15 networkpricingoptimization (tr2002)

8/14/2019 15 NetworkPricingOptimization (TR2002)

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Network Pricing Optimization in Multi-user and Multimodal Context with Elastic Demand

Giuseppe Bellei, Guido Gentile and Natale Papola

Universit 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 INTRODUCTIONDealing 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 othergoods 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


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subject to the optimization of these same individual utilities, taken singly. Any NDP should, in

principle, therefore be formulated as a bileveloptimization 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 networkany 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 Fisks 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


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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 theNPO 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 anEAP 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 FORMULATIONThe NPO problem can be expressed formally in terms of arc variables, as follows:

max f, ts(f, t) s. to:f fUE

(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 fUE

(t) is the map of the equilibrium arc flow vectors.


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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 functionBy 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 iI,

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 i .

On this basis, the objective function, referred to here as Social Surplus, is given by:

S= iIWi/i -E+T , (2)

whereEand Tare 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= +E , (3)

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


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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-travelalternative is

available to users, their satisfaction also takes into account the benefits related to the latent demand

effects.

2.2 The demand modelIn 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 apositive finite number of mutually exclusive travel alternatives constituting theirchoice 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-travelalternative, 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


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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, theNu > 0 users constituting class uU, where Uis 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 setJ(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 jJ(u), uU, is given by the sum of a

finite systematic utility term Vju, and a zero mean random residualj

u.

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 alternativejJ(u), uU, hypothesis b) is formally expressed as:

Vju

=Xju

-

uCju

, (4)

where the scalaru > 0 is the marginal utility of income for class u, whileXj

uis a constant utility term

independent of congestion, and Cju

is the generalized cost ofj. 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 jJ(u), uU, is by definition the

probability ofj being a maximum utility travel alternative:


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Pju

= Pr[Uju

= max kJ(u)Uku] = Pr[Vj

u+j

uVk

u+k

u kJ(u)] (5)

ThesatisfactionWu of class uUis by definition the mean value of the maximum perceived utility:

Wu = E[max kJ(u)Vju+j

u] (6)

The flowFju

of travel alternativejJ(u), uU, is given by multiplying the number of users in the

class by the choice probability of the travel alternative:

Fju

=NuPju

, (7)

while for each class uUwe have, by definition:

jJ(u)Fju

=Nu (8)

Let W= (W1, , Wu, , W|U|)

T

be the (|U| 1) vector of the satisfactions and letN,N

-1

,

and

-1 be the vectors having the same structure as W, whose generic elements are respectivelyNu , 1/Nu ,

u and 1/u . Furthermore, let V= (V1T, , Vu

T, , V|U|

T)

Tbe the (n 1) vector of the systematic

utilities, where n = uU |J(u)| , whose generic component is Vu = (V1u, , Vj

u, , V|J(u)|

u)

T, and letX,

C,Pand Fbe the vectors having the same structure as V, whose generic elements are respectivelyXju,

Cju,Pj

uandFj

u. Finally, let be the (n |U|) alternative-class incidence matrix, whose generic element

ju

is equal to 1 ifjJ(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( )C , (11)

F= diag(N)P , (12)

TF=N (13)

The demand function is obtained from (12) using (9) and (11):

F= diag(N)P(X-diag( )C) = F(C) (14)

The satisfaction function is obtained from (10) using (11):

W= W(X-diag( )C) = (C) (15)

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


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be utilized when characterizing the NPO solutions.

Let H = {hn: h = h, h|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= {Fn: TF=N, F0n} ,

F(C+h) = F(C) hH , (16)

(C+h) = (C) -diag( )h hH, where h|U| : h = h , (17)

(C) is a C2

convex function ,

(C) = -diag( ) diag(N-1) diag(F(C)) , (18)

2u(Cu) = -(u /Nu ) Fu(Cu) u = 1, , |U| (19)

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) hH (20)

Since the satisfaction (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 class-

specific demand function, thus obtaining a diminished matrix definite positive. The formalization of

these properties requires the partitioning of vectors Cand F. To this end, we introduce a convenient

notation.


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LetjJ(u) be a travel alternative of the choice set of class uU. LetIu be the row vector of the

identity matrixI|J(u)| corresponding to travel alternativej and letIu

-be the matrix obtained fromI|J(u)|

by removing the row vectorIu . Finally, letI-andI

be the block diagonal matrices with generic u-th

block, uU, equal toIu-andIu, respectively. Then, once a single travel alternative has been identified

for each class, vectors Cand Fcan be partitioned into the sub-vectors C-

=I- C, C =I Cand

F- =I

- F, F =I F, respectively. Therefore:

C=I- T

C- +I TC , (21)

F=I- T

F- +I TF (22)

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

[I-F(I- TC-

1+I

TC) -I-F(I- TC-2+I

TC)] T(C-1-C

- 2) < 0 C-

1C-

2n-|U| , C|U| , (23)

while each u(Cu) , uU, 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:

iIWi/i = [diag(N) diag( -1) (C)] T1|U| (24)

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

2.3 The supply modelThe 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 aA 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).


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Let Mbe the set of modes. The modal network of the generic mode mMis represented by means

of a sub-hypergraph ofG denoted Gm = (Nm,Am),NmN,AmA.

A hyperpath of the generic modal hypergraph Gm , mM, with initial node oNm and final node

dNm, is a minimal each node has at the most one exiting arc and acyclic hypergraph

r= (N(r),A(r)) such thatA(r)Am,N(r) = {o}[aA(r)HD(a)] and each iN(r) is connected to don r.

In this framework, the route, if any, associated with a given travel alternative no route is

associated with the not-to-travelalternative 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 , mM, 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 yY, 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 yYon mode mM, let caym

andfaym

denote, respectively,

the arc costand the arc flow on the generic arc aAm . The following relations hold:

Cju

= yY, mM, aAmcaymaju

ym, in compact form: C= Tc , (25)

faym

=uU,jJ(u)Fjuaju

ym, in compact form:f= F , (26)

where: ajuym

is the arc-alternative probability that the arc a is utilized by users belonging to class

uUas an element of the hyperpath, if any, associated with the travel alternative jJ(u), ifm is the

mode ofj andy is the category ofu, otherwise ajuym

is equal to zero; is the (n) matrix, with

= mM|Y||Am|, whose elements are the ajuym

; c = (c11T, , cy1

T, , c|Y|1

T, , c1m

T, , cym

T, ,

c|Y|mT, , c1|M|

T, , cy|M|T, , c|Y||M|

T)T is the ( 1) vector whose generic component is the (|Am| 1)

vectorcym = (c1ym

, , caym

, , 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


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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, becomes a

classical arc-path incidence matrix.

The congestion phenomenon is represented through the arc performance function(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 tollvectortfrom the arc performance vector, i.e.:

c = c(f, t) = (f) +t= + t , (27)

where vectors tand 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= T( F) + Tt= (F) + Tt= C(F, t) (28)

In the following, C(F, t) is referred to assupplyfunction.

In section 3, as design variable, besides vectort, we also deal with a travel alternative tollvectorT,

in which case (28) becomes:

C= (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= tTf or T= TTF , (31)

depending on which design variable is considered, torT, respectively.

In the following, all the functions introduced in this subsection are assumed to be C1

.


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2.4 User Equilibrium and System EquilibriumThe 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) = F( Tc) (32)

Because the demand function is C1, the same holds for the network loading map f(c), which ranges in

Sf= {f:f= F, FSF}. This set is compact, convex and non-empty since this is true also forSF.

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:

fUE

= f[c(fUE

, t)] , (33)

cUE

= c[f(cUE

), t] , (34)

FUE

= F[C(FUE

, t)] , (35)

CUE

= C[F(CUE

), 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 ofmarginal social cost, necessary for the definition of the SE. The

Social Cost (3) can be expressed in terms of arc flows:

SC=(f)Tf+E(f) = sc(f) , (37)

or in terms of travel alternative flows:

SC= (F)TF+E( F) = SC(F) (38)

The gradient of the Social Cost can be obtained from (37) with respect tof, yielding:

msc = fsc(f) = (f) +f(f)f+fE(f) = msc(f) , (39)

or from (38) with respect to F, yielding:

MSC= FSC(F) = (F) +F(F)F+FE(F) = MSC(F) (40)

On the basis of both (28) and (26), the following relation holds:

MSC(F) = Tmsc( F) (41)

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


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(33)-(36) respectively with the corresponding marginal social cost functions msc( f ) and MSC(F),

thus obtaining the following equivalent fixed point problems:

fSE

= f[msc(fSE

)] , (42)

cSE= msc[f(cSE)] , (43)

FSE

= F[MSC(FSE

)] , (44)

CSE

= MSC[F(CSE

)] (45)

Let FSE

and CSE

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

problems are equivalent, we have:

CSECSE F(CSE)FSE , (46)

FSEFSE MSC(FSE)CSE (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:

FUE

= F[(FUE

) +T] (48)

Proposition 1. The UE problem (48) has a solution at a FSEFSE if and only ifT= TSE +h , with

TSE

= MSC(FSE

) -(FSE

) and hH.

Proof. Let FSEFSE and h = T-MSC(FSE) +(FSE). Problem (48) becomes:

FUE

= F[(FUE

) +MSC(FSE

) -(FSE

) +h] (49)

Since on the basis of (16) we have F[MSC(F SE

) +h] = F[MSC(F SE

)] hH, the fixed point

problem (49) has a solution at FSE if and only ifhH. 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.


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In order to establish the monotonicity of msc(f) we verify whether its Jacobian is semidefinite. By

differentiating (39), we have:

msc(f) = (f) +(f)T +yY, mM, aAmfaym

2aym

(f) +2E(f) ,

showing that if the Jacobian of( 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 PROBLEMThe 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, tS(f, t) = [diag(N) diag( -1) ( T(f) + Tt)] T1|U| -E(f) +tTf (50)

s. to:f= f[(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, TS(F, T) = [diag(N) diag( -1) ((F) +T)] T1|U| -E(F) +TTF (51)

s. to: F= F[(F) +T]

3.1 The NPO problem in terms of travel alternative tollsThe NPOproblem(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 forTthe expression C-(F), so

that the demand function defines the feasible set. Then, substituting function F(C) for vectorFin the

objective function and using (38), problem (51) becomes:

max CS(C) = [diag(N) diag( -1) (C)] T1|U| +CTF(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) hH .


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Proof. For any hH, using (17) and (16), from (52) we have:

S(C+h) = [diag(N) diag( -1) ((C) -diag( )h)] T1|U| +(C+h) TF(C) -SC[F(C)] ,

where h|U| : h = h . Since [diag(N) h] T1|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 SECSE and hH. 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( )diag(N-1)diag(F(C))diag( -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 uU, expressing the flows in (8) through the demand function

and differentiating with respect to Cku, kJ(u), we have: jJ(u) Fj

u(Cu)/Ck

u= 0 k= 1,, |J(u)|, from

which we have: Fju

(Cu)/Cku

= -j[J(u)-{j}]Fju

(Cu)/Cku

,jJ(u). Then, the generic u-th system (57)


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becomes:

j[J(u)-{j}]Fju(Cu)/Ck

u(xj

u-xj

u) = 0 k= 1, , |J(u)| (58)

Given the strict convexity of the satisfaction function u(Cu) with respect to Cu-, on the basis of

(19) we have that Iu-F u(Cu) Iu

- Tis negative definite and therefore non-singular. Then, the linear

system (58) has only the trivial solution xju

-xju

= 0 j[J(u)-{j}]. It follows then that the solutions to

system (57) are the vectors xu = hu1|J(u)|, where hu is any scalar. Then, the necessary first order

conditions (56) become:xH, which, taking into account (55), means:

C-MSC[F(C)]H (59)

For any hH, on the basis of (16), (59) yields the necessary condition: C-h = MSC[F(C-h)]. Then

vectorC-h must solve the SE problem (45), that is: C-hCSE . 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: TSE = {Tn : T= MSC(FSE) -(FSE), FSESFSE} .

Proposition 4. For a travel alternative toll vector T to solve problem (51) it is necessary that

T= T.SE

+h , where T.SETSE and hH. 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= TSE

+h , where T.SETSE and

hH, 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


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of users with two travel alternatives. In this caseHis the span of vector12 , i.e. the solutions lie on the

parallel to the bisection of plane T1T2 through TSE

.

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

mp(f) = msc(f) -(f) (60)

Proposition 5. For an arc toll vectortto solve problem (50), it is necessary that it solves one of the

systems: Tt= TSE +h , where TSETSE and hH. 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 Tt 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 vectorhHand an SE flow pattern FSEFSE such that

T= TSE

+h , where TSE

= MSC(FSE

) -(FSE

). Then, using (29) and (16), by proposition 2 it follows

that TSE

is also a solution to problem (51). Let tSE

= mp(fSE

) , where fSE

= FSE. Using (41), (28)

and (26), from (60) we have: TtSE = TSE. This implies that the arc toll vector tSE yields a higher

value of the Social Surplus than t, thus contradicting the hypothesis that tis a solution to problem (50).

Then Ttis 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


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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, setHexpresses 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-travelalternativetolls 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 tSE

= mp( fSE

), where fSE

is the solution to

problem (42), is then strictly related to the rank of , 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 T(t-tSE) = 0n as a necessary condition, which implies that no solution ttSE

can exist if the rank of 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 is the arc-path

incidence matrix, two counter-examples of non-uniqueness are: a) an arc is not utilized by any path (

has a zero row); and b) a set of arcs is used exclusively by the same set of paths ( 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


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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 setHdisappear 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 PROBLEMIn 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 anEAP, instead, seeking a

travel alternative flow pattern in the set SF which optimizes the Social Surplus, while tolls are then

determined accordingly:

max FS(F) s. to: FSF (61)

In this kind of problem there is no equilibrium constraint. Consequently, the generic feasible

solution FSF 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 Fas a function ofF

-. To


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this end, let us introduce the functions D(F-) and D(F

-). Using (22), since TI T =I(|U|) , we have:

D(F-) =N- TI- TF- (62)

Then, from (22), using (62), we have:

D(F-) =I

- TF- +I TD(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

)[n -|U||U|]: D(F-) > 0n}

n -|U|,

F(I- T

F -1(F-, C) +I TC) = 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( -1) (I- TF -1(F-, C) +I TC)] T1|U| -E( D(F-)) +

-[(D(F-))

TD(F-) -(F -1(F-, C)TF- +CTD(F-))] (65)

Proposition 7. With reference to the Social Surplus function (65), the following relation holds:

S(F-, C

+h) = S(F

-, C

) h|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 hH. 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 ofC 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)


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becomes:

max F - S(F-) = [diag(N) diag( -1) (I- TF -1(F-, C) +I TC)] T1|U| +

+F-1

(F-, C

)

TF- +C TD(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 uUand j[J(u)-{j}]: a) each flow Fk

u(C), withjk[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 CjuFju(C) 0.

Let us assume, with no loss of generality, that the partition of the current feasible flow vectorFis

such that D(F-) > 0 |U| . This implies that F

-1j

u(F

-, C

) > - for each uUandj[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 -1j

u(F

-, C

) + uU,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 vectorFto 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- TF -1(F-, C) +I TC) +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 sinceI-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:


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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-IMSC(D(F-))] (70)

Assuming again that the partition of the current feasible flow vector Fis such that D(F-) > 0 |U| ,

from (70), using (68), we have:

Fju 0+ S(F-)/Fj

u + uU,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 FF 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 SEC SE . By calculating (64) at I -F SE, because it is

D(I-FSE) = FSE , using (44) we have:

F(I- TF -1(I-FSE, C) +I TC) = F(CSE)

On the basis of (16), it follows that:

F -1(I-FSE, C) =I-(CSE +h) , C =I(CSE +h) , with hH (73)

Since for any hH: Ih = h , from (70) using (73) we have:

S(I-FSE) = 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-IMSC(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

- TI- C +I- TI-MSC(F) -I- TI-IMSC(F) +I TC)

By adding vector -[C -IMSC(F)]H to the argument of the demand function on the right-hand


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side, on the basis of (16) we have:

F= F[MSC(F)] ,

which by (44) contradicts the hypothesis FFSE. 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 Fsuch that D(F-) > 0n , the finite travel alternative toll vector:

T=I- T

F -1(F-, C) +I TC -(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

FSF is feasible, while as the choice model of each class is assumed to be strictly positive, in the

formerFmust 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 CONCLUSIONSWith 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


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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-travelalternative 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.

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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

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Assignment with Elastic Demand. Transpn. Sci. 31, 107-128.

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Smith M. J. (1979) The Marginal Cost Taxation of a Transportation Network. Transpn. Res. 13B,

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FIGURES

Figure 1: The Social Surplus in the case where the SEis unique.

The surface depicts function S(FSE

, T), where F

SEis the unique SEflow 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).

S

TSE

T1

S(FSE, TSE)

T2

TSE

+hh

S(FSE, TSE +h)

15 networkpricingoptimization (tr2002)

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