1.01 0.78 1.20 1.20 1.15 1.04 1.20 0.8319 0.82 1.18 0.73 0.66 1.06 0.86 0.62 1.03 1.20 1.20 1.01 1.00 0.7320 0.54 0.54 0.56 0.73 0.86 0.67 0.54 0.89 1.15 1.20 1.20 1.20 0.5521 0.54 0.62 0.58 0.74 0.82 0.55 0.56 0.79 1.04 1.01 1.20 1.20 1.2022 0.70 0.61 0.54 0.85 0.88 1.00 0.61 0.82 1.20 1.00 1.20 1.20 1.0324 0.67 0.54 0.73 0.54 0.54 0.95 1.20 1.03 0.83 0.73 0.55 1.20 1.03




kw5 ðdw5 Þ ¼ �0:07dw5 þ 80; kw6 ðdw6 Þ ¼ �0:09dw6 þ 70:

For simplicity, the environmental capacity, CEa , is set as follows: CE

a ¼ 45 for a = 1–15, CEa ¼ 55 for a = 16–30 and CE

a ¼ 50 fora = 31–37. After implementing the VI-based trial-and-error scheme, the convergent link toll pattern and the ratio of the equi-librium link flow to the link environmental capacity are summarized in Table 10.

Table 5Estimated link flow ratios and link tolls for the Sioux Falls Network in Example 2.

Link no. a Method based on MP Method based on VI Link no. a Method based on MP Method based on VI

Ratio, va=CEa Toll, ka Ratio, va=CE

a Toll, ka Ratio, va=CEa Toll, ka Ratio, va=CE

a Toll, ka

1 0.19 0.00 0.19 0.00 39 1.00 5.75 1.00 5.712 0.55 0.00 0.56 0.00 40 1.00 4.95 1.00 4.953 0.19 0.00 0.19 0.00 41 0.79 0.00 0.79 0.004 1.00 2.44 1.00 2.44 42 0.89 0.00 0.89 0.005 0.55 0.00 0.56 0.00 43 0.98 0.00 0.98 0.006 0.64 0.00 0.64 0.00 44 0.79 0.00 0.79 0.007 0.32 0.00 0.32 0.00 45 0.35 0.00 0.35 0.008 0.64 0.00 0.64 0.00 46 1.00 0.47 1.00 0.499 0.67 0.00 0.67 0.00 47 0.18 0.00 0.18 0.0010 1.00 0.20 1.00 0.13 48 0.55 0.00 0.54 0.0011 0.67 0.00 0.67 0.00 49 0.43 0.00 0.43 0.0012 0.46 0.00 0.46 0.00 50 0.08 0.00 0.08 0.0013 1.00 0.83 1.00 0.77 51 0.52 0.00 0.52 0.0014 1.00 2.44 1.00 2.44 52 0.43 0.00 0.42 0.0015 0.46 0.00 0.46 0.00 53 1.00 0.71 1.00 0.7116 1.00 2.34 1.00 2.28 54 0.14 0.00 0.14 0.0017 0.41 0.00 0.41 0.00 55 0.08 0.00 0.08 0.0018 0.14 0.00 0.14 0.00 56 0.20 0.00 0.20 0.0019 1.00 2.34 1.00 2.29 57 0.35 0.00 0.35 0.0020 0.41 0.00 0.41 0.00 58 1.00 0.71 0.99 0.7221 0.47 0.00 0.47 0.00 59 0.95 0.00 0.95 0.0022 0.18 0.00 0.18 0.00 60 0.20 0.00 0.20 0.0023 1.00 0.83 1.00 0.78 61 0.95 0.00 0.95 0.0024 0.47 0.00 0.46 0.00 62 0.98 0.00 0.98 0.0025 0.77 0.00 0.77 0.00 63 0.49 0.00 0.49 0.0026 0.77 0.00 0.77 0.00 64 0.98 0.00 0.98 0.0027 0.74 0.00 0.74 0.00 65 0.99 1.26 1.00 1.2628 0.98 0.00 0.98 0.00 66 1.00 0.57 1.00 0.5829 0.55 0.00 0.55 0.00 67 1.00 0.49 1.00 0.4930 0.52 0.00 0.52 0.00 68 0.48 0.00 0.48 0.0031 0.99 0.20 1.00 0.15 69 1.00 1.28 1.00 1.2632 0.74 0.00 0.74 0.00 70 0.88 0.00 0.88 0.0033 0.99 0.22 1.00 0.17 71 0.90 0.00 0.90 0.0034 1.00 4.95 1.00 4.95 72 0.88 0.00 0.88 0.0035 0.32 0.00 0.32 0.00 73 1.00 1.27 1.00 1.2736 0.99 0.22 1.00 0.16 74 1.00 5.75 1.00 5.7137 0.36 0.00 0.36 0.00 75 1.00 0.59 1.00 0.5838 0.36 0.00 0.36 0.00 76 1.00 1.27 1.00 1.26


4.2.1. Discussion of the resultsFig. 8 shows the changes in link toll charges with the number of iterations of the trial experiments for the two examples.

Again, for sake of clarity, only four tolled links for each example are selected. Like the symmetric cases in Examples 1 and 2, a

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1 10 19 28 37 46 55 64 73 82Number of Iterations

Con

verg

ence

Erro

r

Method based on MP

Method based on VI

Fig. 5. Change in the convergence error with iteration for Example 2.

Fig. 6. The network (source: Nagurney, 1984) used for Example 3.

Table 6Asymmetric link travel time functions used in Example 3.

t1ðvÞ ¼ 5� 10�7v41 þ 0:05v1 þ 0:02v2 þ 5 t2ðvÞ ¼ 3� 10�7v4

2 þ 0:04v2 þ 0:04v1 þ 2t3ðvÞ ¼ 5� 10�7v4

3 þ 0:03v3 þ 0:01v4 þ 3:5 t4ðvÞ ¼ 3� 10�7v44 þ 0:06v4 þ 0:03v5 þ 4

t5ðvÞ ¼ 6� 10�7v45 þ 0:06v5 þ 0:04v6 þ 6 t6ðvÞ ¼ 0:07v6 þ 0:03v7 þ 5


8 þ 0:05v8 þ 0:02v9 þ 6:5t9ðvÞ ¼ 10�7v4

9 þ 0:06v9 þ 0:02v10 þ 7 t10ðvÞ ¼ 0:04v10 þ 0:01v12 þ 8t11ðvÞ ¼ 7� 10�7v4

11 þ 0:07v11 þ 0:04v12 þ 6:5 t12ðvÞ ¼ 0:08v12 þ 0:02v13 þ 7t13ðvÞ ¼ 10�7v4



17 þ 0:07v17 þ 0:02v15 þ 4:5 t18ðvÞ ¼ 0:05v18 þ 0:01v16 þ 3t19ðvÞ ¼ 0:08v19 þ 0:03v17 þ 6 t20ðvÞ ¼ 2� 10�7v4

20 þ 0:06v20 þ 0:01v21 þ 3t21ðvÞ ¼ 4� 10�7v4

21 þ 0:04v21 þ 0:01v22 þ 4 t22ðvÞ ¼ 2� 10�7v422 þ 0:06v22 þ 0:01v23 þ 5

t23ðvÞ ¼ 3� 10�7v423 þ 0:09v23 þ 0:02v24 þ 3:5 t24ðvÞ ¼ 2� 10�7v4

24 þ 0:08v24 þ 0:01v25 þ 4t25ðvÞ ¼ 3� 10�7v4

25 þ 0:09v25 þ 0:03v26 þ 4:5 t26ðvÞ ¼ 6� 10�7v426 þ 0:07v26 þ 0:08v27 þ 3


28 þ 0:07v28 þ 6:5

Table 7Environmental capacity for capacity constraint for the network in Example 3.

Link no.a

Environmental capacity,CE

a

Link no.a


a

Link no.a


a

Link no.a


a

1 60 8 60 15 60 22 802 70 9 60 16 80 23 603 75 10 60 17 70 24 604 60 11 60 18 70 25 755 60 12 60 19 60 26 806 150 13 60 20 70 27 757 70 14 60 21 70 28 60

Table 8Estimated link flow ratios and link tolls for the network in Example 3.

Link no. a Ratio, va=CEa Toll, ka Link no. a Ratio, va=CE

a Toll, ka Link no. a Ratio, va=CEa Toll, ka Link no. a Ratio, va=CE

a Toll, ka

1 0.00 0.00 8 1.00 8.77 15 0.29 0.00 22 1.00 7.632 0.85 0.00 9 0.00 0.00 16 1.00 2.49 23 0.61 0.003 1.00 5.96 10 0.00 0.00 17 0.14 0.00 24 0.96 0.004 0.89 0.00 11 0.63 0.00 18 0.86 0.00 25 1.00 10.825 0.54 0.00 12 0.83 0.00 19 0.00 0.00 26 0.81 0.006 1.00 5.57 13 0.36 0.00 20 0.00 0.00 27 1.00 4.207 1.00 3.08 14 0.35 0.00 21 0.54 0.00 28 0.00 0.00


smooth change in link tolls is found with the proposed iterative trial-and-error pricing scheme. In addition, Fig. 9 shows thenice convergence of the aggregate convergence error measures for both examples. The VI-based method performs well withthe asymmetric, non-separable link travel time function.

5. Outstanding political and behavioral issues

To ensure that the basic theory and implementation method are valid, we have made some simplistic and strong assump-tions about the transportation system, such as static networks and O–D demands, deterministic travel behavior and obser-vations. These assumptions cannot be guaranteed in practice, particularly under the proposed sequential toll adjustmentscheme. This section is devoted to a thorough discussion of how these assumptions could be relaxed and the implementationprocedure enhanced.

Fig. 7. The network (source: Nagurney, 1984) used for Example 4.

Table 9Asymmetric link travel time functions used in Example 4.

t1ðvÞ ¼ 5� 10�5v41 þ 5v1 þ 2v2 þ 500 t20ðvÞ ¼ 3� 10�5v4

20 þ 6v20 þ v21 þ 300t2ðvÞ ¼ 3� 10�5v4

2 þ 4v2 þ 4v1 þ 200 t21ðvÞ ¼ 4� 10�5v421 þ 4v21 þ v22 þ 400

t3ðvÞ ¼ 5� 10�5v41 þ 3v3 þ v4 þ 350 t22ðvÞ ¼ 2� 10�5v4

22 þ 6v22 þ v23 þ 500t4ðvÞ ¼ 3� 10�5v4

4 þ 6v4 þ 3v5 þ 400 t23ðvÞ ¼ 3� 10�5v423 þ 9v23 þ 2v24 þ 350


24 þ 8v24 þ v25 þ 400t6ðvÞ ¼ 7v6 þ 3v7 þ 500 t25ðvÞ ¼ 3� 10�5v4

25 þ 9v25 þ 3v26 þ 450t7ðvÞ ¼ 8� 10�5v4



27 þ 8v27 þ 3v28 þ 500t9ðvÞ ¼ 10�5v4

9 þ 6v9 þ 2v10 þ 700 t28ðvÞ ¼ 3� 10�5v428 þ 7v28 þ 650

t10ðvÞ ¼ 4v10 þ v12 þ 800 t29ðvÞ ¼ 3� 10�5v429 þ 3v29 þ v30 þ 450


30 þ 7v30 þ 2v31 þ 600t12ðvÞ ¼ 8v12 þ 2v13 þ 700 t31ðvÞ ¼ 3� 10�5v4

31 þ 8v31 þ v32 þ 750t13ðvÞ ¼ 10�5v4


t14ðvÞ ¼ 8v14 þ 3v15 þ 500 t33ðvÞ ¼ 4� 10�5v433 þ 9v33 þ 2v31 þ 750


34 þ 7v34 þ 3v30 þ 550t16ðvÞ ¼ 8v16 þ 5v12 þ 300 t35ðvÞ ¼ 3� 10�5v4

35 þ 8v35 þ 3v32 þ 600t17ðvÞ ¼ 3� 10�5v4


t18ðvÞ ¼ 5v18 þ v16 þ 300 t37ðvÞ ¼ 6� 10�5v437 þ 5v37 þ v36 þ 350

t19ðvÞ ¼ 8v19 þ 3v17 þ 600


5.1. Short-run and long-run demand changes

The trial-and-error procedure to update link tolls for congestion control was built upon the assumption that changes inequilibrium demand and link flow patterns take place by short-run changes in the price along the same demand curve. Theflow revelation mechanism thus requires a series of price changes to derive the desirable link tolls properly. Politically speak-ing, it is clear that a series of quick changes to a toll system are an unlikely social event, as it would make the newly intro-duced system unsustainable. A plausible toll adjustment by a willing authority would be at annual, semi-annual or at mostquarterly intervals, like the situation in Singapore where toll rate adjustments are made quarterly and more frequently forholidays. Nevertheless, such a socially acceptable long inter-trial period could create a behavioral dilemma for the analysisenvisioned here. On the one hand, it would allow travelers to have sufficient time to learn and adjust their trip-making deci-sions in a new pricing environment, although this might not be a critical issue, as advanced information technology is now-adays available and travelers can be quickly informed of altered toll charges. On the other hand, the background variables,

Table 10Estimated link flow ratios and link tolls for the network in Example 4.

Link no. a Ratio, va=CEa Toll, ka Link no. a Ratio, va=CE

a Toll, ka Link no. a Ratio, va=CEa Toll, ka Link no. a Ratio, va=CE

a Toll, ka

1 1.00 1.93 11 0.35 0.00 21 0.82 0.00 31 0.31 0.002 1.00 0.00 12 0.22 0.00 22 1.00 9.12 32 1.00 11.313 1.00 13.13 13 0.43 0.00 23 0.45 0.00 33 1.00 10.944 0.57 0.00 14 0.57 0.00 24 0.00 0.00 34 0.64 0.005 0.00 0.00 15 0.00 0.00 25 0.00 0.00 35 0.69 0.006 0.00 0.00 16 0.00 0.00 26 0.00 0.00 36 1.00 24.077 0.00 0.00 17 0.00 0.00 27 0.00 0.00 37 0.20 0.008 0.00 0.00 18 0.00 0.00 28 0.00 0.00 – – –9 0.00 0.00 19 0.00 0.00 29 0.59 0.00 – – –10 1.00 1.42 20 0.86 0.00 30 0.04 0.00 – – –

0

2

4

6

8

10

12

1 4 7 10 13 16



(a) For Example 3

0.0

5.0

10.0

15.0

20.0

25.0

30.0

1 2 3 4 5 6 7 8 9 10

Toll on Link 3 Toll on Link 22Toll on Link 32 Toll on Link 36

(b) For Example 4 Number of Iterations Number of Iterations

Link

Tol

l Cha

rge

Link

Tol

l Cha

rge

Fig. 8. Change in the link toll with iteration with asymmetric link travel time functions for Examples 3 and 4.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 3 5 7 9 11 13 15Number of Iterations

Con

verg

ence

Erro

r

Example 3

Example 4

Fig. 9. Change in the convergence error with iteration for Examples 3 and 4.


such as inflation and real income growth, change over such a length of time. Thus, it is imperative to address the issues of theshort-run and long-run differences in behaviors as well as the time lag of user responses in formulating a more practical pric-ing mechanism. In this respect, the current method makes efficient utilization of the traffic counts that are relatively inex-pensive to collect with automatic devices, even on a daily basis with good accuracy. These readily and frequently availabletraffic counts effectively capture the long-run demand shift prior to the next price trial. It is thus interesting to see how theproposed tolling scheme would work under such a changing environment and what modifications should be made to makethe procedure more adaptive to long-run demand shifts.

5.2. Plausible toll development trajectory

The proposed tolling scheme aims at minimizing the convergence process towards the final solution, and the sequentialexperimental process involves both upward and downward adjustment of the starting and intermediate toll levels. Such aprocess should be designed such that it is politically acceptable in the first place. To make the experimental procedure prac-tically feasible, it is thus necessary to optimally consider some plausible bounds for the trajectory of realistic toll evolutionover time, including the starting toll levels, the trajectory towards the final optimum, and the number and size of intervals inbetween.

5.3. Incorporation of stochastic variations and observations

To generate useful, valid, unbiased and converging estimates of the toll levels, it is critical to extend the deterministicbehavioral and optimization models to stochastic conditions. First, choice behavior can be treated through a stochastic equil-ibration approach instead of through the deterministic route choice model considered so far. Second, instead of assuming asingle or deterministic observation value for link flows, which is inconsistent with the proposed experimental approach that


is intended to have an update interval of multiple months, we can incorporate repeated flow observations that are alwayscorrelated stochastic variates, the within-day stochastic route choices and their day-to-day evolution process in a new pric-ing experiment. Thus, it is meaningful to look into a stochastic trial-and-error tolling scheme to handle explicitly the randomvariations that are immanent in the observations.

5.4. Multiple vehicle types and heterogeneous users

The proposed tolling scheme allows for non-separable, asymmetric link cost functions. It should be relatively easy to takeinto account multiple types of vehicles on the network, such as heavy versus light vehicles, although the shared link capacityconstraint could cause difficulty. In this case, we should recognize the fact that different types of vehicles make differentcontributions to traffic congestion and road capacity consumption and they should be subjected to different toll chargesaccording to their observationally distinguishable characteristics. In addition, the tolling scheme makes sequential tolladjustment without requiring the knowledge of the users’ VOTs. This is true under the current assumption of homogeneoususers. Nonetheless, with heterogeneous users having different VOTs, it is unclear whether the proposed tolling schemewould continue to work. In this case, the tolling scheme must be anonymous in the sense that the same amount of toll islevied on each link for all user classes, because users differ from one another in VOT only, which is observationally indistin-guishable, and thus toll differentiation across user classes is unrealistic in reality.

6. Conclusions

Recognizing the widespread practical difficulty in determining appropriate toll charges for mitigating traffic congestionand reducing environmental damage, we developed an attractive and convergent trial-and-error implementation scheme foremerging traffic restraint and road pricing problems. Distinguished from the traditional modeling approach, the proposedscheme allows traffic planners to estimate link tolls easily from observed link flows only, without resorting to both link tra-vel time and demand functions and users’ VOTs. As demonstrated through extensive numerical experiments, the proposedmethod applies to the generalized traffic equilibrium problem with asymmetric link flow interactions, and thus offers a pow-erful practical method for the network toll design problem for effective congestion management.

In spite of the appealing features of the proposed tolling scheme, by no means is the problem well solved. A number ofconcerns for practical implementation have been identified and have to be addressed in future research.

Acknowledgements

The research described in this paper was substantially supported by a grant from the Research Grants Council of the HongKong Special Administrative Region, China (Project No. HKUST6215/06E). The second author was partially supported by theNSF of China (Project Nos. 70571033 and 70831002) and the third author was partially supported by the Cultivation Fund ofthe Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708044). The authors wish to expresstheir thanks to four anonymous reviewers for their very constructive comments on an earlier version of the paper.

Appendix A. Proofs of Theorems

Proof of Theorem 1. First setting x :¼ uþ ab �v � Cð Þ and z :¼ u� in (26), we get

kuðaÞ � u�k26 kuþ abð�v � CÞ � u�k2 � kuþ abð�v � CÞ � uðaÞk2

:

Substitute this into (34) leads to

HðaÞP ku� uðaÞk2 � 2abðu� u�ÞTð�v � CÞ � 2abðuðaÞ � uÞTð�v � CÞ: ð45Þ

In view of the similarly to the proof of Eq. (19), we have

ð�u� u�ÞTðv� � �vÞP 0;

which implies

ð�u� u�ÞTbðC � �vÞP ð�u� u�ÞTbðC � v�ÞP 0;

and, consequently, it yields

ðu� u�ÞTbðC � �vÞP ðu� �uÞTbðC � �vÞ: ð46Þ

Substituting (46) into (45), we obtain

HðaÞP ku� uðaÞk2 � 2abðuðaÞ � �uÞTð�v � CÞ ð47Þ


Now, we observe the second term in the right-hand-side of the inequality above. By setting x :¼ uþ bðv � CÞ and z :¼ uðaÞin (24), we have:

ðuðaÞ � �uÞT ½uþ bðv � CÞ � �u� 6 0;

that is,

bðuðaÞ � �uÞTð�v � CÞ 6 ð�u� uðaÞÞT gðv ;uÞ:

From (47), we thus arrive at:

HðaÞP ku� uðaÞk2 þ 2aðuðaÞ � �uÞT gðv; uÞ ¼ ku� uðaÞ � agðv ;uÞk2 þ 2aðu� �uÞT gðv ;uÞ � a2kgðv; uÞk2

P 2aðu� �uÞT gðv;uÞ � a2kgðv ;uÞk2:

The theorem thus holds. h

Proof of Theorem 2. From (38), we know that

kuðkþ1Þ � u�k26 kuðkÞ � u�k2 � c0kuðkÞ � �uðkÞk; ð48Þ

where c0 ¼ gð2� gÞð1� cÞa�. This means that the sequence fuðkÞg is Fejer monotone with respect to the solution set. There-fore, fuðkÞg is bounded and

limk!1kuðkÞ � �uðkÞk ¼ 0: ð49Þ

Consequently, the sequence f�uðkÞg is also bounded. Since we can always find a b > 0 such that fbkg � ½b;þ1Þ), it then followsfrom Lemma 2.2 in He and Liao (2002) that

keð�uðkÞ;bÞk 6 keð�uðkÞ;bkÞk ¼ k�uðkÞ � PRjAjþjWjþ½�uðkÞ þ bkð�v ðkÞ � CÞ�k

¼ kPRjAjþjWjþ½yðkÞ þ bkðv ðkÞ � CÞ� � P

RjAjþjWjþ½�uðkÞ þ bkð�v ðkÞ � CÞ�k 6 kðuðkÞ � �uðkÞÞ þ bkðv ðkÞ � �v ðkÞÞk

6 ð1þ cÞkuðkÞ � �uðkÞk ð50Þ

and thus

limk!1

eð�uðkÞ; bÞ ¼ 0: ð51Þ

Let u� be a cluster point of f�uðkÞg and the sub-sequence f�uðkjÞg converges to u�. Since eðu; bÞ is a continuous function of u, itfollows from (51) that

eðu�; bÞ ¼ limj!1

eð�uðkjÞ;bÞ ¼ 0: ð52Þ

Therefore, u� is a solution point. Note that inequality (48) is true for any solution point. We thus have

kuðkþ1Þ � u�k26 kuðkÞ � u�k2

; 8 k P 0: ð53Þ

As f�uðkjÞg ! u� and kuðkÞ � �uðkÞk ! 0, for any given e > 0, there is an l > 0 such that

k�uðklÞ � u�k2< e=2 and kuðklÞ � �uðklÞk < e=2: ð54Þ

Thus, for any k P kl, it follows from (53) and (54) that

kuðkÞ � u�k 6 kuðklÞ � u�k 6 kuðklÞ � �uðklÞk þ k�uðklÞ � u�k < e

and thereby the sequence fuðkÞg converges to u�. h

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