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Optimal road tolls under conditions of queueing and congestion
Trunspn Res.-A. Vol. 30, No. 5, pp. 319-332, 1996 Copyright 6 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0965~8564/96 St 5.00 + 0.00
Pergamon
SO%5-8564(%)00003-1
OPTIMAL ROAD TOLLS UNDER CONDITIONS AND CONGESTION
OF QUEUEING
HAI YAN Department
of
Civil and Structural Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
and WILLIAM H. K. LAM Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong (Received 10 August 1994: in revisedform
I November
1995)
Ah&act-Urban road networks in Hong Kong are highly congested, particularly during peak periods. Long vehicle queues at bottlenecks. such as the harbor tunnels, have become a daily occurrence. At present, tunnel tolls are charged in Hong Kong as one means to reduce traffic congestion. In general, flow pattern and queue length on a road network are highly dependent on traffic control and road pricing. An efficient control scheme must, therefore, take into account the effects of traffic control and road pricing on network flow. In this paper, we present a bi-level programming approach for determination of road toll pattern. The lower-level problem represents a queueing network equilibrium model that describes users’ route choice behavior under conditions of both queueing and congestion. The upper-level problem is to determine road tolls to optimize a given system’s performance while considering users’ route choice behavior. Sensitivity analysis is also performed for the queueing network equilibrium problem to obtain the derivatives of equilibrium link flows with respect to link tolls. This derivative information is then applied to the evaluation of alternative road pricing policies and to the development of heuristic algorithms for the bi-level road pricing problem. The proposed model and algorithm are illustrated with numerical examples. Copyright 0 1996 Elsevier Science Ltd
I. INTRODUCTION
Road pricing is considered to be one of the most efficient approaches to reducing traffic congestion, and is now implemented in many road network systems (Small, 1992). Congestion road pricing also provides a basis for investment decision in transport infrastructure, and assists in traffic control and management (Choi, 1986; Lam et al., 1994). In urban Hong Kong, traffic exhibits high levels of congestion during peak periods, with long vehicle queueing at bottlenecks such as the harbor tunnels. Currently, tunnel tolls - a form of road pricing - are charged as a primary means of affecting traffic distribution and mitigating traffic congestion. For a :ong time, the congestion road pricing problem has attracted considerable attention from both economists and transportation researchers, and has been examined extensively in different ways. In general, peak period congestion is due to an inefficient choice of departure times and travel routes. The structure of optimal road tolls should, therefore, reflect the distribution of demand in both time and space. However, most previous studies are restricted to hypothetical, idealized situations using analytical tools. The complications of traffic assignment and road network structure are avoided, and the time and space dimensions of road pricing have been treated separately. Conventional studies on the space dimension of road pricing are represented by traditional marginalcost pricing theory. It has long been recognized that road users using congested roads should pay a toll equal to the difference between the marginal social and the marginal private 319
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Hai Yan and William H. K. Lam
cost in order to achieve a system-optimal flow pattern. Beckman (1965) pointed out that tolls are economically optimal if they induce efficient use of the available road capacity. Dafermos and Sparrow (197 1) showed that, by the means of cong,stion tolls, individual users can be influenced to choose their travel paths leading to a system optimal resource allocation. A simple network with two paths was used to illustrate this result. Dafermos (1973) further proved that the marginal cost pricing theory is applicable to mixed traffic where travel costs vary from one type of vehicle to another. In addition, using a simple example, Smith (1979) indicated that the marginal-cost pricing principle also holds when there are link flow interactions between the cost of travel along one link and flow along other links. These previous analytical works made simplified assumptions that a perfect pricing system could be followed precisely and a system-optimal equilibrium established. They are less detailed as far as the network is concerned, and inadequate for treating the real issues at stake. In reality, road pricing is subject to various kinds of constraints, and a system-optimal equilibrium is thus not always possible. Therefore, it is desirable to establish constraints and then seek optimal road pricing within these limitations and to examine how the benefit obtained may fall short of the ideal. In an attempt to investigate this impact, Lam (1988) examined how road tolls affect decisions in transport investment and the role of road pricing in network design. In contrast to the traditional theoretical approach that seeks a system-optimum equilibrium by charging marginal-cost tolls, Lam et al. (1994) considered realistic road pricing issues in Hong Kong and presented heuristic optimization methods that enable the analysis of coordinated tunnel toll and petrol tax policies by minimizing an “objective function”, while satisfying the associated constraints. Recently, the time dimension of road pricing has been examined in terms of time-dependent toll patterns on simple road networks, typically with one or two routes in parallel. Arnott et al. (1990) investigated the efficiency gains from a time-dependent toll due to reduced congestion brought about by changing departure times. Laih (1994) considered toll systems designated to alleviate queueing problems that result from road bottlenecks. He presented an analysis of the multi-step tolls and the amount of queueing delay removal that can be achieved by varying the step tolls. Although the theoretical foundation of marginal-cost pricing has been well established, it is difficult to deal satisfactorily with the effects of road pricing in highly congested networks like Hong Kong, where long vehicle queues regularly form at some tunnels and demand an explicit remedy. In general, traffic flow and queue size on a road network depend on road toll pattern (and also traffic control, but this is outside the scope of this paper). An efficient pricing scheme should therefore take into account the effects of the altered network flow pattern and queueing due to road pricing to achieve a global optimal solution. This requires development of an efficient procedure for calculating optimal toll patterns in general road networks while anticipating driver response in terms of route choice. The procedure should be able to estimate queueing delay and queue length, both of which are critical in queue management in congested urban road networks. In this paper, we deal with combined traffic assignment and road pricing in general road networks with both queueing and congestion. An efficient strategy is developed to find a coordinated link toll pattern for reducing peak-hour traffic congestion. The main purpose of this work is: (1) to show that a bi-level programming can be used as an integrated approach to the problem of traffic assignment and road pricing in general congested queueing networks; (2) to implement sensitivity analysis of alternative road pricing policies; and (3) to develop heuristic algorithms to find an efficient set of link toll solutions for the bi-level congestion road pricing problem. The next section will present a bi-level programming formulation of the congestion road pricing problem. The lower-level problem is a queueing network equilibrium model that describes users’ route choice behavior under conditions of both queueing and congestion for a given link toll pattern. The upper-level problem determines toll pattern to optimize system performance while taking into account users’ reactions in response to alternative road tolls. In Section 3, heuristic algorithms are presented using the derivatives of equilibrium
321
Optimal road toils
link flows and equilibrium queueing delays with respect to link tolls. The derivative information is obtained by performing sensitivity analysis for the queueing network equilibrium problem under toll charging. Section 4 presents some simple examples to explain the proposed models and the algorithms. Conclusions and suggestions for future studies are summarized in Section 5. 2. A H-LEVEL
PROGRAMMING
APPROACH
We do not seek to establish an exact system optimum equilibrium by charging a marginalcost toll on every Iink in a road network. Instead, our purpose is to optimize system performances by choosing optimal tolls for a subset of links within the realistic constraints. The congestion road pricing problem considered here can be represented as a leader-follower, or Stackelbcrg, game where the system manager is the leader, and the network users are the followers. It is assumed that the system manager can influence, but cannot control, the users’ route choice behavior by choosing alternative toll policies. In light of any toll decision, the road users make their route choice decisions in a user-optimal manner. This interaction game can be represented as the following bi-ievel programming problem (Yang & Yagar, 1994, 1995): min F(u, v(u)) subject to G(u, v(u)) < 0, where v(u) is implicitly defined by minfiu, v) subject to g(u, v) I 0, where: F = objective function of upper-level decision-maker (system manager); u = decision vector of upper-level decision-maker (road toll pattern); G = constraint set of upper-level decision vector;S = objective function of lower-level decision-maker (network users); v = decision vector of lower-level decision-maker (network flow pattern); and g = constraint set of lower-level decision vector. It is assumed that the system manager selects feasible values for his decision variables, u, in an attempt to optimize his objective function F. The network users, after and with complete knowledge of the system manager’s decision, make route choice decisions in an attempt to minimize their travel cost, resulting in an aggregate network flow pattern, v(u). Furthermore, it is assumed that for any given toll pattern, u, there is a unique equilibrium flow distribution, v(u), obtained from the lower-level problem. v(u) is also called the response or reaction function. An efficient toll pattern, u, will greatly depend on how to evaluate the reaction function, v(u), or, in other words, how to predict route changes of users in response to alternative toll charges. Network equilibrium with queueing
In the urban road network of Hong Kong, traffic flow exhibits a high level of congestion. Especially long vehicle queues are formed at bottlenecks, as typified by the two harbor tunnels connecting Kowloon and Hong Kong Island in the morning and evening peak periods. Queueing delays constitute a significant component of travel cost. It is therefore necessary to explicitly consider queueing behavior in order to derive an efficient pricing policy. This paper adopts the queueing network equilibrium approach as already employed by Yang (1995a) and Yang and Yagar (1994, 1995). Queues only form when capacity is reached; below capacity, link travel time will solely depend on flow. Additionally, we assume that the o~~n~estination (O-D) demands are given and fixed, and hence steady-state traffic conditions prevail in the road network system.
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Hai Yan and William H. K. Lam
Under these assumptions, the feasible flow pattern through the road network is given by:
(2)
fr 2
0, rtzR,
(3)
where: W = the set of O-D pairs; A = the set of links in the network; R = the set of routes in the network; R,,. = the set of routes between O-D pair WEW,D,,. = the demand between O-D pair weW, v,, = flow on link aeA;f, = flow on route E-R; and S,, = 1 if route r uses link a, and 0 otherwise. Equation (1) is the O-D demand constraint, eqn (2) is the flow conservation constraint, and eqn (3) is a non-negativity constraint. As already pointed out, vehicle queueing at bottleneck links is explicitly considered. It is evident that vehicle queue length, Q,, or queueing delay, d,, at link aeA will grow if v,, > C,, and decrease if v, < C,, where C, is the capacity of link aeA (C, may be interpreted as saturation flow or exit capacity because queues arise at intersections where capacity is least). Since we consider a steady-state situation, v, I C, will always be satisfied, and Q, and d, will take certain values. Furthermore, it is obvious that there will be a vehicle queue (d, 2 0) only if v, = C,. These relationships can be expressed as: v, I C,, aeA d, = 0 if v,, < C,
(4) aeA
d, 2 0 if v, = C,
(5) ’
Here, we assume that only a subset of links (e.g. road tunnels) is subject to toll charges. The cost for a link consists of a flow-dependent running time, any delay due to queueing, and any tolls (for toll links only). Consistent with the assumption that each driver traveling from an origin to a destination will have perfect knowledge of the travel costs and queues via all routes, and will choose the lowest cost, the following equilibrium relationships are satisfied:
where: A* = a subset of toll links in the network; c,,.= minimal travel cost between O-D pair WEW; t, = flow-dependent running time on link aeA; and u,, = charge of toll to traverse link aeA* in equivalent minutes. For a given toll pattern, u, the steady-state traffic assignment in a congested road network with queueing is a specification of the vector of link flows, v, and a set of link queues, d, satisfying flow conservation conditions and queueing equilibrium conditions (l)-(6). Let c,( v,,u,) = t,(v,) + u, if aeA* and c,(v,, u,) = t,(v,,) otherwise; then the queueing network equilibrium problem for a given toll pattern, u, is equivalent to the following mathematical program: min!mizetiz
j:c,(x,
u,,)dx,
(74
subject to (7b)
3
.fA = v,,,ad
(7c) (74
Optimal road tolls
323
f, 2 0, MR.
(74
This problem is distinguished from the standard network equilibrium problem by imposing capacity constraint (7d) explicitly. As long as the road network has sufficient capacity, as a whole, to handle total demand, a feasible solution is guaranteed. Note that since the constraint set is convex and the objective function is strictly convex with respect to link flow variables, the equilibrium link flow pattern wilf be determined uniquely. Furthermore, the Lagrange multiplier associated with capacity constraint (7d) equals the ~uilibrium queueing delay. It should be pointed out that the Lagrange multipliers or queueing delays may not be unique. A necessary and sufficient condition for the queueing delays to be unique is that all the link capacity constraints that bind are linearly independent (Bell, 1994). The bi-level programming model
We have shown that users’ route choice behavior under conditions of queueing and congestion in a road network for any given toll pattern can be represented by the mathematical programming model (7). Global evaluation of the likely effects of road pricing thus becomes possible. We are now able to formulate our model to find an optimal set of link tolls such that a particular system performance criterion is optimized. The performance should include the global effects of pricing actions for given demands. Many alternative performance measures or system objective functions can be chosen. A meaningful objective is to minimize the total network cost or to maximize total revenue raised from toll charges. The total network travel cost, denoted as F;, is the sum of travel times and queueing delays experienced by all vehicles:
The total revenue, denoted as Fz, arising from toll charges can be expressed as:
A third objective function considered here is to maximize the ratio, denoted as F3, of the total revenue to total cost: F;=
I:
(10)
mA*
Note that the final results are quite sensitive to the chosen objective function. In general. F, could be adopted if the road pricing is intended to mitigate traffic congestion; F2 could be chosen if the road pricing is used as a means of generating revenue for infrastructure investment; F3 in eqn (IO) can be considered if the road pricing is intended to solve the problems of both congestion control and investment financing simultaneously. Other objective functions that better approximate the decision process, such as maximization of consumers’ surplus, could also be considered. Furthermore, since queues may form at links operating at capacity, an acceptable road pricing policy should consider queue length constraints. The number of queueing vehicles, Q,, queueing delay, d,, and the link exit flow, v,. are related by the following equation:
Q,, = v,d,,, ad. Maximum allowable storage capacity at link a~,4 of Qy queue length constraints expressed as: v,,d, S Qy,
UCA.
(111 (vehicles) can be enforced by (12)
Since queues cause a wasteful loss of time for users, and result in inefficient utilization of the road facility, an optimal road pricing might be required to eliminate queues completely. In this case, Q,“” may be set to be zero. On the other hand, a road network may involve constant returns to scale in practice, and hence a budget constraint should be imposed in determining road tolls so as to optimize a system performance:
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Hai Yan and William H. K. Lam &
(13)
uuvu 2 G,
where G is a given constant. Note that this constraint should not be imposed when the objective functions F2 or Fj are employed. Finally, the toll charges of all toll links should fit into given lower and upper bounds: Uzin I u, I Uy,
aeA*,
(14)
where ci” is the lower bound of toll charges and might be set to be zero or predetermined values for revenue consideration; I!J~ is the upper bound of toll charges and determined by what users may consider to be reasonable. In summary, the congestion road pricing problem under queueing network equilibrium conditions can be formulated as the following bi-level mathematical program: minimize F(u, v(u), d(u)),
(154
subject to
2
acA*
u,,v, 2 G -
(15c) (154
U,m’“
where v(u) and d(u) are obtained by solving: minimize 2
UCA
(15e)
subject to (15f.l
(1%) v, 5 C,,, aEA f;. 2 0, reR.
UW (15i)
This problem is to find an optimal toll pattern u* such that a system objective function F(u*,v(u*),d(u*)) is an optimum (the objective function may be selected either as F,, F2, or as FJ. Note that the above problem, like any other form of bi-level mathematical programming problems, is intrinsically non-convex, and hence it might be difficult to solve for a global optimum (Friesz et al., 1990; Yang et al., 1994). It should be pointed out that the aforementioned bi-level congestion pricing model assumes a fixed travel demand pattern. However, congestion pricing might have important effects on the demand for travel beyond route choice. Ir this case, the elasticity of travel demand can be explicitly incorporated by employing an elastic-demand network equilibrium model with queues as the lower-level model, instead of a fixed-demand equilibrium model. 3. SENSITIVITY ANALYSIS BASED ALGORITHMS
The key point in solving the global optimal toll charging problem (I 5) lies in how to evaluate the equilibrium link flow, v(u), and equilibrium queueing delay, d(u), which are implicitly defined by the lower-level queueing network equilibrium problem. Recently, Yang et al. (1994) and Yang and Yagar (1994) developed efficient heuristic algorithms for solving bi-level transportation system optimization problems, based on sensitivity analysis results for equilib-
Optimal road tolls
325
rium network flows. These algorithms have been successfully applied to optimal ramp metering in general freeway-arterial corridor systems (Yang & Yagar, 1994) traffic signal control in saturated road networks (Yang & Yagar, 1995) and congested O-D matrix estimation problems (Yang, 1995b). A detailed description of the development of the sensitivity analysis based algorithm is given in Yang and Yagar (1994). Here, we adopt the sensitivity analysis-based algorithm developed by Yang and Yagar (1994) for solving the bi-level congestion road pricing problem. Because the upper-level objective function and constraints are implicit, non-linear functions of decision variable u, local linear approximations using Taylor’s formula are implemented based on the derivatives of the reaction functions with respect to link tolls. The derivative information is obtained by implementing sensitivity analysis for a given solution of the queueing network equilibrium problem. The sensitivity analysis based (SAB) algorithm is outlined below. SAB algorithm: Step 0. Determine an initial set of link toll pattern II’(‘).Set n = 0. Step I. Solve the lower-level queueing network equilibrium problem for given II(“) using the inner penalty function method, and hence get v(“)and d(“). Step 2. Calculate the derivatives, %‘%u and &V&I, using the sensitivity analysis method. Step 3. Formulate local linear approximations of the upper-level objective function and constraints using the derivative information and solve the resulting linear program to obtain an auxiliary solution y. Step 4. Compute u”’+ ‘) = u(“)+ (Y(‘l)(y-uO’)) where (Y’“)is given by: Ly(“’= pl( 1 + !?)Y, where p, y are parameters (p > 0, y 2 1). Step 5. If Itll;+” -- u!“)lI E for all aeA*, then stop where E is a predetermined Otherwise, let n: = n + 1 and return to Step 1.
tolerance.
At Step I, the lower-level queueing network equilibrium problem with explicit capacity constraints is solved by an inner penalty function method (Yang & Yagar, 1994, 1995). After proper transformation of the augmented objective function, the penalty function method gives rise to a sub-problem structure similar to the Frank-Wolfe convex combination algorithm (Sheffi, 1985) and hence can be easily implemented, even for large networks. In the final output, the algorithm generates information necessary for implementing sensitivity analysis, including a complete set of link flow patterns, queueing delays on a subset of links, and a subset of minimum time paths used by the users between each O-D pair. At Step 2, sensitivity analysis is implemented to compute the derivatives of equilibrium link flows and queueing delays with respect to link tolls for a given solution of the queueing network equilibrium problem. The sensitivity analysis method for the network equilibrium problem with capacity constraints has been developed and incorporated into the bi-level traffic control problem (Yang & Yagar, 1994, 1995; Yang, 1995a). At Step 3, the derivative information obtained from sensitivity analysis is used to formulate local linear approximations of the reaction functions and hence the upper-level objective function and constraints. This approximation will result in a linear programming problem, to which the solution would be located at the farthest point in the descent direction of the upper-level objective function. At Step 4, the step size, II(“), is determined a priori, and decreases monotonically with the number of iterations. It should be noted that any standard one-dimensional search method might be adopted, but it will require repeated evaluation of the upper-level objection function along its descent direction. Each of these evaluations involves an equilibrium assignment in order to determine link flow and queueing delay. Thus, the computational requirements of the one-dimensional search may be prohibitively expensive, and will not be adopted here. The above SAB algorithm is applicable when the upper-level problem contains non-linear constraints (15b) and (15~). In some situations, these non-linear constraints might not be
Hai Yan and William H. K. Lam
326
involved, for example, constraint (1%) should be deleted when the upper-Ievel objective function is a revenue maxim~tion. When only the constant fl5d) is included in the upperleve1 problem, the following simple equ~Iib~~ decomposition opt~~tion (EDO) algorithm proposed by Suwansirikul et al. (1987) and Friesz et al. (1990) may be employed. ED0 algorithm: Step 0. Set uA”’= 4(u:in + Uy), ,QO’= tJ:in, u:O’ = Uy and set n = 0. Step 1. Solve the lower-level queueing network equilibrium problem
for given u(“’ using the inner penalty function method, and hence get v(“)and d(“). Step 2. Calculate the derivatives, @‘Y&r and &P%3u, using the sensitivity analysis method. Step 3. Calculate the derivatives, V,A4(u), using sensitivity analysis information and the following chain rule: am au,
8F dvh (---+--
_ x bd
&,
au,
at;
ad,
ad,
ihi<,’ +
__aF au<*’ aeA*7
where M(u) = F(v(u),d(u),u). Step 4. Calculate Z,(”+ ‘) and iJ(” + ‘) according to if aM/&
if aM/&
< 0 3 set L(“+‘) = us”‘, U(n+‘) (10 LI = u(“) ” 1 D
> 0
set T
U(n+‘l = uj,“), L(“+‘) ‘I
”
= L(“) (1
.
Step 5. Compute uln*‘)as UP+‘)= t (U:+” + Lp+‘)), ad*. Step 6. If Iuy+l) -uj;“‘lI E for all ad*, then stop. Otherwise let n: = n + 1 and go to
Step 1.
4. NUMERICAL
EXAMPLES
In this section, we present three examples using hypothetical networks. The first example is presented to illustrate the sensitivity analysis procedure and its applications. The second example examines the traditional marginal-cost pricing theory under ideal conditions with the bi-level road pricing model. The last example is used to investigate the congestion road pricing under realistic constraints. Example I. Sensitivity anaiysis of congestion pricing
This example illustrates the ~nsitivity analysis procedure for a given road toll pattern. The road network, as shown in Fig. 1, consists of seven nodes and 11 links, of which links l-4 are subject to toil charges. The following link cost function is used
Fig. I. Example network I.
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321
Table 1. Input data for example network I Link a
to ?
1
2
3
4
5
6
7
8
9
10
II
200 6
2005
200 6
2Ozl
100 4
150 3
150 5
150 IO
200 16
200 15
100 17
t,(v,) = fZ{ 1.0 + 0.15(+}.
(16)
(I
Link free-flow travel time, c, and link capacity, C,, are given in Table 1. It is assumed that there are three O-D pairs (1+7, 2+7 and 3+7) and the demands are fixed to be D,,= 250, D?,= 200 and D,,= 200, respectively. Sensitivity analysis is used to predict changes in equilibrium link flow pattern, queueing delay, and system objective function in response to any small variation in toll charges, Table 2 presents the derivatives of link flows, queueing delays and the system objective functions (total network travel cost, F,, total revenue F2,and their ratio F3)with respect to link tolls at U, = uz = u3 = uq = 1.0 (link tolls are expressed in equivalent travel time units). Note that only derivatives of queueing delays at queued links (links 3, 4 and 5 as indicated in the figure and table) are included in Table 2, while derivatives for other links without queues will be zero. These results show the variations of flows through the road network, queueing delays at queued links, and the variations of system performance functions when the charge at one of the toll links is increased by 1 unit. It should be mentioned that a small change in tolls that are charged at a congested link with strictly positive queue will have no effect on the network link flow pattern because the effect of toll change is cancelled by reduction of the equivalent amount of queueing delay at the same link. This has been demonstrated in Table 2, where all the derivatives of link flows with respect to toll on a queued link are equal to zero, and the derivative of queueing delay at a queued link with respect to the toll on the same link is equal to -1.0. The derivative information has many important implications in both the operational efficiency of congested road networks and the economic appraisal of varied road pricing policies. The derivatives indicate that the link toll charge to which the congested equilibrium flow pattern is the most sensitive, and, therefore, is the link toll that deserves the most consideration to improve the efficiency of the congested road system. The derivative information also gives the “direction” in which the queueing network equilibrium flow pattern should move so that a system optimum could be achieved if alternative road tolls are adopted. It is interesting to see, from Table 2, that the link toll interaction is symmetric. Namely,
Table 2. Derivatives of link flows, queueing delays and objective values with respect lo link tolls at u, = uz = q = u4 = 1.O for example network I Variable
a(‘,/aU, -3 I ,622 28.675 0.000 0.000 0.000 -28.675 31.622 0.000 28.675 -3 1.622 2.948 -0.616 0.050 -0.146 -195.547 146.291 0.013
a(.)fauz 28.675 44.096 0.000 0.000 0.000 44.096 -28.675 0.000 44.096 28.675 15.421 0.558 -0.078 -0.776 145.132 106.43I 0.008
3. m3 0.000
0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
-1.00 0.000 0.000 -199.998 199.998 0.018
mau4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
-1.000 0.000 -199.998 199.998 0.018
Hai Yan and William H. K. Lam
328
av,_- au& 84
for all a, beA*
B
(17)
This means that the marginal effect of one link toll, say u,, on the link flow on any other toll link, say v,,, is equal to the marginal effect of uh on v,. It can be proved that this conclusion holds for the standard network equilibrium problem in general networks. The derivative information also allows one to estimate nearby solutions for any combination of changes in link tolls, once a queueing equilibrium solution has been obtained. Table 3 presents the impacts on the link flows when the link tolls are varied slightly by Su, = Su, = &+= au, = 0.1 with respect to a reference toll charge U, = u2 = U) = u4 = 1.0. The estimation is made using the following linear approximation based on the derivative information in Table 2. v(u) =v(uo) +
$(” - UJ.
(18)
The impacts of the toll adjustments on the queueing delays and system performance functions are also indicated in Table 3. These results could allow traffic engineers and planners to evaluate alternative road tolls, and hence to develop improvement strategies for systems operations. Example 2. First-best roud pricing
The genera1 theory of first-best optimal pricing, i.e. the theory of marginal cost pricing in economics, is well established. In the traffic assignment literature (Beckmann et al., 1956), congestion tolls have been proposed to drive a user equilibrium pattern toward a “system optimum”. Namely, an exact system optimum equilibrium can be achieved if on every link of a network a toll (marginal-cost toll, MCT) is charged, which equals the difference between marginal social and marginal private costs: MCT
= ‘I
4vax t<,(v,,)) _ VU
t,,(v,)
=
v, ___ dt,(v,) for all aeAA.
(19)
V”
This idea1 MCT provides an exact optimal solution under perfect charging conditions, and hence could be used to verify our mode1 and algorithm. Traditional marginal cost pricing theory uses the network equilibrium model that does not consider capacity constraint (15h) and hence does not consider queues explicitly. For the purpose of comparison, the standard network equilibrium model is used here as the lowerlevel problem, and the non-linear constraints (15b) and (15~) are deleted in the upper-level problem. Table 3. Estimated and exact solutions for perturbed link tolls at ut = u2 = uj = uq = 1.0 for example network I Perturbed with &q = &I? = 6q = &44= 0.10 Solution variable 1’
“2
“3 v4 v5 v6 v-i V8 v9 “IO VII d3 d4 d, FI F2 F3
Unperturbed solution 149.238 121.852 199.998 199.998 99.998 78.145 50.760 100.002 21.857 149.242 78.908 2.778 2.985 2.685 12.062.850 67 1.086 0.056
Exact 148.693 120.797 199.997 199.997 99.997 79.198 51.305 lOQ.003 20.804 148.696 80.509 2.198 2.247 2.312 Il,766.800 736.433 0.063
Estimated 148.944 120.310 199.998 199.998 99.998 79.687 51.055 100.002 20.315 148.947 80.745 2.672 2.882 2.592 12,017.810 736.358 0.061
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Fig. 2. Example network 2
Figure 2 shows the example network consisting of six nodes and seven links, of which link 1 and link 2 are toll links. The same link cost function (16) is used with link free-flow travel time and link capacity provided in Table 4. It is assumed that there are only two O-D pairs (143 and 2+4) and the demands are fixed to be D,, = D14 = 30.0. The system-optimum flow pattern and the corresponding marginal-cost tolls can be obtained using the network equilibrium assignment with a marginal cost curve, instead of an average cost function. The results obtained for example network 2 are shown in Table 5. Now consider the solutions produced by the proposed bi-level pricing model when all links are subject to toll charges. The ED0 algorithm is used with the lower and upper bounds: 0.0 < U, 5 5.0 for a = 1,2 and 0.0 I U, I 2.0 for u = 3-7. Note that the upper bounds of link tolls have been set higher than the corresponding MCTs so that at least one optimal solution leading to a system optimum is included in the feasible region. The convergence was achieved in 13 iterations with a convergence tolerance E = 0.001. From Table 5, it can be seen that the optimal toll solution obtained by the ED0 algorithm is different from the marginal-cost tolls, while it gives rise to the same total network travel cost F, = 628.60. This demonstrated that there is usually more than one link toll pattern that leads to the system optimum. In fact, different optimal solutions have been found when different upper bounds of link tolls are employed. This observation indicates that there is a great degree of flexibility in a road network for setting link tolls to satisfy various kinds of constraints, while at the same time leading to a system-optimum equilibrium. For instance, a link toll pattern may be determined subject to a given budget constraint, yet still lead to a systemoptimum equilibrium. Example 3. Second-best road pricing
The first-best road pricing assumed that a perfect charging system could be followed precisely and a system-optimal equilibrium established. However, for whatever the reasons. Table 4. Input data for example network 2 Link u
I
2
3
4
5
6
7
20.0 8.0
9.0 20.0
2.0 20.0
40.0 6.0
3.0 20.0
3.0 25.0
4.0 25.0
to &
Table 5. System-optimal flow pattern, marginal-cost toll and optimal toll solution for example network 2 Link number System optimum flow Marginal-cost toll Optimal toll solution
I
2
3
4
5
6
7
17.916 3.091 3.820
18.486 3.941 4.265
12.084 0.160 0.472
23.598 0.436 0.476
1I.514 0.198 0.294
12.084 0.098 0.472
II.514 0.108 0.294
Total network cost in system optimum = 628.60 Total network cost corresponding to optimal link toll solution = 628.60 Total network cost in user-equilibrium without toll charge = 720.09
Hai Yan and William H. K. Lam
330
the rule of setting marginal-cost toll may not be applied. For example, the marginal cost pricing rule cannot be applied when only a subset of liiks (e.g. harbor tunnels in Hong Kong) in the network is subject to toll charges, and/or the network involves a budget constraint, as considered in our b&level pricing model. Now consider the same example network 2. Suppose that only link 1 and link 2 are subject to toll charges. For the moment, we omit the queue length constraints (15b) and budget constraint (15c), and assume the bounds: 0.0 I u, < 5.0, 0.0 < z.+5. 5.0, the ED0 algorithm is applied to solve the b&level pricing model with alternative upper-level objective functions. From the results presented in Table 6, it is easily seen that different upper-level objective function would generally result in different road toll pattern. It can also be observed that the solution for minimizing the total network travel cost F, is obviously a global optimum since it associates the objective function with the same value as in system optimum (& = 628.60). This indicates that a system optima equilib~um can be (though is not necessarily) achieved if we impose proper tolls, even only on a subset of network links. Now we compare this result with the situation where the marginal-cost pricing rule is applied, but only for link 1 and link 2. This rule yields the following results: u, = 2.577, u, = 3.390, F = 629.28. Although this partial marginal-cost pricing rule results in a reduction in total network cost (compared to the total cost 720.09 in user-equilibrium without toll charges as shown in Table 5), the resulting toll pattern does not represent the most desirable one (u, = 2.399, u2 = 3.200 shown in Table S), i.e. one that leads to a system optimum. So far, we have not yet tested the model and algorithm with queue length constraint (1 Sb). This constraint may become active, or vehicle queues may occur when there is a heavy demand during peak periods. Therefore, it might be necessary to impose queue constraints in optimizing a system performance function. In this case, the ED0 algo~thm is no longer applicable, and instead the SAB algorithm should be applied. Using a higher traffic demand, & = &, = 35.0, which, when assigned to the non-toll example network 2, produces the following vehicle queues at links 1 and 2: Q, = v, x d, = 19.990 X 1.999 = 39.96 (veh) QZ = v? X d7 = 19.993 X 2.691 = 53.80 (veh). Here, we assume that the toll bounds 0.0 < u, < 5.0, 0.0 5 Ed?I 5.0 are imposed. and no vehicle queues are allowed in optimizing system objectives. Namely, maximum queue length is set to be: Qy = 0.0, which means that the link tolls are dete~ined so as to at least eliminate queue. Table 6. Solutions with alternative upper-level objective functions for example network 2
For minimizing F, For maximizing FZ For maximizing F3
Link toll UI
Link toll U?
2.399 3.164 3.125
3.200 4.063 3.828
Objective value 628.60 107.15 0.168
Main iteration number 13 I3 I3
Note: The EJDO algorithm is applied with E = 0.001 and initial values ,i”’ = ut”’ = 2.5.
Table 7. Solutions with alternative objective functions and queue length constraint for example network 2
For minimizing F, For maximizing F2 For maximizing Fj
Link toll Ul
Link toll
2.398 3.871 2.401
3.316 4.853 3.342
u2
Objective value 766.91 136.95 0.146
Main iteration number 49 I6 51
Note: The SAB al orithm is applied with fixed step size a = l/(1 + n) and convergence criterion E = 0.05. The initial values are u!) - @ - 2.5, and maximum queue length Qy = 0.0 for all aA.
Optimal road tolls
331
The solutions for alternative objective functions with explicit queue constraint (c = 0.0) are shown in Table 7. It can be seen that a large number of iterations are required to achieve the convergence when using the SAB algorithm. Note that, although not reported in Table 7, the queue length constraint (15b) is active in the final convergent solution when maximizing F2 and F3. 5. CONCLUSIONS
Some developments in model formulations and solution procedures for the congestion road pricing problem under queueing network equilibrium conditions have been presented. The methodology proposed in this paper is characteristic in two aspects. Firstly, the bi-level model considers a subset of links subject to toll charges, and the toll charges are subject to given lower and upper bounds. This characteristic is a significant departure from the conventional marginal-cost pricing theory that imposes prices on every link exactly equal to the difference between marginal social cost and marginal private cost. Secondly, the queueing network equilibrium model predicts queueing delay explicitly, and hence makes it easier to implement queue management in congested urban areas by implementing different toll pricing policies. Furthermore, a sensitivity analysis has been implemented to obtain the derivatives of link flows and queueing delays with respect to link tolls, and hence has produced the “direction” in which the queueing network equilibrium pattern can move if the toll pattern is changed. This information has been used efficiently to determine optimal road tolls such that total network travel time is minimized or total toll revenue is maximized. While the examples are very simple, our numerical results reveal some interesting properties of congestion road pricing and demonstrate that the sensitivity analysis based-algorithms perform well in finding the optimal road tolls. The bi-level pricing model is being applied to coordination of tunnel toll patterns in the Hong Kong road network, and the numerical results will be reported in a future paper. Finally, we should acknowledge that there are still many theoretical and practical issues to be discussed. We have only examined the space dimension of congestion road pricing. As pointed out in the introduction, time-dependent road pricing is a complementary and equally critical topic for further research. This would involve modeling the combination of choice of departure time and choice of route under time-dependent toll charges. AcknoM?/edgemenr.s-The authors wish to thank two anonymous referees for their helpful comments and suggestions for improvement, This research has been supported with a UGC Research Infrastructure Grant (R194/95.EGOl) from The Hong Kong University of Science and Technology.
REFERENCES Arnott R., Palma A. de and Lindsey R. (1990) Departure time and route choice for the morning commute. Trunspn Res. 24B, 209-228. Eeckmann M. J. (1965) On optimal tolls for highways, tunnels and bridges. In: Vehicubr Truj% Science (Edie L. C., Herman R. and Rothery R. eds). pp. 331- 341. Elsevier, New York. Beckmann M. J., Mcguire C. B. and Winsten B. (1956) S/dies in /he Economics of Trunsporta/ion. Yale University Press, New Haven, Connecticut. Bell M. G. H. (1994) Stochastic user equilibrium assignment in networks with queues. Transpn Res. 29B, 125-137. Choi Y. L. (1986) Land use transport optimization (LUTO) model for strategic planning in Hong Kong. Asian Geogr. 5, 155-176. Dafermos S. C. (1973) Toll patterns for multiclass-user transportation networks. Transpn Sri. 7, 21 l-223. Dafermos S. C. and Sparrow F. T. (1971) Optimal resource allocation and toll patterns in user-optimized transport network. J. Trunsp. Econ. Policy 5, 198-200. Friesz T. L., Tobin R. L., Cho H. J. and Mehta N. J. (1990) Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints. Mat/r. Progm. 48, 265-284. Laih C. H. (1994) Queueing at a bottleneck with single- and multi-step tolls. Transpn Res. 28A, 197-208. Lam W. H. K. (1988) Effects of road pricing on system performance. Tru@c Engng Conrrol29,631-635. Lam W. H. K., Poon A. C. K. and Ye R. J. (1994) Coordination of road pricing policies. Proceerfings of fhe VII Internutionul Conference on Truvel Behavior, Valle Nevado, Chile, June 1994, pp. 307-318. Sheffi Y. (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Cliffs. New Jersey. Small K. A. (1992) Special issue: Congestion pricing. Trunsporfufion 19, 287-291.
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Smith M. J. (1979) The marginal cost taxation of a transportation network. Trunspn Res. 13B, 237-242. Suwansirikul C., Friesz T. L. and Tobin R. L. (1987) Equilibrium decomposed optimization: A heuristic for the continuous equilibrium network design problem. Transpn Sci. 21, 254-263. Yang H. (1995a) Sensitivity analysis for queueing equilibrium network flow and its application to traffic control. Math. Compt. Model. 22, 242-258.
Yang H. (199513)Heuristic algorithms for the bi-level origin-destination
matrix estimation problem. Trnnspn
Rex 29B, 23 I-241.
Yang H. and Yagar S. (1994) Traffic assignment and traffic control in general freeway-arterial corridor systems. Transpn Res. 28B, 463-486. Yang H. and Yagar S. (1995) Traffic assignment and signal control in saturated road networks. Trnnspn Rex 29A, 125-139.
Yang H., Yagar S., Iida Y. and Asakura Y. (1994) An algorithm for the inflow control problem on urban freeway networks with user-optimal flows. Transprl Res. 28B, 123-139.
Pergamon
SO%5-8564(%)00003-1
OPTIMAL ROAD TOLLS UNDER CONDITIONS AND CONGESTION
OF QUEUEING
HAI YAN Department
of
Civil and Structural Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
and WILLIAM H. K. LAM Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong (Received 10 August 1994: in revisedform
I November
1995)
Ah&act-Urban road networks in Hong Kong are highly congested, particularly during peak periods. Long vehicle queues at bottlenecks. such as the harbor tunnels, have become a daily occurrence. At present, tunnel tolls are charged in Hong Kong as one means to reduce traffic congestion. In general, flow pattern and queue length on a road network are highly dependent on traffic control and road pricing. An efficient control scheme must, therefore, take into account the effects of traffic control and road pricing on network flow. In this paper, we present a bi-level programming approach for determination of road toll pattern. The lower-level problem represents a queueing network equilibrium model that describes users’ route choice behavior under conditions of both queueing and congestion. The upper-level problem is to determine road tolls to optimize a given system’s performance while considering users’ route choice behavior. Sensitivity analysis is also performed for the queueing network equilibrium problem to obtain the derivatives of equilibrium link flows with respect to link tolls. This derivative information is then applied to the evaluation of alternative road pricing policies and to the development of heuristic algorithms for the bi-level road pricing problem. The proposed model and algorithm are illustrated with numerical examples. Copyright 0 1996 Elsevier Science Ltd
I. INTRODUCTION
Road pricing is considered to be one of the most efficient approaches to reducing traffic congestion, and is now implemented in many road network systems (Small, 1992). Congestion road pricing also provides a basis for investment decision in transport infrastructure, and assists in traffic control and management (Choi, 1986; Lam et al., 1994). In urban Hong Kong, traffic exhibits high levels of congestion during peak periods, with long vehicle queueing at bottlenecks such as the harbor tunnels. Currently, tunnel tolls - a form of road pricing - are charged as a primary means of affecting traffic distribution and mitigating traffic congestion. For a :ong time, the congestion road pricing problem has attracted considerable attention from both economists and transportation researchers, and has been examined extensively in different ways. In general, peak period congestion is due to an inefficient choice of departure times and travel routes. The structure of optimal road tolls should, therefore, reflect the distribution of demand in both time and space. However, most previous studies are restricted to hypothetical, idealized situations using analytical tools. The complications of traffic assignment and road network structure are avoided, and the time and space dimensions of road pricing have been treated separately. Conventional studies on the space dimension of road pricing are represented by traditional marginalcost pricing theory. It has long been recognized that road users using congested roads should pay a toll equal to the difference between the marginal social and the marginal private 319
320
Hai Yan and William H. K. Lam
cost in order to achieve a system-optimal flow pattern. Beckman (1965) pointed out that tolls are economically optimal if they induce efficient use of the available road capacity. Dafermos and Sparrow (197 1) showed that, by the means of cong,stion tolls, individual users can be influenced to choose their travel paths leading to a system optimal resource allocation. A simple network with two paths was used to illustrate this result. Dafermos (1973) further proved that the marginal cost pricing theory is applicable to mixed traffic where travel costs vary from one type of vehicle to another. In addition, using a simple example, Smith (1979) indicated that the marginal-cost pricing principle also holds when there are link flow interactions between the cost of travel along one link and flow along other links. These previous analytical works made simplified assumptions that a perfect pricing system could be followed precisely and a system-optimal equilibrium established. They are less detailed as far as the network is concerned, and inadequate for treating the real issues at stake. In reality, road pricing is subject to various kinds of constraints, and a system-optimal equilibrium is thus not always possible. Therefore, it is desirable to establish constraints and then seek optimal road pricing within these limitations and to examine how the benefit obtained may fall short of the ideal. In an attempt to investigate this impact, Lam (1988) examined how road tolls affect decisions in transport investment and the role of road pricing in network design. In contrast to the traditional theoretical approach that seeks a system-optimum equilibrium by charging marginal-cost tolls, Lam et al. (1994) considered realistic road pricing issues in Hong Kong and presented heuristic optimization methods that enable the analysis of coordinated tunnel toll and petrol tax policies by minimizing an “objective function”, while satisfying the associated constraints. Recently, the time dimension of road pricing has been examined in terms of time-dependent toll patterns on simple road networks, typically with one or two routes in parallel. Arnott et al. (1990) investigated the efficiency gains from a time-dependent toll due to reduced congestion brought about by changing departure times. Laih (1994) considered toll systems designated to alleviate queueing problems that result from road bottlenecks. He presented an analysis of the multi-step tolls and the amount of queueing delay removal that can be achieved by varying the step tolls. Although the theoretical foundation of marginal-cost pricing has been well established, it is difficult to deal satisfactorily with the effects of road pricing in highly congested networks like Hong Kong, where long vehicle queues regularly form at some tunnels and demand an explicit remedy. In general, traffic flow and queue size on a road network depend on road toll pattern (and also traffic control, but this is outside the scope of this paper). An efficient pricing scheme should therefore take into account the effects of the altered network flow pattern and queueing due to road pricing to achieve a global optimal solution. This requires development of an efficient procedure for calculating optimal toll patterns in general road networks while anticipating driver response in terms of route choice. The procedure should be able to estimate queueing delay and queue length, both of which are critical in queue management in congested urban road networks. In this paper, we deal with combined traffic assignment and road pricing in general road networks with both queueing and congestion. An efficient strategy is developed to find a coordinated link toll pattern for reducing peak-hour traffic congestion. The main purpose of this work is: (1) to show that a bi-level programming can be used as an integrated approach to the problem of traffic assignment and road pricing in general congested queueing networks; (2) to implement sensitivity analysis of alternative road pricing policies; and (3) to develop heuristic algorithms to find an efficient set of link toll solutions for the bi-level congestion road pricing problem. The next section will present a bi-level programming formulation of the congestion road pricing problem. The lower-level problem is a queueing network equilibrium model that describes users’ route choice behavior under conditions of both queueing and congestion for a given link toll pattern. The upper-level problem determines toll pattern to optimize system performance while taking into account users’ reactions in response to alternative road tolls. In Section 3, heuristic algorithms are presented using the derivatives of equilibrium
321
Optimal road toils
link flows and equilibrium queueing delays with respect to link tolls. The derivative information is obtained by performing sensitivity analysis for the queueing network equilibrium problem under toll charging. Section 4 presents some simple examples to explain the proposed models and the algorithms. Conclusions and suggestions for future studies are summarized in Section 5. 2. A H-LEVEL
PROGRAMMING
APPROACH
We do not seek to establish an exact system optimum equilibrium by charging a marginalcost toll on every Iink in a road network. Instead, our purpose is to optimize system performances by choosing optimal tolls for a subset of links within the realistic constraints. The congestion road pricing problem considered here can be represented as a leader-follower, or Stackelbcrg, game where the system manager is the leader, and the network users are the followers. It is assumed that the system manager can influence, but cannot control, the users’ route choice behavior by choosing alternative toll policies. In light of any toll decision, the road users make their route choice decisions in a user-optimal manner. This interaction game can be represented as the following bi-ievel programming problem (Yang & Yagar, 1994, 1995): min F(u, v(u)) subject to G(u, v(u)) < 0, where v(u) is implicitly defined by minfiu, v) subject to g(u, v) I 0, where: F = objective function of upper-level decision-maker (system manager); u = decision vector of upper-level decision-maker (road toll pattern); G = constraint set of upper-level decision vector;S = objective function of lower-level decision-maker (network users); v = decision vector of lower-level decision-maker (network flow pattern); and g = constraint set of lower-level decision vector. It is assumed that the system manager selects feasible values for his decision variables, u, in an attempt to optimize his objective function F. The network users, after and with complete knowledge of the system manager’s decision, make route choice decisions in an attempt to minimize their travel cost, resulting in an aggregate network flow pattern, v(u). Furthermore, it is assumed that for any given toll pattern, u, there is a unique equilibrium flow distribution, v(u), obtained from the lower-level problem. v(u) is also called the response or reaction function. An efficient toll pattern, u, will greatly depend on how to evaluate the reaction function, v(u), or, in other words, how to predict route changes of users in response to alternative toll charges. Network equilibrium with queueing
In the urban road network of Hong Kong, traffic flow exhibits a high level of congestion. Especially long vehicle queues are formed at bottlenecks, as typified by the two harbor tunnels connecting Kowloon and Hong Kong Island in the morning and evening peak periods. Queueing delays constitute a significant component of travel cost. It is therefore necessary to explicitly consider queueing behavior in order to derive an efficient pricing policy. This paper adopts the queueing network equilibrium approach as already employed by Yang (1995a) and Yang and Yagar (1994, 1995). Queues only form when capacity is reached; below capacity, link travel time will solely depend on flow. Additionally, we assume that the o~~n~estination (O-D) demands are given and fixed, and hence steady-state traffic conditions prevail in the road network system.
322
Hai Yan and William H. K. Lam
Under these assumptions, the feasible flow pattern through the road network is given by:
(2)
fr 2
0, rtzR,
(3)
where: W = the set of O-D pairs; A = the set of links in the network; R = the set of routes in the network; R,,. = the set of routes between O-D pair WEW,D,,. = the demand between O-D pair weW, v,, = flow on link aeA;f, = flow on route E-R; and S,, = 1 if route r uses link a, and 0 otherwise. Equation (1) is the O-D demand constraint, eqn (2) is the flow conservation constraint, and eqn (3) is a non-negativity constraint. As already pointed out, vehicle queueing at bottleneck links is explicitly considered. It is evident that vehicle queue length, Q,, or queueing delay, d,, at link aeA will grow if v,, > C,, and decrease if v, < C,, where C, is the capacity of link aeA (C, may be interpreted as saturation flow or exit capacity because queues arise at intersections where capacity is least). Since we consider a steady-state situation, v, I C, will always be satisfied, and Q, and d, will take certain values. Furthermore, it is obvious that there will be a vehicle queue (d, 2 0) only if v, = C,. These relationships can be expressed as: v, I C,, aeA d, = 0 if v,, < C,
(4) aeA
d, 2 0 if v, = C,
(5) ’
Here, we assume that only a subset of links (e.g. road tunnels) is subject to toll charges. The cost for a link consists of a flow-dependent running time, any delay due to queueing, and any tolls (for toll links only). Consistent with the assumption that each driver traveling from an origin to a destination will have perfect knowledge of the travel costs and queues via all routes, and will choose the lowest cost, the following equilibrium relationships are satisfied:
where: A* = a subset of toll links in the network; c,,.= minimal travel cost between O-D pair WEW; t, = flow-dependent running time on link aeA; and u,, = charge of toll to traverse link aeA* in equivalent minutes. For a given toll pattern, u, the steady-state traffic assignment in a congested road network with queueing is a specification of the vector of link flows, v, and a set of link queues, d, satisfying flow conservation conditions and queueing equilibrium conditions (l)-(6). Let c,( v,,u,) = t,(v,) + u, if aeA* and c,(v,, u,) = t,(v,,) otherwise; then the queueing network equilibrium problem for a given toll pattern, u, is equivalent to the following mathematical program: min!mizetiz
j:c,(x,
u,,)dx,
(74
subject to (7b)
3
.fA = v,,,ad
(7c) (74
Optimal road tolls
323
f, 2 0, MR.
(74
This problem is distinguished from the standard network equilibrium problem by imposing capacity constraint (7d) explicitly. As long as the road network has sufficient capacity, as a whole, to handle total demand, a feasible solution is guaranteed. Note that since the constraint set is convex and the objective function is strictly convex with respect to link flow variables, the equilibrium link flow pattern wilf be determined uniquely. Furthermore, the Lagrange multiplier associated with capacity constraint (7d) equals the ~uilibrium queueing delay. It should be pointed out that the Lagrange multipliers or queueing delays may not be unique. A necessary and sufficient condition for the queueing delays to be unique is that all the link capacity constraints that bind are linearly independent (Bell, 1994). The bi-level programming model
We have shown that users’ route choice behavior under conditions of queueing and congestion in a road network for any given toll pattern can be represented by the mathematical programming model (7). Global evaluation of the likely effects of road pricing thus becomes possible. We are now able to formulate our model to find an optimal set of link tolls such that a particular system performance criterion is optimized. The performance should include the global effects of pricing actions for given demands. Many alternative performance measures or system objective functions can be chosen. A meaningful objective is to minimize the total network cost or to maximize total revenue raised from toll charges. The total network travel cost, denoted as F;, is the sum of travel times and queueing delays experienced by all vehicles:
The total revenue, denoted as Fz, arising from toll charges can be expressed as:
A third objective function considered here is to maximize the ratio, denoted as F3, of the total revenue to total cost: F;=
I:
(10)
mA*
Note that the final results are quite sensitive to the chosen objective function. In general. F, could be adopted if the road pricing is intended to mitigate traffic congestion; F2 could be chosen if the road pricing is used as a means of generating revenue for infrastructure investment; F3 in eqn (IO) can be considered if the road pricing is intended to solve the problems of both congestion control and investment financing simultaneously. Other objective functions that better approximate the decision process, such as maximization of consumers’ surplus, could also be considered. Furthermore, since queues may form at links operating at capacity, an acceptable road pricing policy should consider queue length constraints. The number of queueing vehicles, Q,, queueing delay, d,, and the link exit flow, v,. are related by the following equation:
Q,, = v,d,,, ad. Maximum allowable storage capacity at link a~,4 of Qy queue length constraints expressed as: v,,d, S Qy,
UCA.
(111 (vehicles) can be enforced by (12)
Since queues cause a wasteful loss of time for users, and result in inefficient utilization of the road facility, an optimal road pricing might be required to eliminate queues completely. In this case, Q,“” may be set to be zero. On the other hand, a road network may involve constant returns to scale in practice, and hence a budget constraint should be imposed in determining road tolls so as to optimize a system performance:
324
Hai Yan and William H. K. Lam &
(13)
uuvu 2 G,
where G is a given constant. Note that this constraint should not be imposed when the objective functions F2 or Fj are employed. Finally, the toll charges of all toll links should fit into given lower and upper bounds: Uzin I u, I Uy,
aeA*,
(14)
where ci” is the lower bound of toll charges and might be set to be zero or predetermined values for revenue consideration; I!J~ is the upper bound of toll charges and determined by what users may consider to be reasonable. In summary, the congestion road pricing problem under queueing network equilibrium conditions can be formulated as the following bi-level mathematical program: minimize F(u, v(u), d(u)),
(154
subject to
2
acA*
u,,v, 2 G -
(15c) (154
U,m’“
where v(u) and d(u) are obtained by solving: minimize 2
UCA
(15e)
subject to (15f.l
(1%) v, 5 C,,, aEA f;. 2 0, reR.
UW (15i)
This problem is to find an optimal toll pattern u* such that a system objective function F(u*,v(u*),d(u*)) is an optimum (the objective function may be selected either as F,, F2, or as FJ. Note that the above problem, like any other form of bi-level mathematical programming problems, is intrinsically non-convex, and hence it might be difficult to solve for a global optimum (Friesz et al., 1990; Yang et al., 1994). It should be pointed out that the aforementioned bi-level congestion pricing model assumes a fixed travel demand pattern. However, congestion pricing might have important effects on the demand for travel beyond route choice. Ir this case, the elasticity of travel demand can be explicitly incorporated by employing an elastic-demand network equilibrium model with queues as the lower-level model, instead of a fixed-demand equilibrium model. 3. SENSITIVITY ANALYSIS BASED ALGORITHMS
The key point in solving the global optimal toll charging problem (I 5) lies in how to evaluate the equilibrium link flow, v(u), and equilibrium queueing delay, d(u), which are implicitly defined by the lower-level queueing network equilibrium problem. Recently, Yang et al. (1994) and Yang and Yagar (1994) developed efficient heuristic algorithms for solving bi-level transportation system optimization problems, based on sensitivity analysis results for equilib-
Optimal road tolls
325
rium network flows. These algorithms have been successfully applied to optimal ramp metering in general freeway-arterial corridor systems (Yang & Yagar, 1994) traffic signal control in saturated road networks (Yang & Yagar, 1995) and congested O-D matrix estimation problems (Yang, 1995b). A detailed description of the development of the sensitivity analysis based algorithm is given in Yang and Yagar (1994). Here, we adopt the sensitivity analysis-based algorithm developed by Yang and Yagar (1994) for solving the bi-level congestion road pricing problem. Because the upper-level objective function and constraints are implicit, non-linear functions of decision variable u, local linear approximations using Taylor’s formula are implemented based on the derivatives of the reaction functions with respect to link tolls. The derivative information is obtained by implementing sensitivity analysis for a given solution of the queueing network equilibrium problem. The sensitivity analysis based (SAB) algorithm is outlined below. SAB algorithm: Step 0. Determine an initial set of link toll pattern II’(‘).Set n = 0. Step I. Solve the lower-level queueing network equilibrium problem for given II(“) using the inner penalty function method, and hence get v(“)and d(“). Step 2. Calculate the derivatives, %‘%u and &V&I, using the sensitivity analysis method. Step 3. Formulate local linear approximations of the upper-level objective function and constraints using the derivative information and solve the resulting linear program to obtain an auxiliary solution y. Step 4. Compute u”’+ ‘) = u(“)+ (Y(‘l)(y-uO’)) where (Y’“)is given by: Ly(“’= pl( 1 + !?)Y, where p, y are parameters (p > 0, y 2 1). Step 5. If Itll;+” -- u!“)lI E for all aeA*, then stop where E is a predetermined Otherwise, let n: = n + 1 and return to Step 1.
tolerance.
At Step I, the lower-level queueing network equilibrium problem with explicit capacity constraints is solved by an inner penalty function method (Yang & Yagar, 1994, 1995). After proper transformation of the augmented objective function, the penalty function method gives rise to a sub-problem structure similar to the Frank-Wolfe convex combination algorithm (Sheffi, 1985) and hence can be easily implemented, even for large networks. In the final output, the algorithm generates information necessary for implementing sensitivity analysis, including a complete set of link flow patterns, queueing delays on a subset of links, and a subset of minimum time paths used by the users between each O-D pair. At Step 2, sensitivity analysis is implemented to compute the derivatives of equilibrium link flows and queueing delays with respect to link tolls for a given solution of the queueing network equilibrium problem. The sensitivity analysis method for the network equilibrium problem with capacity constraints has been developed and incorporated into the bi-level traffic control problem (Yang & Yagar, 1994, 1995; Yang, 1995a). At Step 3, the derivative information obtained from sensitivity analysis is used to formulate local linear approximations of the reaction functions and hence the upper-level objective function and constraints. This approximation will result in a linear programming problem, to which the solution would be located at the farthest point in the descent direction of the upper-level objective function. At Step 4, the step size, II(“), is determined a priori, and decreases monotonically with the number of iterations. It should be noted that any standard one-dimensional search method might be adopted, but it will require repeated evaluation of the upper-level objection function along its descent direction. Each of these evaluations involves an equilibrium assignment in order to determine link flow and queueing delay. Thus, the computational requirements of the one-dimensional search may be prohibitively expensive, and will not be adopted here. The above SAB algorithm is applicable when the upper-level problem contains non-linear constraints (15b) and (15~). In some situations, these non-linear constraints might not be
Hai Yan and William H. K. Lam
326
involved, for example, constraint (1%) should be deleted when the upper-Ievel objective function is a revenue maxim~tion. When only the constant fl5d) is included in the upperleve1 problem, the following simple equ~Iib~~ decomposition opt~~tion (EDO) algorithm proposed by Suwansirikul et al. (1987) and Friesz et al. (1990) may be employed. ED0 algorithm: Step 0. Set uA”’= 4(u:in + Uy), ,QO’= tJ:in, u:O’ = Uy and set n = 0. Step 1. Solve the lower-level queueing network equilibrium problem
for given u(“’ using the inner penalty function method, and hence get v(“)and d(“). Step 2. Calculate the derivatives, @‘Y&r and &P%3u, using the sensitivity analysis method. Step 3. Calculate the derivatives, V,A4(u), using sensitivity analysis information and the following chain rule: am au,
8F dvh (---+--
_ x bd
&,
au,
at;
ad,
ad,
ihi<,’ +
__aF au<*’ aeA*7
where M(u) = F(v(u),d(u),u). Step 4. Calculate Z,(”+ ‘) and iJ(” + ‘) according to if aM/&
if aM/&
< 0 3 set L(“+‘) = us”‘, U(n+‘) (10 LI = u(“) ” 1 D
> 0
set T
U(n+‘l = uj,“), L(“+‘) ‘I
”
= L(“) (1
.
Step 5. Compute uln*‘)as UP+‘)= t (U:+” + Lp+‘)), ad*. Step 6. If Iuy+l) -uj;“‘lI E for all ad*, then stop. Otherwise let n: = n + 1 and go to
Step 1.
4. NUMERICAL
EXAMPLES
In this section, we present three examples using hypothetical networks. The first example is presented to illustrate the sensitivity analysis procedure and its applications. The second example examines the traditional marginal-cost pricing theory under ideal conditions with the bi-level road pricing model. The last example is used to investigate the congestion road pricing under realistic constraints. Example I. Sensitivity anaiysis of congestion pricing
This example illustrates the ~nsitivity analysis procedure for a given road toll pattern. The road network, as shown in Fig. 1, consists of seven nodes and 11 links, of which links l-4 are subject to toil charges. The following link cost function is used
Fig. I. Example network I.
Optimal road tolls
321
Table 1. Input data for example network I Link a
to ?
1
2
3
4
5
6
7
8
9
10
II
200 6
2005
200 6
2Ozl
100 4
150 3
150 5
150 IO
200 16
200 15
100 17
t,(v,) = fZ{ 1.0 + 0.15(+}.
(16)
(I
Link free-flow travel time, c, and link capacity, C,, are given in Table 1. It is assumed that there are three O-D pairs (1+7, 2+7 and 3+7) and the demands are fixed to be D,,= 250, D?,= 200 and D,,= 200, respectively. Sensitivity analysis is used to predict changes in equilibrium link flow pattern, queueing delay, and system objective function in response to any small variation in toll charges, Table 2 presents the derivatives of link flows, queueing delays and the system objective functions (total network travel cost, F,, total revenue F2,and their ratio F3)with respect to link tolls at U, = uz = u3 = uq = 1.0 (link tolls are expressed in equivalent travel time units). Note that only derivatives of queueing delays at queued links (links 3, 4 and 5 as indicated in the figure and table) are included in Table 2, while derivatives for other links without queues will be zero. These results show the variations of flows through the road network, queueing delays at queued links, and the variations of system performance functions when the charge at one of the toll links is increased by 1 unit. It should be mentioned that a small change in tolls that are charged at a congested link with strictly positive queue will have no effect on the network link flow pattern because the effect of toll change is cancelled by reduction of the equivalent amount of queueing delay at the same link. This has been demonstrated in Table 2, where all the derivatives of link flows with respect to toll on a queued link are equal to zero, and the derivative of queueing delay at a queued link with respect to the toll on the same link is equal to -1.0. The derivative information has many important implications in both the operational efficiency of congested road networks and the economic appraisal of varied road pricing policies. The derivatives indicate that the link toll charge to which the congested equilibrium flow pattern is the most sensitive, and, therefore, is the link toll that deserves the most consideration to improve the efficiency of the congested road system. The derivative information also gives the “direction” in which the queueing network equilibrium flow pattern should move so that a system optimum could be achieved if alternative road tolls are adopted. It is interesting to see, from Table 2, that the link toll interaction is symmetric. Namely,
Table 2. Derivatives of link flows, queueing delays and objective values with respect lo link tolls at u, = uz = q = u4 = 1.O for example network I Variable
a(‘,/aU, -3 I ,622 28.675 0.000 0.000 0.000 -28.675 31.622 0.000 28.675 -3 1.622 2.948 -0.616 0.050 -0.146 -195.547 146.291 0.013
a(.)fauz 28.675 44.096 0.000 0.000 0.000 44.096 -28.675 0.000 44.096 28.675 15.421 0.558 -0.078 -0.776 145.132 106.43I 0.008
3. m3 0.000
0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
-1.00 0.000 0.000 -199.998 199.998 0.018
mau4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
-1.000 0.000 -199.998 199.998 0.018
Hai Yan and William H. K. Lam
328
av,_- au& 84
for all a, beA*
B
(17)
This means that the marginal effect of one link toll, say u,, on the link flow on any other toll link, say v,,, is equal to the marginal effect of uh on v,. It can be proved that this conclusion holds for the standard network equilibrium problem in general networks. The derivative information also allows one to estimate nearby solutions for any combination of changes in link tolls, once a queueing equilibrium solution has been obtained. Table 3 presents the impacts on the link flows when the link tolls are varied slightly by Su, = Su, = &+= au, = 0.1 with respect to a reference toll charge U, = u2 = U) = u4 = 1.0. The estimation is made using the following linear approximation based on the derivative information in Table 2. v(u) =v(uo) +
$(” - UJ.
(18)
The impacts of the toll adjustments on the queueing delays and system performance functions are also indicated in Table 3. These results could allow traffic engineers and planners to evaluate alternative road tolls, and hence to develop improvement strategies for systems operations. Example 2. First-best roud pricing
The genera1 theory of first-best optimal pricing, i.e. the theory of marginal cost pricing in economics, is well established. In the traffic assignment literature (Beckmann et al., 1956), congestion tolls have been proposed to drive a user equilibrium pattern toward a “system optimum”. Namely, an exact system optimum equilibrium can be achieved if on every link of a network a toll (marginal-cost toll, MCT) is charged, which equals the difference between marginal social and marginal private costs: MCT
= ‘I
4vax t<,(v,,)) _ VU
t,,(v,)
=
v, ___ dt,(v,) for all aeAA.
(19)
V”
This idea1 MCT provides an exact optimal solution under perfect charging conditions, and hence could be used to verify our mode1 and algorithm. Traditional marginal cost pricing theory uses the network equilibrium model that does not consider capacity constraint (15h) and hence does not consider queues explicitly. For the purpose of comparison, the standard network equilibrium model is used here as the lowerlevel problem, and the non-linear constraints (15b) and (15~) are deleted in the upper-level problem. Table 3. Estimated and exact solutions for perturbed link tolls at ut = u2 = uj = uq = 1.0 for example network I Perturbed with &q = &I? = 6q = &44= 0.10 Solution variable 1’
“2
“3 v4 v5 v6 v-i V8 v9 “IO VII d3 d4 d, FI F2 F3
Unperturbed solution 149.238 121.852 199.998 199.998 99.998 78.145 50.760 100.002 21.857 149.242 78.908 2.778 2.985 2.685 12.062.850 67 1.086 0.056
Exact 148.693 120.797 199.997 199.997 99.997 79.198 51.305 lOQ.003 20.804 148.696 80.509 2.198 2.247 2.312 Il,766.800 736.433 0.063
Estimated 148.944 120.310 199.998 199.998 99.998 79.687 51.055 100.002 20.315 148.947 80.745 2.672 2.882 2.592 12,017.810 736.358 0.061
Optimal road tolls
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Fig. 2. Example network 2
Figure 2 shows the example network consisting of six nodes and seven links, of which link 1 and link 2 are toll links. The same link cost function (16) is used with link free-flow travel time and link capacity provided in Table 4. It is assumed that there are only two O-D pairs (143 and 2+4) and the demands are fixed to be D,, = D14 = 30.0. The system-optimum flow pattern and the corresponding marginal-cost tolls can be obtained using the network equilibrium assignment with a marginal cost curve, instead of an average cost function. The results obtained for example network 2 are shown in Table 5. Now consider the solutions produced by the proposed bi-level pricing model when all links are subject to toll charges. The ED0 algorithm is used with the lower and upper bounds: 0.0 < U, 5 5.0 for a = 1,2 and 0.0 I U, I 2.0 for u = 3-7. Note that the upper bounds of link tolls have been set higher than the corresponding MCTs so that at least one optimal solution leading to a system optimum is included in the feasible region. The convergence was achieved in 13 iterations with a convergence tolerance E = 0.001. From Table 5, it can be seen that the optimal toll solution obtained by the ED0 algorithm is different from the marginal-cost tolls, while it gives rise to the same total network travel cost F, = 628.60. This demonstrated that there is usually more than one link toll pattern that leads to the system optimum. In fact, different optimal solutions have been found when different upper bounds of link tolls are employed. This observation indicates that there is a great degree of flexibility in a road network for setting link tolls to satisfy various kinds of constraints, while at the same time leading to a system-optimum equilibrium. For instance, a link toll pattern may be determined subject to a given budget constraint, yet still lead to a systemoptimum equilibrium. Example 3. Second-best road pricing
The first-best road pricing assumed that a perfect charging system could be followed precisely and a system-optimal equilibrium established. However, for whatever the reasons. Table 4. Input data for example network 2 Link u
I
2
3
4
5
6
7
20.0 8.0
9.0 20.0
2.0 20.0
40.0 6.0
3.0 20.0
3.0 25.0
4.0 25.0
to &
Table 5. System-optimal flow pattern, marginal-cost toll and optimal toll solution for example network 2 Link number System optimum flow Marginal-cost toll Optimal toll solution
I
2
3
4
5
6
7
17.916 3.091 3.820
18.486 3.941 4.265
12.084 0.160 0.472
23.598 0.436 0.476
1I.514 0.198 0.294
12.084 0.098 0.472
II.514 0.108 0.294
Total network cost in system optimum = 628.60 Total network cost corresponding to optimal link toll solution = 628.60 Total network cost in user-equilibrium without toll charge = 720.09
Hai Yan and William H. K. Lam
330
the rule of setting marginal-cost toll may not be applied. For example, the marginal cost pricing rule cannot be applied when only a subset of liiks (e.g. harbor tunnels in Hong Kong) in the network is subject to toll charges, and/or the network involves a budget constraint, as considered in our b&level pricing model. Now consider the same example network 2. Suppose that only link 1 and link 2 are subject to toll charges. For the moment, we omit the queue length constraints (15b) and budget constraint (15c), and assume the bounds: 0.0 I u, < 5.0, 0.0 < z.+5. 5.0, the ED0 algorithm is applied to solve the b&level pricing model with alternative upper-level objective functions. From the results presented in Table 6, it is easily seen that different upper-level objective function would generally result in different road toll pattern. It can also be observed that the solution for minimizing the total network travel cost F, is obviously a global optimum since it associates the objective function with the same value as in system optimum (& = 628.60). This indicates that a system optima equilib~um can be (though is not necessarily) achieved if we impose proper tolls, even only on a subset of network links. Now we compare this result with the situation where the marginal-cost pricing rule is applied, but only for link 1 and link 2. This rule yields the following results: u, = 2.577, u, = 3.390, F = 629.28. Although this partial marginal-cost pricing rule results in a reduction in total network cost (compared to the total cost 720.09 in user-equilibrium without toll charges as shown in Table 5), the resulting toll pattern does not represent the most desirable one (u, = 2.399, u2 = 3.200 shown in Table S), i.e. one that leads to a system optimum. So far, we have not yet tested the model and algorithm with queue length constraint (1 Sb). This constraint may become active, or vehicle queues may occur when there is a heavy demand during peak periods. Therefore, it might be necessary to impose queue constraints in optimizing a system performance function. In this case, the ED0 algo~thm is no longer applicable, and instead the SAB algorithm should be applied. Using a higher traffic demand, & = &, = 35.0, which, when assigned to the non-toll example network 2, produces the following vehicle queues at links 1 and 2: Q, = v, x d, = 19.990 X 1.999 = 39.96 (veh) QZ = v? X d7 = 19.993 X 2.691 = 53.80 (veh). Here, we assume that the toll bounds 0.0 < u, < 5.0, 0.0 5 Ed?I 5.0 are imposed. and no vehicle queues are allowed in optimizing system objectives. Namely, maximum queue length is set to be: Qy = 0.0, which means that the link tolls are dete~ined so as to at least eliminate queue. Table 6. Solutions with alternative upper-level objective functions for example network 2
For minimizing F, For maximizing FZ For maximizing F3
Link toll UI
Link toll U?
2.399 3.164 3.125
3.200 4.063 3.828
Objective value 628.60 107.15 0.168
Main iteration number 13 I3 I3
Note: The EJDO algorithm is applied with E = 0.001 and initial values ,i”’ = ut”’ = 2.5.
Table 7. Solutions with alternative objective functions and queue length constraint for example network 2
For minimizing F, For maximizing F2 For maximizing Fj
Link toll Ul
Link toll
2.398 3.871 2.401
3.316 4.853 3.342
u2
Objective value 766.91 136.95 0.146
Main iteration number 49 I6 51
Note: The SAB al orithm is applied with fixed step size a = l/(1 + n) and convergence criterion E = 0.05. The initial values are u!) - @ - 2.5, and maximum queue length Qy = 0.0 for all aA.
Optimal road tolls
331
The solutions for alternative objective functions with explicit queue constraint (c = 0.0) are shown in Table 7. It can be seen that a large number of iterations are required to achieve the convergence when using the SAB algorithm. Note that, although not reported in Table 7, the queue length constraint (15b) is active in the final convergent solution when maximizing F2 and F3. 5. CONCLUSIONS
Some developments in model formulations and solution procedures for the congestion road pricing problem under queueing network equilibrium conditions have been presented. The methodology proposed in this paper is characteristic in two aspects. Firstly, the bi-level model considers a subset of links subject to toll charges, and the toll charges are subject to given lower and upper bounds. This characteristic is a significant departure from the conventional marginal-cost pricing theory that imposes prices on every link exactly equal to the difference between marginal social cost and marginal private cost. Secondly, the queueing network equilibrium model predicts queueing delay explicitly, and hence makes it easier to implement queue management in congested urban areas by implementing different toll pricing policies. Furthermore, a sensitivity analysis has been implemented to obtain the derivatives of link flows and queueing delays with respect to link tolls, and hence has produced the “direction” in which the queueing network equilibrium pattern can move if the toll pattern is changed. This information has been used efficiently to determine optimal road tolls such that total network travel time is minimized or total toll revenue is maximized. While the examples are very simple, our numerical results reveal some interesting properties of congestion road pricing and demonstrate that the sensitivity analysis based-algorithms perform well in finding the optimal road tolls. The bi-level pricing model is being applied to coordination of tunnel toll patterns in the Hong Kong road network, and the numerical results will be reported in a future paper. Finally, we should acknowledge that there are still many theoretical and practical issues to be discussed. We have only examined the space dimension of congestion road pricing. As pointed out in the introduction, time-dependent road pricing is a complementary and equally critical topic for further research. This would involve modeling the combination of choice of departure time and choice of route under time-dependent toll charges. AcknoM?/edgemenr.s-The authors wish to thank two anonymous referees for their helpful comments and suggestions for improvement, This research has been supported with a UGC Research Infrastructure Grant (R194/95.EGOl) from The Hong Kong University of Science and Technology.
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