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Network Pricing Optimization in Multi-user and Multimodal Context with Elastic Demand Giuseppe Bellei, Guido Gentile and Natale Papola Università degli Studi di Roma “La Sapienza” ABSTRACT Network Pricing Optimization is formulated first as a Network Design Problem where the design variables are tolls, the objective function is the Social Surplus and the equilibrium constraint is any current multi-user multimodal stochastic traffic assignment model with elastic demand up to trip generation and asymmetric arc cost function Jacobian. Network Pricing Optimization is then formulated also as an Efficient Allocation Problem, where an optimal flow pattern, the System Optimum, is sought and tolls are consistently determined. Necessary and sufficient conditions for the solutions to both problems are stated, showing the validity of the marginal pricing principle in the context considered. Key words: toll optimization, Network Design Problem, marginal pricing, System Optimum, System Equilibrium, Social Surplus 1

INTRODUCTION

Dealing with the Network Design Problem (NDP) involves seeking a transportation network supply configuration and demand flow pattern which maximize a given objective function of the social type, while satisfying the equilibrium constraint. In this paper the equilibrium is formalized as a fixed point problem using, on the demand side, hierarchical choice models based on random utility theory, and on the supply side, congested networks with asymmetric arc cost function Jacobian in a multi-user and multimodal context. On the basis of discrete choice analysis, it is possible to derive the travel demand model within the framework of the microeconomic analysis where users may be considered consumers of trips – just as they are consumers of other goods – who make travel choices by optimizing their own individual utility. This is referred to as the trip consumer approach (Oppenheim, 1995). The NDP then consists in achieving the optimization of a function of the individual utilities by measuring the social welfare

1

subject to the optimization of these same individual utilities, taken singly. Any NDP should, in principle, therefore be formulated as a bilevel optimization problem where the individual utility plays a decisive role at both levels, so that once the neoclassical microeconomic theory is accepted, this constitutes the framework in which the internal consistency of the problem should be analyzed. The Network Pricing Optimization (NPO) problem is a special case of NDP where the tolls are assumed to constitute a complete and unconstrained set of design variables. This means that if, as usual, the design variables are arc tolls, then the problem requires the possibility of charging each arc of the network any real valued toll. Pricing is one of the best tools for improving the efficiency of highly congested transportation networks. The methodological contributions concerning this topic can be classified into two groups. The first is founded on the welfare-economics principle of marginal pricing whereby, when external effects are present, an optimal equilibrium is achievable from a social point of view by charging each decision-maker the difference between marginal social costs and average individual costs. In transportation, this idea was first applied to the traffic assignment problem by Beckmann (1965). Dafermos and Sparrow (1971) proved the optimality of marginal pricing with respect to total user costs in a mono-user monomodal deterministic context with fixed demand and separable arc cost functions. Dafermos (1973) extended this result to a multi-user context, while Smith (1979) generalized it to the case of elastic demand and asymmetric arc cost function Jacobian. In these papers the NPO is formulated as an Efficient Allocation Problem (EAP), actually seeking an optimal flow pattern generally referred to as System Optimum (SO). Tolls are then determined accordingly. So far, the stochastic case has been addressed only with reference to the Logit choice model. In Delle Site, Filippi and Papola (1997) a multi-user multimodal equilibrium with elastic demand is determined by employing as performance functions the marginal social costs instead of the average individual costs. Yang (1999) proved the optimality of marginal pricing with reference to an SO formulation based on Fisk’s integral (1980), in a mono-user monomodal context with rigid demand and separable arc cost functions. In the second group of contributions, the assumption on the feasible toll set characterizing the NPO problem is relaxed, thus formulating a more general NDP, as in Yang (1997), Ferrari (1999) and 2

Clune, Smith and Xiang (1999), where sensitivity analysis, polynomial approximation of the objective function, and specific bilevel optimization techniques for variational inequality constrained problems, respectively, are applied in the solution algorithm. In this paper, referring specifically to the NPO problem, the validity of the marginal pricing principle is extended to the case where the equilibrium constraint is any current multi-user multimodal stochastic traffic assignment model with elastic demand up to trip generation and asymmetric arc cost function Jacobian, thus generalizing previous results obtained with reference to deterministic or Logit formulations. It will be seen that a sufficient condition for achieving this extension of principle is to assume the Social Surplus as the objective function of the NPO, expressing the social welfare in monetary terms, as is consistent with the microeconomic consumer theory. In section 2 the NPO problem is introduced and an expression of the Social Surplus is determined. The demand and supply models utilized in the formulation of the equilibrium constraint are presented and certain properties useful for the characterization of the solutions to the NPO are recalled. Finally, User Equilibrium (UE) and System Equilibrium (SE) are formalized as fixed point problems, and sufficient conditions for the existence and uniqueness of the solution are stated on the basis of the results obtained in Cantarella (1997). In section 3 the NPO is formulated as an NDP in terms of both travel alternative tolls and arc tolls. Necessary and sufficient conditions for the solutions to both problems are stated, and some insights into their uniqueness are presented. In section 4 the NPO is formulated as an EAP in terms of travel alternative flows. Necessary and sufficient conditions for the solutions to this problem are stated and the existence of a solution to the NPO problem is proved. 2

MODEL FORMULATION

The NPO problem can be expressed formally in terms of arc variables, as follows: max f, t s( f, t)

s. to: f ∈ f UE(t)

(1)

where s( f, t) is a suitably defined social welfare function, f is the arc flow vector, t is the arc toll vector and f UE(t) is the map of the equilibrium arc flow vectors. 3

The structure and complexity of the problem depend on the form of the objective function and on how demand and supply are modeled to formulate the equilibrium constraint. 2.1

The objective function By adopting the trip consumer approach, the objective function is obtained here, consistently with

the utilitarian social welfare function (Luenberger, 1995), by summing up the monetizations of all the effects produced by the supply modification. As we know, whenever the supply modification affects generic users only through a generalized cost modification of their travel alternatives, the Equivalent Variation (EV) yields a rigorous measure in monetary terms of the effects on the individual utility (specifically, the EV is the budget variation that determines the same variation of the indirect utility as that caused by the generalized costs modification considered, evaluated using as reference the final indirect utility – Varian, 1992). Jara-Diaz and Farah (1988), with reference to a Logit demand model, shown how, assuming suitable hypotheses, the EV can be approximated to the User Surplus, concluding that “the log-sum formula is a fairly well-founded form of valuating” the effects connected with the network modification. Bellei, Gentile and Papola (2000a) verify how such a result holds with reference to any current choice model based on random utility theory, and prove that the EV of the generic user i∈I, where I is the set of users, is obtained by dividing the variation of their so-called satisfaction Wi (Sheffi, 1985) by their marginal utility of income γ i . On this basis, the objective function, referred to here as Social Surplus, is given by: S = ∑ i∈I Wi /γ i -E +T ,

(2)

where E and T are respectively the monetary value of transport externalities (e.g. environmental costs and accident costs), and the toll revenue. The status quo terms of the Social Surplus are not included in (2) because they do not influence the optimization process. Often in previous works on toll optimization the objective function to be minimized represents the Social Cost (SC), defined here as: SC = C +E -T = Ĉ +E ,

(3)

where C and Ĉ are the total user costs, respectively including and not including tolls. The Social Cost 4

is used in subsection 2.4 when defining the System Equilibrium. The Social Surplus differs from the Social Cost because, when adopting probabilistic choice models, individual travel disutilities are given by the opposite of the satisfaction instead of by the generalized cost of the chosen travel alternative. It is to be noted that if the not-to-travel alternative is available to users, their satisfaction also takes into account the benefits related to the latent demand effects. 2.2

The demand model In modelling travel demand we follow the behavioural approach based on random utility theory,

where it is assumed that users are rational decision-makers who, when making their travel choice: a) consider a positive finite number of mutually exclusive travel alternatives constituting their choice set; b) associate with each travel alternative of their choice set a perceived utility, not known with certainty and thus regarded by the analyst as a random variable; and c) select a maximum utility travel alternative. We also assume that the travel choice process can be broken down into a sequence of mobility choices (e.g. to travel or not to travel, by which mode, to which destination and following which route) represented by a choice tree (see, for instance: Ben Akiva and Lerman, 1985; Oppenheim, 1995; Cascetta, 2001). A travel alternative is then a path on the user choice tree where the choice is specified at each level: trip generation, distribution, modal split and assignment. In particular, each travel alternative (except for the not-to-travel alternative, when present) is associated with a single route connecting the origin of the trip to its specific destination on its specific modal network. This representation has its counterpart on the supply side, as shown in subsection 2.3. In this paper the hierarchic structure of the choice model (i.e. the correlations between the travel alternatives available to users) is implicit in the form of the joint probability density function of the travel alternative perceived utilities. The perceived utility of the generic travel alternative is clearly the sum of the utilities associated with each level, along the corresponding path of the choice tree. A choice model based on the concept of travel alternative can then both support a fully elastic travel demand model and at the same time be formalized in a very simple way, perfectly similar to that 5

of the fixed demand case. The users are grouped into classes. All individuals belonging to a same class are assumed to be identical with respect to any characteristics influencing travel behaviour and externalities. More specifically, the N u > 0 users constituting class u∈U, where U is the set of classes, share: a) the same set of individual attributes characterizing the user as a trip consumer (e.g. age, marginal utility of income, value of time, and purpose of trip); b) the same set of attributes specifying the production of trip externalities, such as congestion and pollution (e.g. vehicle type and occupancy rate); and c) the same choice set J(u) of travel alternatives. Our definition of class combines the common specification of user and vehicle class with the spatial and modal identification of the trip. This enables us to associate simplicity of the notation and generality of the formalization when dealing with elastic demand in a multi-user and multimodal context. The perceived utility Uju of the generic travel alternative j∈J(u), u∈U, is given by the sum of a finite systematic utility term Vju, and a zero mean random residual εju. The family of choice models considered in this paper is defined by the following properties: a) the random residuals have non-zero finite variance and their joint probability density function is independent of the systematic utilities, continuous, and strictly positive (probabilistic, additive, continuous and strictly-positive choice model); and b) the systematic utility of the generic travel alternative is linearly decreasing with respect to the generalized cost of the associated route. With reference to the generic travel alternative j∈J(u), u∈U, hypothesis b) is formally expressed as: Vju = Xju -γ u ⋅Cju ,

(4)

where the scalar γ u > 0 is the marginal utility of income for class u, while Xju is a constant utility term independent of congestion, and Cju is the generalized cost of j. In the following we assume that Cju is equal to the generalized cost of the route associated with j and is equal to zero for the not-to-travel alternative. The choice probability Pju of the generic travel alternative j∈J(u), u∈U, is by definition the probability of j being a maximum utility travel alternative:

6

Pju = Pr[Uju = max k∈J(u) Uku] = Pr[Vju+εju ≥ Vku+εku ∀k∈J(u)]

(5)

The satisfaction Wu of class u∈U is by definition the mean value of the maximum perceived utility: Wu = E[max k∈J(u) Vju+εju]

(6)

The flow Fju of travel alternative j∈J(u), u∈U, is given by multiplying the number of users in the class by the choice probability of the travel alternative: Fju = N u ⋅Pju ,

(7)

while for each class u∈U we have, by definition: ∑ j∈J(u) Fju = N u

(8)

Let W = (W1, … , Wu, … , W|U|)T be the (|U| × 1) vector of the satisfactions and let N, N -1, Γ and

Γ -1 be the vectors having the same structure as W, whose generic elements are respectively Nu , 1/N u , γ u and 1/γ u . Furthermore, let V = (V1T, … , VuT, … , V|U|T)T be the (n × 1) vector of the systematic utilities, where n = ∑ u∈U |J(u)| , whose generic component is Vu = (V1u, … , Vju, … , V|J(u)|u)T, and let X, C, P and F be the vectors having the same structure as V, whose generic elements are respectively Xju, Cju, Pju and Fju. Finally, let ∆ be the (n × |U|) alternative-class incidence matrix, whose generic element

δ ju is equal to 1 if j∈J(u), and 0 otherwise, thus generalizing the path-OD incidence matrix. In compact form, equations (4), (5), (6), (7) and (8) can be expressed, in that order, as follows: P = P(V) ,

(9)

W = W(V) ,

(10)

V = X -diag(∆⋅Γ )⋅C ,

(11)

F = diag(∆⋅N )⋅P ,

(12)

∆T⋅F = N

(13)

The demand function is obtained from (12) using (9) and (11): F = diag(∆⋅N )⋅P(X-diag(∆⋅Γ )⋅C ) = F(C )

(14)

The satisfaction function is obtained from (10) using (11): W = W(X-diag(∆⋅Γ )⋅C ) = Ŵ(C )

(15)

We now recall the properties of both the demand function and the satisfaction function which will

7

be utilized when characterizing the NPO solutions. Let H = {h∈ℜn: h = ∆⋅h’, h’∈ℜ|U|}. Having assumed that the choice model is probabilistic, additive, and continuous, and that the linearity assumption (11) holds, then the following properties also hold (for details, see Cantarella, 1997): F(C) is a C1 function ranging in a compact, convex and not empty set SF = {F∈ℜn: ∆T⋅F = N, F ≥ 0n} , F(C+h) = F(C ) ∀h∈H ,

(16)

Ŵ(C +h) = Ŵ(C ) -diag(Γ )⋅h’ ∀h∈H, where h’∈ℜ|U| : ∆⋅h’ = h ,

(17)

Ŵ(C) is a C2 convex function , ∇Ŵ(C ) = -diag(∆⋅Γ ) ⋅diag(∆⋅N -1) ⋅diag(F(C )) ⋅∆ ,

(18)

∇2 Ŵu(Cu) = -(γ u /N u ) ⋅∇Fu(Cu)

(19)

u = 1, … , |U|

As the choice model is assumed to be also strictly positive, it can easily be proved that (16) holds also as a necessary condition: F(C+h) = F(C) ⇔ h∈H

(20)

Since the satisfaction Ŵ(C ) is a C2 convex function, on the basis of (18) the demand function F(C ) can easily be proved to be monotone decreasing, while by (19) the Jacobian of each class-specific demand function ∇Fu(Cu) is negative semidefinite. At the same time, owing to the additivity of the choice model, the satisfaction is not strictly convex, so the demand function is not strictly monotone and its class-specific Jacobians are not negative definite. However, if on the one hand each class-specific demand function has one degree of freedom due to the additivity of the choice model, on the other it must satisfy the consistency constraint (8). On this basis, in the case of strictly positive choice models, the strict convexity of each class-specific satisfaction function and the strict monotonicity of each class-specific demand function can be established with reference to a subspace having dimension equal to the cardinality of the choice set minus one. Moreover, one row and one column can be removed from the Jacobian of each classspecific demand function, thus obtaining a “diminished” matrix definite positive. The formalization of these properties requires the partitioning of vectors C and F. To this end, we introduce a convenient notation. 8

Let j °∈J(u) be a travel alternative of the choice set of class u∈U. Let Iu° be the row vector of the identity matrix I|J(u)| corresponding to travel alternative j ° and let Iu- be the matrix obtained from I|J(u)| by removing the row vector Iu° . Finally, let I - and I ° be the block diagonal matrices with generic u-th block, u∈U, equal to Iu- and Iu°, respectively. Then, once a single travel alternative has been identified for each class, vectors C and F can be partitioned into the sub-vectors C - = I - ⋅C , C ° = I ° ⋅C and F - = I - ⋅F , F ° = I ° ⋅F, respectively. Therefore: C = I - T ⋅C - +I ° T ⋅C ° ,

(21)

F = I - T ⋅F - +I ° T ⋅F °

(22)

Using this notation, the strict monotonicity of the demand function can be expressed as follows: 1

2

1

2

1

2

[I -⋅F(I - T⋅C - +I ° T⋅C °) -I -⋅F(I - T⋅C - +I ° T⋅C °)] T⋅(C - -C - ) < 0 ∀ C - ≠ C - ∈ℜn-|U| , ∀C °∈ℜ|U| , (23) while each Ŵu(Cu) , u∈U, is strictly convex with respect to Cu-. These results will be used when characterizing the solution to the NPO problem. Finally, after grouping the users into classes, the first term of (2) becomes: ∑ i∈I Wi /γ i = [diag(N) ⋅diag(Γ -1) ⋅Ŵ(C)] T⋅1|U|

(24)

It will be utilized in this form when formalizing the NPO problem in section 3. 2.3

The supply model The multimodal network of infrastructures and services which constitutes the transport supply is

modeled here through an oriented hypergraph G = (N, A), where N is the set of nodes, each node representing a spatial location and possibly a state of the trip (e.g. arriving at a road intersection, start waiting at a transit stop, and end waiting at a transit stop), and A is the set of arcs, each arc representing a specific phase of the trip (e.g. driving throughout a road link, and waiting for a line vehicle at a transit stop). The generic arc a∈A is identified by its tail TL(a)∈N and by its head HD(a)⊆N: a = (TL(a), HD(a)). The generic arc is defined here as a one-to-many relationship between nodes. However, in practice all the arcs are ordinary one-to-one relationships, except for the transit waiting arcs, where the tail is a stop node (e.g. a bus stop) and the head is a set of line nodes (Nguyen, Pallottino and Gendreau, 1998).

9

Let M be the set of modes. The modal network of the generic mode m∈M is represented by means of a sub-hypergraph of G denoted Gm = (Nm, Am), Nm ⊆ N, Am ⊆ A. A hyperpath of the generic modal hypergraph Gm , m∈M, with initial node o∈Nm and final node d∈Nm, is a minimal – each node has at the most one exiting arc – and acyclic hypergraph r = (N(r), A(r)) such that A(r)⊆Am, N(r) = {o}∪[∪a∈A(r) HD(a)] and each i∈N(r) is connected to d on r. In this framework, the route, if any, associated with a given travel alternative – no route is associated with the not-to-travel alternative – is represented in general through a hyperpath connecting on its specific modal hypergraph the origin of the trip to its specific destination. Each modal hypergraph Gm , m∈M, is assumed to be strongly connected (there exists at least one hyperpath joining each node to any other node). The user classes are grouped into categories, assumed to be homogeneous with respect to any characteristic influencing the supply side of the equilibrium problem. More specifically, the classes constituting category y∈Y, where Y is the set of categories, share: a) the same set of individual attributes characterizing users as a trip consumer; and b) the same set of attributes specifying the externalities of the trip, but unlike the user classes they do not share the same choice set. With reference to the users of category y∈Y on mode m∈M, let ca ym and fa ym denote, respectively, the arc cost and the arc flow on the generic arc a∈Am . The following relations hold: Cju = ∑ y∈Y, m∈M, a∈Am ca ym⋅πaju ym , in compact form: C = Π T⋅c ,

(25)

fa ym = ∑ u∈U, j∈J(u) Fju⋅πaju ym , in compact form: f = Π ⋅F ,

(26)

where: πaju ym is the arc-alternative probability that the arc a is utilized by users belonging to class u∈U as an element of the hyperpath, if any, associated with the travel alternative j∈J(u), if m is the mode of j and y is the category of u, otherwise πaju ym is equal to zero; Π is the (ν × n) matrix, with

ν = ∑ m∈M |Y|⋅|Am|, whose elements are the πaju ym; c = (c11T, … , cy1T, … , c|Y|1T, … , c1mT, … , cymT, … , c|Y|mT, … , c1|M|T, … , cy|M|T, … , c|Y||M|T)T is the (ν × 1) vector whose generic component is the (|Am| × 1) vector cym = (c1 ym , … , ca ym , … , c|Am| ym)T; f has the same structure as c. The main advantage of using this hypergraph-based network representation is the possibility of

10

expressing in linear form the travel alternative generalized costs in terms of the arc costs through (25) and the arc flows in terms of the travel alternative flows through (26), also in the case of adaptive choices. It is to be noted that where there is no need to represent adaptive choices, Π becomes a classical arc-path incidence matrix. The congestion phenomenon is represented through the arc performance function ĉ( f ), defined for non-negative arc flows, which is introduced here together with the arc cost function c( f, t) in order to separate the arc toll vector t from the arc performance vector ĉ, i.e.: c = c( f , t) = ĉ( f ) +t = ĉ + t ,

(27)

where vectors t and ĉ both have the same structure as c. As (27) is simply a formal expression, it enables one, in principle, to model the congestion in a multi-user multimodal context in any desired way. A specification of the arc performance model (27) representing the main congestion phenomena affecting transit performance (the interaction between private cars and transit line vehicles using the same road link and the various effects of line capacity) is presented in Bellei, Gentile and Papola (2000b). Using (26) and (27), (25) becomes: C = Π T⋅ĉ(Π ⋅F ) +Π T⋅t = Ĉ(F) +Π T⋅t = C(F, t)

(28)

In the following, C(F, t) is referred to as supply function. In section 3, as design variable, besides vector t, we also deal with a travel alternative toll vector T, in which case (28) becomes: C = Ĉ(F) +T = C(F, T )

(29)

Finally, with reference to the last two terms on the right hand side of (2), the monetary value of transport externalities is expressed through the following function: E = E( f ) ,

(30)

while the toll revenue is simply given as follows: T = t T⋅f

or

T = T T⋅F ,

(31)

depending on which design variable is considered, t or T, respectively. In the following, all the functions introduced in this subsection are assumed to be C1.

11

2.4

User Equilibrium and System Equilibrium The UE and the SE are formalized here as fixed point problems. On this basis, the relationship

among UE, tolls and SE is analyzed. Existence and uniqueness conditions are also stated. By combining (26), (14) and (25), we obtain the network loading map, which yields the arc flows as a function of the arc costs: f(c) = Π ⋅F(Π T⋅c)

(32)

Because the demand function is C1, the same holds for the network loading map f(c), which ranges in Sf = { f ∈ℜν: f = Π ⋅F, F∈SF}. This set is compact, convex and non-empty since this is true also for SF . For a given value of the arc tolls t, a UE flow and cost pattern is determined by solving one of the following fixed point problems: f UE = f[c( f UE, t)] ,

(33)

cUE = c[f(cUE), t] ,

(34)

F UE = F[C(F UE, t)] ,

(35)

C UE = C[F(C UE), t] ,

(36)

which are all equivalent, as F(C), and consequently also f(c), are here point-to-point maps. We now introduce the concept of marginal social cost, necessary for the definition of the SE. The Social Cost (3) can be expressed in terms of arc flows: SC = ĉ( f )T⋅f +E( f ) = sc( f ) ,

(37)

or in terms of travel alternative flows: SC = Ĉ(F)T⋅F +E(Π⋅F) = SC(F)

(38)

The gradient of the Social Cost can be obtained from (37) with respect to f , yielding: msc = ∇f sc( f ) = ĉ( f ) +∇f ĉ( f )⋅f +∇f E( f ) = msc( f ) ,

(39)

or from (38) with respect to F , yielding: MSC = ∇F SC(F) = Ĉ(F) +∇F Ĉ(F)⋅F +∇F E(Π⋅F) = MSC(F)

(40)

On the basis of both (28) and (26), the following relation holds: MSC(F) = Π T⋅msc(Π⋅F)

(41)

An SE flow and cost pattern is determined by replacing the cost functions c( f, t) and C(F, t) in 12

(33)-(36) respectively with the corresponding marginal social cost functions msc( f ) and MSC(F), thus obtaining the following equivalent fixed point problems: f SE = f[msc( f SE)] ,

(42)

c SE = msc[f(c SE)] ,

(43)

F SE = F[MSC(F SE)] ,

(44)

C SE = MSC[F(C SE)]

(45)

Let F

SE

and C

SE

be the set of solutions to problems (44) and (45), respectively. Since these two

problems are equivalent, we have: C SE∈C SE ⇒ F(C SE)∈F SE ,

(46)

F SE∈F SE ⇒ MSC(F SE)∈C SE

(47)

We are now in a position to establish which travel alternative tolls are to be charged in order to obtain an SE from a UE. Using (29), problem (35) becomes: F UE = F[Ĉ(F UE) +T ]

(48)

Proposition 1. The UE problem (48) has a solution at a F SE∈F SE if and only if T = T SE +h , with T SE = MSC(F SE) -Ĉ(F SE) and h∈H . Proof. Let F SE∈F SE and h = T -MSC(F SE) +Ĉ(F SE). Problem (48) becomes: F UE = F[Ĉ(F UE) +MSC(F SE) -Ĉ(F SE) +h] Since on the basis of (16) we have F[MSC(F

(49) SE

) +h] = F[MSC(F

SE

)] ⇔ h∈H, the fixed point

problem (49) has a solution at F SE if and only if h∈H . The result follows.  It can easily be proved that theorems 1 and 2 in Cantarella (1997), which state sufficient conditions for the existence and uniqueness, respectively, of a multi-user and multimodal UE with elastic demand and asymmetric arc cost function Jacobian, hold also with reference to the hypergraph-based formalization introduced here. Since the SE is, by definition, a particular UE where the arc cost function is given by the arc marginal social cost function, the existence of an SE pattern is assured, on the basis of (39), by the assumption that all the functions defining the supply model in section 2.3 are C1. Moreover, if the arc marginal social cost function msc( f ) is monotone non-decreasing, then the SE is unique. 13

In order to establish the monotonicity of msc( f ) we verify whether its Jacobian is semidefinite. By differentiating (39), we have: ∇msc( f ) = ∇ĉ( f ) +∇ĉ( f )T +∑ y∈Y, m∈M, a∈Am fa ym ⋅∇2 ĉa ym( f ) +∇2E( f ) , showing that if the Jacobian of ĉ( f ) is positive semidefinite and each of its components, as well as E( f ), are convex C2 functions (i.e. have a positive semidefinite Hessian), then ∇msc( f ) is positive semidefinite. 3

THE NETWORK PRICING OPTIMIZATION PROBLEM

The NPO problem (1) is formalized here with reference to the case where the objective function is the Social Surplus and the equilibrium constraint is expressed by means of the fixed point formulation developed in subsection 2.4. Using (24), (25), (27), (30), and (31) to specify the Social Surplus, and using (33) with (27) to express the equilibrium constraint, we obtain the NPO in terms of arc tolls: max f, t S( f, t) = [diag(N ) ⋅diag(Γ -1) ⋅Ŵ(Π T⋅ĉ( f ) +Π T⋅t)] T⋅1|U| -E( f ) +t T⋅f

(50)

s. to: f = f[ĉ( f ) +t] Using (24), (29), (30), (26), and (31) to specify the Social Surplus, and (36) with (29) to express the equilibrium constraint, we obtain the NPO in terms of travel alternative tolls: max F, T S(F, T ) = [diag(N ) ⋅diag(Γ -1) ⋅Ŵ(Ĉ(F ) +T )] T⋅1|U| -E(Π⋅F ) +T T⋅F

(51)

s. to: F = F[Ĉ(F ) +T] 3.1

The NPO problem in terms of travel alternative tolls The NPO problem (51) can be transformed into an unconstrained optimization problem in terms of

travel alternative costs. To this end, on the basis of (29), we substitute for T the expression C -Ĉ(F), so that the demand function defines the feasible set. Then, substituting function F(C ) for vector F in the objective function and using (38), problem (51) becomes: max C S(C ) = [diag(N ) ⋅diag(Γ -1) ⋅Ŵ(C )] T⋅1|U| +C T⋅F(C ) -SC[F(C )]

(52)

Proposition 2. With reference to the Social Surplus function in (52), the following relation holds: S(C +h) = S(C ) ∀h∈H . 14

Proof. For any h∈H , using (17) and (16), from (52) we have: S(C +h) = [diag(N ) ⋅diag(Γ -1) ⋅(Ŵ(C ) -diag(Γ )⋅h’)] T⋅1|U| +(C +h) T⋅F(C ) -SC[F(C )] , where h’∈ℜ|U| : ∆⋅h’ = h . Since [diag(N ) ⋅h’] T⋅1|U| = hT⋅ F(C ), the result follows.  In section 4 it will be proved that problem (51) has a solution, which implies the existence of a solution to problem (52). We now characterize the solutions to this problem, given the existence. Proposition 3. (necessary and sufficient conditions) For a travel alternative generalized cost vector C to solve problem (52) it is necessary that C = C SE +h , where C SE∈C SE and h∈H. If the SE is unique, then the condition is also sufficient. Proof. As problem (52) is unconstrained and its objective function is C1, its solutions satisfy the necessary first order conditions: ∇S(C ) = 0n

(53)

Using (18) and (40) to perform the differentiation of the objective function S(C ), we have: ∇S(C ) = [-diag(∆⋅Γ )⋅diag(∆⋅N -1)⋅diag(F(C ))⋅∆⋅diag(Γ -1)⋅diag(N )]⋅1|U| +F(C ) + +∇F(C )⋅C -∇F(C )⋅MSC[F(C )] Since the first term on the right-hand side equals -F(C ), we have: ∇S(C ) = ∇F(C )⋅(C -MSC[F(C )])

(54)

Then, setting for short: C -MSC[F(C)] = x ,

(55)

the necessary first order conditions (53) become: ∇F(C )⋅x = 0n

(56)

Since the Jacobian of the demand function is a block diagonal matrix, where each block refers to a specific class of users, (56) can be written as |U| systems: ∇Fu(Cu)⋅xu = 0 |J(u)| u = 1, … , |U|

(57)

With reference to the generic class u∈U, expressing the flows in (8) through the demand function and differentiating with respect to Cku, k∈J(u), we have: ∑j∈J(u) ∂Fju(Cu)/∂Cku = 0 k = 1,…, |J(u)|, from which we have: ∂Fj°u(Cu)/∂Cku = -∑ j∈[J(u)-{j°}] ∂Fju(Cu)/∂Cku , j°∈J(u). Then, the generic u-th system (57)

15

becomes: ∑ j∈[J(u)-{j°}] ∂Fju(Cu)/∂Cku ⋅(xju -xj°u) = 0 k = 1, … , |J(u)|

(58)

Given the strict convexity of the satisfaction function Ŵu(Cu) with respect to Cu-, on the basis of (19) we have that Iu-⋅∇F u(C u) ⋅Iu- T is negative definite and therefore non-singular. Then, the linear system (58) has only the trivial solution xju -xj°u = 0 j∈[J(u)-{j°}]. It follows then that the solutions to system (57) are the vectors xu = hu⋅1|J(u)|, where hu is any scalar. Then, the necessary first order conditions (56) become: x∈H , which, taking into account (55), means: C -MSC[F(C)]∈H

(59)

For any h∈H, on the basis of (16), (59) yields the necessary condition: C-h = MSC[F(C-h)]. Then vector C-h must solve the SE problem (45), that is: C-h∈C SE . The necessity assertion follows. Let us now assume that the SE is unique. In this case, the set of the points satisfying the necessary conditions is a connected set, where, by proposition 2, the Social Surplus assumes a same value. Then, because problem (52) has a solution, each point of this set is a solution. This proves the sufficiency.  In order to state the necessary and sufficient conditions for the toll vectors solving the NPO problem, let us introduce the set: T SE = {T∈ℜn : T = MSC(F SE) -Ĉ(F SE), F SE∈SFSE} . Proposition 4. For a travel alternative toll vector T to solve problem (51) it is necessary that T = T.SE +h , where T.SE∈T SE and h∈H . If the SE is unique, then the condition is also sufficient. Proof. By the necessity assertion of proposition 3, using (16), on the basis of (46) it follows that the UE travel alternative flow vector at a solution to problem (51) solves the SE problem (44). Then, by the necessity assertion of proposition 1, the necessity follows. By the sufficiency assertion of proposition 1, the toll vectors T = T SE +h , where T.SE∈T SE and h∈H, yield a UE travel alternative flow vector solving the SE problem (44) and, using (29), on the basis of (47) the corresponding generalized cost vector solves the SE problem (45). Then, assuming that the SE is unique, by the sufficiency assertion of proposition 3, the sufficiency follows.  Proposition 4 states that, if the SE is unique, problem (51) has an infinite number of solutions constituting a connected set, as depicted in figure 1 with reference to the elementary case of one class 16

of users with two travel alternatives. In this case H is the span of vector 12 , i.e. the solutions lie on the parallel to the bisection of plane T1T2 through T SE. 3.2

The NPO problem in terms of arc tolls In order to characterize problem (50) we introduce the marginal pricing toll function :

mp( f ) = msc( f ) -ĉ( f )

(60)

Proposition 5. For an arc toll vector t to solve problem (50), it is necessary that it solves one of the systems: Π T⋅t = T SE +h , where T SE∈T SE and h∈H. If the SE is unique, then the condition is also sufficient. Moreover, for each arc toll vector that solves problem (50) there is another solution vector which leads to the same SE flow pattern, obtained by calculating the marginal pricing toll function at that point. Proof. Let us assume by contradiction that t is a solution to problem (50), while Π T⋅t is not a solution to problem (51). In this case, the solutions to problem (51) yield a Social Surplus value greater than the solutions to problem (50). Let T be a solution to problem (51). By the necessity assertion of proposition 4 there must exist a vector h∈H and an SE flow pattern F SE∈F SE such that T = T SE +h , where T SE = MSC(F SE) -Ĉ(F SE). Then, using (29) and (16), by proposition 2 it follows that T SE is also a solution to problem (51). Let t SE = mp( f SE) , where f SE = Π⋅F SE. Using (41), (28) and (26), from (60) we have: Π T⋅t SE = T SE. This implies that the arc toll vector t SE yields a higher value of the Social Surplus than t, thus contradicting the hypothesis that t is a solution to problem (50). Then Π T⋅t is a solution to problem (51). The necessity is proved by means of the necessity assertion of proposition 4. The sufficiency follows immediately by the sufficiency assertion of proposition 4. The last part of the proposition follows on the basis of the arguments used to prove the necessity.  Proposition 5 shows that any travel alternative toll vector solving problem (51) yields an SE and a corresponding Social Surplus that can be achieved in terms of arc tolls by adopting the corresponding marginal pricing. Vice versa, any solution in terms of arc tolls can be attained straightaway in terms of travel alternative tolls. Consequently, the existence of a solution to problem (50) is assured by the 17

existence of a solution to problem (51). 3.3

On the uniqueness of the solution to the NPO problem By propositions 4 and 5, it is clear that, even if the SE is unique, the solution to the NPO problem

is not unique in terms of tolls. This result is perfectly consistent with the additivity of the choice model. Indeed, set H expresses the intrinsic possibility of freely fixing the toll of one travel alternative without modifying the flow pattern and consequently the congestion on the network. However, we should make a distinction between the multiplicity of solutions related to the presence of degrees of freedom and that related to a non-convexity of the problem. In order to concentrate on the first type of multiplicity, in this subsection we assume that the SE is unique. Let us consider the case in which the travel demand is elastic up to the trip generation; assuming the not-to-travel alternative tolls equal to zero, we have: H = {0n}. Then, by proposition 4, the NPO problem has a unique solution in terms of travel alternative tolls. The existence of other solution vectors t besides t

SE

= mp( f

SE

), where f

SE

is the solution to

problem (42), is then strictly related to the rank of Π, i.e., it requires the possibility of implementing the optimal travel alternative toll pattern in terms of arc tolls in an infinite number of ways. Indeed, by proposition 5, we have Π T⋅(t -t SE) = 0n as a necessary condition, which implies that no solution t ≠ t SE can exist if the rank of Π is equal to the number of arc toll variables. In practical terms, this means that the non-uniqueness of the solution can occur only in particular situations having no relevance from an operational point of view. In particular, if Π is the arc-path incidence matrix, two counter-examples of non-uniqueness are: a) an arc is not utilized by any path (Π has a zero row); and b) a set of arcs is used exclusively by the same set of paths (Π has a set of equal rows). In these and other similar cases the uniqueness of the solution can be restored by dropping the “superfluous” arc tolls from the formulation. Hearn and Ramana (1998), and Dial (1999), exploited the multiplicity of the solution in terms of arc tolls in order to meet a second criterion, other than social welfare (bicriteria approach). More specifically, they show that a preferred solution may be obtained either by minimizing the toll revenue or by charging tolls only on a limited set of arcs. In this regard, it should be noted that they consider a 18

deterministic choice model, so that only a limited set of paths is utilized at the equilibrium, especially when the congestion level is low. Since the toll of an unutilized path can be set at an arbitrary level without modifying the value of the objective function, the solution to the deterministic version of the NPO problem has a higher number of degrees of freedom than in the stochastic case, where all the travel alternatives are utilized. Therefore, in a stochastic framework, such as ours, the bicriteria approach is not promising. Finally, it is to be emphasized that the degrees of freedom in the solution to the NPO problem coupled with set H disappear when the problem is considered in terms of the flow pattern. Proposition 6. For a toll pattern to solve the NPO problem, either in terms of travel alternative variables or in terms of arc variables, it is necessary for it to yield an SE flow pattern. If the SE is unique, then the condition is also sufficient and the solution is unique in terms of flow pattern. Proof. This result follows immediately, on the basis of propositions 4 and 5, by proposition 1.  4

THE SYSTEM OPTIMUM PROBLEM

In section 3 the NPO is formulated as an NDP and it is proved that the tolls maximizing the Social Surplus lead to an SE flow pattern. In this section the NPO is formulated as an EAP, instead, seeking a travel alternative flow pattern in the set SF which optimizes the Social Surplus, while tolls are then determined accordingly: max F S(F)

s. to: F∈SF

(61)

In this kind of problem there is no equilibrium constraint. Consequently, the generic feasible solution F∈SF does not necessarily belong to the demand function, which implies that the Social Surplus (2), based on the equivalent variations, cannot be directly used as an objective function. The Social Surplus must then be formalized in terms of the inverse of the demand function, introduced below with reference to our alternative-based formulation. As the feasible flow vectors satisfy the consistency constraint (13), the SO problem (61) can be conveniently analysed in a subspace having dimension n-|U| . Once a single travel alternative has been identified for each class, vectors F and C can be consistently partitioned through (21) and (22), respectively. On the basis of (13), it is possible to express F ° and therfore F as a function of F -. To 19

this end, let us introduce the functions D°(F -) and D(F -). Using (22), since ∆ T⋅I ° T = I(|U|) , we have: D°(F -) = N -∆ T ⋅I - T ⋅F -

(62)

Then, from (22), using (62), we have: D(F -) = I - T ⋅F - +I ° T ⋅D°(F -)

(63)

In the following we assume that C ° is a given finite valued vector. Then, since the demand function is a C1 point-to-point map, on the basis of (23) function I -⋅F(C) can be inverted with respect to C -. The inverse of the demand function is defined on the set of positive values of the flows D(F -), where it is a C1 point-to-point map, and ranges in the subspace of vectors C - . Formally: F -1: {(F -, C °)∈[ℜn -|U| × ℜ|U|]: D(F -) > 0n} → ℜn -|U| , F(I - T ⋅F -1(F -, C °) +I ° T ⋅C °) = D(F -)

(64)

In order to monetize the individual utilities corresponding to any given feasible flow pattern, we first calculate the Social Surplus corresponding to the inverse of the demand function, and then subtract from this quantity the difference between the two measures of the total user costs obtained by multiplying the given flows, once by the supply function, and then by the inverse of the demand function. The Social Surplus can then be expressed in the following form: S(F -, C °) = [diag(N ) ⋅diag(Γ -1) ⋅Ŵ(I - T ⋅F -1(F -, C °) +I ° T ⋅C °)] T⋅1|U| -E(Π⋅D(F -)) + -[Ĉ(D(F -))T⋅D(F -) -(F -1(F -, C °)T⋅F - +C °T⋅D°(F -))]

(65)

Proposition 7. With reference to the Social Surplus function (65), the following relation holds: S(F -, C ° +h’) = S(F -, C °) ∀h’∈ℜ|U| Proof. Since the right-hand side of (64) is independent of C °, if an algebraic increment h’ is summed up to C °, on the basis of (16) the argument of the demand function on the left-hand side of (64) must vary by the quantity ∆⋅h’∈H . We then have: F -1(F -, C ° +h’) = F -1(F -, C °) +I -⋅∆ ⋅h’

(66)

On the basis of (66), proceeding as in proposition 2, the result follows.  Since, by proposition 7 the arbitrary choice of C ° does not affect the problem formulation, using (38) to express the objective function (65) and (63) to express the set SF , the SO problem (61)

20

becomes: max F - S(F -) = [diag(N) ⋅diag(Γ -1) ⋅Ŵ(I - T ⋅F -1(F -, C °) +I ° T ⋅C °)] T⋅1|U| + +F -1(F -, C °)T⋅F - +C ° T⋅D°(F -) -SC(D(F -))

(67)

s. to: D(F -) ≥ 0n Proposition 8. (existence) The SO problem (61) has a solution. Proof. Recalling that C ° is a given finite valued vector, it is obviously true and can be formally proved that, when Cju → +∞, with u∈U and j∈[J(u)-{j°}]: a) each flow Fku(C), with j ≠ k∈[J(u)-{j°}], tends to assume the value corresponding to the case where the travel alternative j is not present at all; b) the same holds for the satisfaction Wu ; c) it is Cju⋅Fju(C) → 0. Let us assume, with no loss of generality, that the partition of the current feasible flow vector F is such that D°(F -) > 0 |U| . This implies that F -1ju(F -, C °) > -∞ for each u∈U and j∈[J(u)-{j°}]. Then, as the choice model of each class is assumed to be strictly positive, on the basis of the above limit properties of the demand function, we have: Fju → 0+ ⇔ F -1ju(F -, C °) → +∞

u∈U, j∈[J(u)-{j°}]

(68)

Moreover, it follows that, despite the fact that function F -1(F -, C °) is not defined on the boundary of the feasible set of the SO problem (67), its objective function is continuous there. Then, as the feasible set of this problem is compact and not empty, on the basis of Weierstrass’ theorem it has a solution.  Proposition 9. (necessary and sufficient conditions) For a travel alternative flow vector F to solve the SO problem (61) it is necessary for it to solve the SE problem (44). If the SE is unique, then the condition is also sufficient and the solution is unique. Proof. Proceeding so as to obtain (54) from (52), the differentiation of (67) yields: ∇F - S(F -, C °) = -∇F - F -1(F -, C °) ⋅I - ⋅F(I - T ⋅F -1(F -, C °) +I ° T ⋅C °) +∇F - F -1(F -, C °) ⋅F - + +F -1(F -, C °) +∇D°(F -) ⋅C ° -∇D(F -) ⋅MSC[D(F -)]

(69)

On the basis of (64) and since I - ⋅D(F -) = F - , the first two terms on the right-hand side of (69) can be dropped. Since by differentiating (62) and (63) we have respectively: 21

∇D°(F -) = -I -⋅∆ , ∇D(F -) = I - -I -⋅∆ ⋅I ° , then (69) becomes: ∇F - S(F -, C °) = [F -1(F -, C °) -I -⋅∆⋅C °] -[I -⋅MSC(D(F -)) -I -⋅∆⋅I °⋅MSC(D(F -))]

(70)

Assuming again that the partition of the current feasible flow vector F is such that D°(F -) > 0 |U| , from (70), using (68), we have: Fju → 0+ ⇒ ∂S(F -)/∂Fju → +∞

u∈U, j∈[J(u)-{j°}]

(71)

It follows then that problem (67) has only zero gradient solutions within the feasible set, i.e.: ∇S(F -) = 0n , D(F -) > 0n

(72)

We now prove that these necessary conditions are satisfied only by the SE travel alternative flow patterns. To this end, we prove, first, that for any given flow pattern F SE∈F SE both relations in (72) are satisfied, and then, that with any given flow pattern F ∉F SE, such that D(I -⋅F) > 0n , we have ∇S(I -⋅F) ≠ 0n . Since the choice model is assumed to be strictly positive, the second condition in (72) is satisfied. Let C

SE

= MSC(F

SE

) , by (47) it is: C

SE

∈C

SE

. By calculating (64) at I -⋅F

SE

, because it is

D(I -⋅F SE) = F SE , using (44) we have: F(I - T ⋅F -1(I -⋅F SE, C °) +I ° T ⋅C °) = F(C SE) On the basis of (16), it follows that: F -1(I -⋅F SE, C °) = I -⋅(C SE +h) , C ° = I °⋅(C SE +h)

, with h∈H

(73)

Since for any h∈H : ∆ ⋅I °⋅h = h , from (70) using (73) we have: ∇S(I -⋅F SE) = 0n This proves the first assertion. We now prove the second assertion. By contradiction, let us assume that ∇S(I -⋅F) = 0n . From (70) we have: F -1(I -⋅F, C °) = I -⋅∆⋅C ° +I -⋅MSC(D(I -⋅F)) -I -⋅∆⋅I °⋅MSC(D(I -⋅F))

(74)

Since D(I -⋅F) = F, by taking both members of (74) as arguments of the demand function and using (64) we have: F = F(I - T⋅I -⋅∆ ⋅C ° +I - T⋅I -⋅MSC(F) -I - T⋅I -⋅∆⋅I °⋅MSC(F) +I ° T⋅C °) By adding vector -∆⋅[C ° -I °⋅MSC(F)]∈H to the argument of the demand function on the right-hand

22

side, on the basis of (16) we have: F = F[MSC(F)] , which by (44) contradicts the hypothesis F∉F SE. This proves the second assertion; then the necessity follows. Assuming that the SE is unique, the sufficiency and the uniqueness are a direct consequence of both the necessity and the existence proposition 8.  We now analyze the relation between the EAP (61) and the NDP (51). On the basis of (64), given any flow pattern F such that D(F -) > 0n , the finite travel alternative toll vector: T = I - T ⋅F -1(F -, C °) +I ° T ⋅C ° -Ĉ(F) ,

(75)

is such that the corresponding UE problem, which appears as a constraint in the NDP (51), has a solution in F. This implies that the NDP (51) can be addressed in terms of flows. Since using (75) in (51) we have (65), the NDP (51) actually differs from the EAP (61) only because in the latter any flow F∈SF is feasible, while as the choice model of each class is assumed to be strictly positive, in the former F must be such that D(F -) > 0n . However, in proposition 9 we proved that the EAP (67) has only solutions within the feasible set. In terms of flows, the solutions to the two problems coincide, as also clearly evidenced by propositions 6 and 9. The existence of a solution to the NPO problem, assumed in subsection 3.1, is thus proved by proposition 8. Since by propositions 6 and 9 any solution to the NPO problem is an SE, by proposition 1, the consistency of marginal pricing with the flow pattern optimizing the Social Surplus is proved, thus generalizing the notion of SO, current in the literature, to the more general context here considered. 5

CONCLUSIONS

With regard to the NPO problem, the validity of the marginal pricing principle is extended to the case in which the equilibrium constraint is any current multi-user multimodal stochastic traffic assignment model with elastic demand up to trip generation and asymmetric arc cost function Jacobian, thus generalizing previous results obtained with reference to deterministic or Logit formulations. This extension of principle requires the problem to be defined with respect to a specific 23

objective function, referred to as Social Surplus, expressing in monetary terms the social welfare, consistently with the microeconomic consumer theory, and thus generalizing the Social Cost function – often used in previous works on toll optimization – to contexts where the behavioral model is based on random utility theory. With reference to the NPO formulated as an NDP, it is proved that any toll solution vector must lead to an SE flow pattern. In cases where the design variables are the travel alternative tolls, the solutions to the problem are marginal path toll vectors. When the SE is unique, if the toll of one travel alternative for each class of users – possibly the not-to-travel alternative – is fixed, then the solution is unique. Otherwise, due to the additivity of the choice model, an infinite number of solutions constitute a connected set. When the design variables are the arc tolls, the solutions to the problem are arc toll vectors yielding a marginal path toll solution vector. The analysis of the NPO from the point of view of economic efficiency leads to the well-known result that the optimal flow pattern is an SE, which enables us to generalize the notion of SO current in the literature to the case of stochastic equilibrium. To this end the invertibility of the demand function is investigated and the existence of a solution to the NPO problem is proved. REFERENCES Beckmann M. (1965) On Optimal Tolls for Highway, Tunnels and Bridges. Vehicular Traffic Science, Elsevier, New York, 331-341. Bellei G. , Gentile G. , Papola N. (2000a) Ottimizzazione del Trasporto Urbano in Contesto Multiutente e Multimodo Mediante l’Introduzione di Pedaggi. In Metodi e Tecnologie dell’Ingegneria dei Trasporti, ed.s G. Cantarella, F. Russo, Franco Angeli s.r.l. , Milano, Italia. Bellei G. , Gentile G. , Papola N. (2000b) Transit Assignment with Variable Frequencies and Congestion Effects. In Proceedings of the 8th Meeting of the Euro Working Group Transportation EWGT, ed.s M. Bielli, P. Carotenuto, Roma, Italia. Ben Akiva M. , Lerman S. (1985) Discrete Choice Analysis: Theory and Application to Travel Demand. MIT Press, Cambridge, Mass. Cantarella G. E. (1997) A General Fixed-Point Approach to Multimode Multi-User Equilibrium 24

Assignment with Elastic Demand. Transpn. Sci. 31, 107-128. Cascetta E. (2001) Transportation Systems Engineering : Theory and Methods. Kluwer Academic Publishers. Clune A. , Smith M. , Xiang Y. (1999) A Theoretical Basis for Implementation of a Quantitative Decision Support System Using Bilevel Optimisation. Proceedings of the 14th International Symposium on Transportation and Traffic Flow Theory, Jerusalem, Israel. Dafermos S. C. , Sparrow F. T. (1971) Optimal Resource Allocation and Toll Patterns in UserOptimized Transport Network. Journal of Transport Economics and Policy 5, 198-200. Dafermos S. C. (1973) Toll Patterns for Multiclass-User Transportation Networks. Transpn. Sci. 7, 211-223. Delle Site P. , Filippi F. , N. Papola n. (1997) Optimization of Public Transport Services and Central Area Car Pricing. Proceeding of the 5th International Conference on Competition and Ownership Land Passenger Transport, Leeds. Dial R. B. (1999) Minimal-Revenue Congestion Pricing Part I : a Fast Algorithm for the SingleOrigin Case. Transpn. Res. 33B, 189-202. Ferrari P. (1999) A Model of Urban Transport Management.Transpn. Res. 33B, 43-61. Fisk C. (1980) Some Developments in Equilibrium Traffic Assignment. Transpn. Res. 14B, 243255. Hearn D. W. , Ramana M. V. (1998) Solving Congestion Toll Pricing Models. in Equilibrium and Advanced Transportation Modelling. Pergamon. Jara-Diaz S. R. , Farah M. (1988) Valuation of Users’ Benefits in Transport Systems. Transport Reviews 8, 197-218. Luenberger D. G. (1995) Microeconomic Therory. McGraw-Hill, Inc. Nguyen S. , Pallottino S. , Gendreau M. (1998) Implicit Enumeration of Hyperpaths in a Logit Model for Transit Networks. Transpn. Sci. 32, 54-64. Oppenheim N. (1995) Urban Travel Demand Modelling. Wiley-Interscience, New York. Sheffi Y. (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Cliffs, NJ. 25

Smith M. J. (1979) The Marginal Cost Taxation of a Transportation Network. Transpn. Res. 13B, 237-242. Varian H. (1992) Microeconomic Analysis, 3rd ed. Norton, New York. Yang H. (1997) Sensitivity Analysis for the Elastic-Demand Network Equilibrium Problem with Applications. Transpn. Res. 31B, 55-70. Yang H. (1999) System Optimum, Stochastic User Equilibrium, and Optimal Link Tolls. Transpn. Sci. 33, 354-360.

26

FIGURES

S

T2

S(F SE, T SE) S(F SE, T SE +h)

T SE

h

T SE +h

T1

Figure 1: The Social Surplus in the case where the SE is unique. The surface depicts function S(F SE, T ), where F SE is the unique SE flow pattern, with reference to the elementary case of one class of users with two travel alternatives. The thick line is the solution set of problem (51).

27