A multiclass, multicriteria traffic network equilibrium model with elastic demand

Transportation Research Part B 36 (2002) 445–469 www.elsevier.com/locate/trb

A multiclass, multicriteria traffic network equilibrium model with elastic demand Anna Nagurney a

a,*

, June Dong

b

Department of Finance and Operations Management, Isenberg School of Management, University of Massachusetts, Amherst, MA 01003, USA b School of Business, State University of New York at Oswego, Oswego, NY 13126, USA Received 1 March 2000; received in revised form 14 December 2000; accepted 28 February 2001

Abstract In this paper, we develop a multiclass, multicriteria traffic network equilibrium model in which travelers of a class perceive their travel disutility or generalized cost on a route as a weighting of travel time and travel cost, each of which is flow-dependent. In addition, the weights are not only class-dependent but also link-dependent. The model is an elastic demand model and allows the demand function for each class and origin/destination (O/D) pair to depend, in general, upon the disutilities of all classes at all O/D pairs. The formulation of the governing equilibrium conditions, as well as the qualitative analysis, and the computational procedure, are based on finite-dimensional variational inequality theory. The model provides an alternative to existing multimodal and multicriteria traffic network equilibrium models and has location choice implications as well. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction Multicriteria traffic network models were introduced by Quandt (1967) and Schneider (1968) and explicitly consider that travelers may be faced with several criteria, notably, travel time and travel cost, in selecting their optimal routes of travel. The ideas were further developed by Dial (1979) who proposed an uncongested model and Dafermos (1981) who introduced congestion effects and derived an infinite-dimensional variational inequality formulation of her multiclass, multicriteria traffic network equilibrium problem, along with some qualitative properties. Recently, there has been renewed interest in the formulation, analysis, and computation of multicriteria traffic network equilibrium problems. For the convenience of the reader, we have *

Corresponding author. Tel.: +1-413-545-5635; fax: +1-413-545-3858. E-mail address: nagurney@gbfin.umass.edu (A. Nagurney).

0191-2615/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 0 1 ) 0 0 0 1 3 - 3

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identified in Table 1 some principal contributions, since those of Quandt (1967) and Schneider (1968), to the multicriteria traffic network equilibrium literature which reflect conceptual/modeling/methodological advances. We believe that such a tabularization is useful since some of the contributions appear as unpublished manuscripts or in proceedings volumes and, hence, may not be as accessible to the general audience. In particular, Table 1 provides a chronology of citations, which highlights: the number of criteria treated, typically, travel time and travel cost; whether or not these functions are allowed to be flow-dependent or not, and the form (separable or general) handled. In addition, Table 1 notes the type of demand considered, that is, fixed or elastic, and whether the demand is class-dependent, and, if elastic, what form the demand functions take. Moreover, Table 1 provides the type of methodology used in the formulation and analysis such as, for example, an optimization approach, a finite-dimensional variational inequality (fin.-dim. VI) approach, or infinite-dimensional (inf.-dim. VI) approach, along with whether the citation contains algorithmic contributions and/ or qualitative ones. We note that, in the case of infinite-dimensional variational inequality formulations, the number of classes is, usually, infinite, whereas in the case of finite-dimensional formulations, the number of classes is assumed to be finite. Additional citations, including literature exploring multicriteria traffic models used in practice, may be found in the book chapter by Leurent (1998). This paper, in turn, focuses on the development of a new multiclass, multicriteria network equilibrium model with elastic demand. The model has the following novel and what we believe are significant features: 1. It includes weights associated with the two criteria of travel time and travel cost which are not only class-dependent but also, explicitly, link-dependent. Such weights may incorporate such subjective factors as the relative safety or risk associated with particular links, the relative comfort, or even the view. None of the citations in Table 1 have this feature. 2. It treats demand functions (rather than their inverses) which are very general and not separable functions. Specifically, the demand associated with a class and origin/destination (O/D) pair can depend not only on the travel disutility of different classes traveling between the particular O/D pair but can also be influenced by the disutilities of the classes traveling between other O/D pairs. Hence, the model has implications for locational choice (see, e. g., Beckmann et al., 1956; Boyce, 1980; Boyce et al., 1983). Not one of the citations in Table 1 considers such general demand functions. Moreover, for completeness, we include the case of known O/D pair travel disutility (or inverse demand) functions at a similar level of generality. This paper not only develops such a model, but provides qualitative properties, as well as a computational procedure, accompanied by convergence results and numerical examples.

2. The model In this section, we develop the multiclass, multicriteria traffic network equilibrium model with elastic travel demands. The model permits each class of traveler to perceive the travel cost on a link and the travel time in an individual manner, each of which is flow-dependent. Moreover, the

Table 1 Some major multicriteria traffic network equilibrium contributions post-1968 # of criteria

Flow dependence

Type of demand

Formulation

Algorithm

Qualitative properties

Dial (1979) Dafermos (1981)

2 2

No Yes; separable functions

Fixed Fixed; class-dependent

Optimization inf.-dim. VI

Leurent (1993a)

2

Optimization

2

fin.-dim. VI

Yes

Yes

Marcotte et al. (1996); see also Marcotte and Zhu (1994); Leurent (1996)

2

Elastic; separable not classs-dependent Elastic; separable not class-dependent Fixed; class-dependent

No Existence; uniqueness in special case Existence; uniqueness

Leurent (1993b)

Time only; separable functions Time only; general functions Yes; general functions

Yes Yes; for network of special structure Yes

inf.-dim. VI

Yes

Yes

2

Elastic; not class-dependent

fin.-dim. VI

Yes; optimization case only

Existence; uniqueness

Dial (1996)

2

Time only; general functions Yes; separable functions

Fixed; class dependent

Yes; for special case

Yes

Marcotte and Zhu (1997)

2

Yes; general fucntions

Fixed; class-dependent

Optimization; inf.- and fin.-dim. VI inf.-dim VI

Yes

Marcotte (1998)

2 2

Fixed; class-dependent Fixed; class-dependent

Yes

Dial (1999)

Yes; general functions Yes; separable functions

Existence uniqueness in special case Existence

No

Yes

Nagurney (2000)

2

Yes; general functions

Fixed; class-dependent

Yes

Existence; uniqueness in special case

inf.-dim. VI; fin.-dim. VI Optimization; inf.- and fin.-dim. VI fin.-dim. VI

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Citation

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weights are both class- and link-dependent. The equilibrium conditions are then shown to satisfy a finite-dimensional variational inequality problem. Consider a general network G ¼ ½N; L, where N denotes the set of nodes in the network and L the set of directed links. Let a denote a link of the network connecting a pair of nodes and let p denote a path, assumed to be acyclic, consisting of a sequence of links connecting an O/D pair of nodes. There are n links in the network and nP paths. Let W denote the set of J O/D pairs. The set of paths connecting the O/D pair x is denoted by Px and the entire set of paths in the network by P. Assume that there are k classes of travelers in the network with a typical class denoted by i. Let fai denote the flow of class i on link a and let xip denote the nonnegative flow of class i on path p. The relationship between the link loads by class and the path flows is: X fai ¼ xip dap 8i; 8a; ð1Þ p2P

where dap ¼ 1, if link a is contained in path p, and 0, otherwise. Hence, the load of a class of traveler on a link is equal to the sum of the flows of the class on the paths that contain that link. In addition, let fa denote the total flow on link a, where fa ¼

k X

fai

8a 2 L:

ð2Þ

i¼1

Group the class link loads into the kn-dimensional column vector f~ with components: ffa1 ; . . . ; fn1 ; . . . ; fak ; . . . ; fnk g and the total link loads: ffa ; . . . ; fn g into the n-dimensional column vector f. Also, group the class path flows into the knP -dimensional column vector ~x with components: fx1p1 ; . . . ; xkpn g. P We are now ready to describe the functions associated with the links. We assume, as given, a travel time function ta associated with each link a in the network, where ta ¼ ta ðf Þ

8a 2 L;

ð3Þ

and a travel cost function ca associated with each link a, that is, ca ¼ ca ðf Þ 8a 2 L;

ð4Þ

with both these functions assumed to be continuous. Note that here we allow for the general situation in which both the travel time and the travel cost can depend on the entire link load pattern, whereas in Dafermos (1981) it was assumed that these functions were separable. We assume that each class of traveler i has his own perception of the trade-off between travel time and travel cost which are represented by the nonnegative weights wi1a and wi2a . Here wi1a denotes the weight associated with class i’s travel time on link a and wi2a denotes the weight associated with class i’s travel cost on link a. The weights wi1a and wi2a are link-dependent and, hence, can incorporate such link-dependent factors as safety, comfort, and view. For example, in the case of a pleasant view on a link, travelers may weight the travel cost higher than the travel time on such a link. However, if a link has a rough surface or is noted for unsafe road conditions such as ice in the winter, travelers may then assign a higher weight to the travel time than the travel cost. Link-dependent weights provide a greater level of generality and flexibility in modeling travel decision-making than weights that are identical for the travel time and for the travel cost on all links for a given class (see Nagurney, 2000 and the other citations in Table 1).

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We then construct the generalized cost/disutility of class i associated with link a, and denoted by uia , as uia ¼ wi1a ta þ wi2a ca

8i; 8a:

ð5Þ

In view of (2)–(4), we may write uia ¼ uia ðf~Þ

8i; 8a;

ð6Þ

and group the link generalized costs into the kn-dimensional row vector u with components: fu1a ; . . . ; u1n ; . . . ; uka ; . . . ; ukn g. Observe that a possible weighting scheme would be: wi1a ¼ wia and wi2a ¼ ð1  wia Þ with wia lying in the range from 0 to 1 with wia ¼ 1 denoting a class of traveler who is only concerned with the travel time on a particular link a, and with wia ¼ 0 denoting a class of traveler only concerned about travel cost on link a; with weights within the range reflecting classes who perceive travel time and travel cost as per the disutility functions accordingly. Dafermos (1981) proposed such a weighting scheme in which wi1a ¼ wi and wi2a ¼ ð1  wi Þ for all links a and classes i. Such a weighting scheme has an interpretation of a weighted average, but is not link-dependent. Let vip denote the generalized cost of class i associated with traveling on path p, where X uia ðf~Þdap 8i; 8p: ð7Þ vip ¼ a2L

Hence, the generalized cost, as perceived by a class, associated with traveling on a path is the sum of the generalized link costs on links comprising the path. Let dxi denote the travel demand of class i traveler between O/D pair x, and let kix denote the travel disutility associated with class i traveler traveling between the O/D pair x. We group the travel demands into a kJ-dimensional row vector d and the O/D pair travel disutilities into a kJdimensional column vector k. The path flow vector ~x induces the demand vector d with components X xip 8i; 8x: ð8Þ dxi ¼ p2Px

We assume that the travel demands are determined by the O/D travel disutilities, that is, dxi ¼ dxi ðkÞ

8i; 8x;

ð9Þ

and denote the kJ-dimensional row vector of demand functions by dðkÞ. Note that the travel demand function (9) is quite general and has choice location implications as well. For example, it allows the demand for a class associated with an O/D pair to depend not only on the travel disutilities of different classes associated with that O/D pair, but also on those associated with other O/D pairs. 2.1. Traffic network equilibrium conditions The traffic network equilibrium conditions in the case of elastic travel demands (see Beckmann et al., 1956, Dafermos and Nagurney, 1984, and Nagurney, 1999), in the generalized context of the multiclass, multicriteria traffic network equilibrium problem take on the form:

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For each class i, for all O/D pairs x 2 W , and for all paths p 2 Px , the flow pattern ~x is said to be in equilibrium if the following conditions hold: ( if xi ¼ ki p > 0; x ð10Þ vip ðf~ Þ i P kx if xip ¼ 0; and

( dxi ðk Þ

P ¼ p2Px xi p P 6 p2Px xi p

if ki x > 0; if ki x ¼ 0:

ð11Þ

In other words, all utilized paths by a class connecting an O/D pair have equal and minimal generalized path costs. Meanwhile, if the travel disutility associated with traveling between O/D pair x of class i is positive, then the market clears for this O/D pair and this class; that is, the sum of the path flows of this class of traveler on paths connecting this O/D pair is equal to the demand associated with this O/D pair; if the travel disutility is zero, then the sum of the path flows can exceed the demand of this class of traveler. Remark 1. Hence, in the elastic demand framework, different classes of travelers can also choose their O/D pairs, in addition to their paths. Thus, this model allows one to capture the relative attractiveness of different O/D pairs as perceived by the distinct classes of travelers through the travel disutilities. Note that none of the citations given in Table 1 consider elastic demand functions of such a level of generality. Moreover, even the fixed demand models cited in Table 1 have generalized user link cost functions which are not link-dependent. Finally, the fact that we consider travel time and travel cost functions which can depend on the entire total link load pattern further demonstrates the generality of our model. We define the feasible set K underlying the problem as K fðf~; d; kÞ j k P 0 and 9~x P 0, such that, (1) and (2) and (8), hold}. Theorem 1 (Variational inequality formulation). A multiclass, multicriteria link load, travel demand, and O/D travel disutility pattern ðf~ ; d ; k Þ 2 K is a traffic network equilibrium, that is, satisfies equilibrium conditions (10) and (11) if and only if it satisfies the variational inequality problem: k X k X    X  X  i  i uia f~  fai  fai  ki x  dx  dx i¼1 x2W

i¼1 a2L

k X X    i  dx  dxi ðk Þ  kix  ki þ x P0

8ðf~; d; kÞ 2 K;

ð12Þ

i¼1 x2W

equivalently, in standard form: hF ðX ÞT ; X  X i P 0

8X 2 K;

where F ðu; k ; d  dðkÞÞ and X ðf~; d; kÞ. T

ð13Þ

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Proof. Assume that ðf~ ; d ; k Þ satisfies equilibrium conditions (10) and (11). Then we have from (10) that, for a fixed class i, O/D pair x, and path p 2 Px :     i i  x  x P 0 8xip P 0; vip ðf~ Þ  ki ð14Þ p p x and from (11), that, for a fixed class i and O/D pair x: ! X    kix  ki xi  dxi ðk Þ  8kix P 0: p x P0

ð15Þ

p2Px

Summing inequalities (14) over all classes of travelers, all O/D pairs x 2 W , and all paths p 2 Px , and summing (15) over all classes and all O/D pairs, and adding the two resulting inequalities, yields k X X    X i i vip ðf~ Þ  ki  x  x p p x i¼1 x2W p2Px



k X X

dxi ðk Þ

i¼1 x2W



X

! xi p

   kix  ki x P0

kJ P 8~x 2 Rkn þ ; k 2 Rþ ;

ð16Þ

p2Px

kJ P where Rkn þ and Rþ denote the nonnegative real spaces in knP and kJ dimensions, respectively. Using (1), (7) and (8), the inequality (16), after algebraic manipulation, simplifies to k X k X X   X  i  i uia ðf~ Þ  fai  fai  ki x  dx  dx i¼1 x2W

þ

i¼1 x2W

k X X 

   dxi  dxi ðk Þ  kix  ki x P0

8ðf~; d; kÞ 2 K;

ð17Þ

i¼1 x2W

which is the variational inequality (12). Assume now that ðf~ ; d ; k Þ 2 K is a solution to variational inequality (12). Note that variational inequality (12) is equivalent to variational inequality (16). If all dxi ¼ 0 8i; x, then xi p ¼ 0 8i; p. Substituting k ¼ k into (16), one must have that k X X X i ðvip ðf~ Þ  ki 8xip P 0; 8i; p; x Þxp P 0 i¼1 x2W p2Px

and, hence, vip ðf~ Þ P ki x

8i; x:

Otherwise, we fix a class of traveler j and an O/D pair v, such that dvj > 0 and construct a feasible path flow pattern as follows: Set xip ¼ xip for all i 6¼ j, x 6¼ v, and p 2 Px . Select a path r 2 Pv such that xjr > 0 (such a path must exist since dvj > 0) and any path q 2 Pv 6¼ r. Let xjr ¼ xjr  d and xjq ¼ xjq þ d, for some small d > 0 such that xjr ; xjq P 0, with all other path flows for this class and O/D pair equal to the solution path flows, that is, xjp ¼ xjp for p 6¼ q [ r, and k ¼ k . Substitution of the feasible path flow pattern thus constructed into inequality (16) yields:   ð18Þ vjq ðf~ Þ  vjr ðf~ Þ d P 0:

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Now, if xjq > 0, we can construct another feasible path flow pattern identical to the one as above, but with a reallocation of d from path q to path r yielding:   ð19Þ vj ðf~ Þ  vj ðf~ Þ d 6 0; q

r

which implies that if both (18) and (19) are to hold, then vjq ðf~ Þ ¼ vjr ðf~ Þ





with both xjq > 0 and xjr > 0. On the other hand, from (18) we can conclude that if xjq ¼ 0, then vjq P vjr . Since these constructions hold for any class and O/D pair and such pairs of paths, we have established that a solution to variational inequality (16) also satisfies equilibrium conditions (10). j Let xip ¼ xi p 8i; p. First, fix a class of traveler j and an O/D pair x such that kx > 0. Let i i j j j kx ¼ kx 8i 6¼ j, and kx ¼ kx þ d, with d P 0 small enough so that kx  d P 0. Substitution of this feasible solution into inequality (16) yields ! X d P 0: ð20Þ xj  dxj ðk Þ  p p2Px

Now if kjx ¼ kj x  d, then we have ! X ðdÞ P 0: xj  dxj ðk Þ  p

ð21Þ

p2Px

Therefore, for (20) and (21) to hold it follows that X dxj ðk Þ ¼ xj if kj p x > 0: p2Px

In view of (20), if kj x ¼ 0, then X j j xp : dx ðk Þ 6

ð22Þ

p2Px

Since the constructions hold for any class of traveler and O/D pair, we can conclude that a solution to variational inequality (12) also satisfies equilibrium conditions (11).  Remark 2. Dafermos and Nagurney (1984) presented a variational inequality formulation (in link loads), but without a proof, of a multimodal traffic network equilibrium model with elastic demand and known demand functions. The structure of their variational inequality is identical to (12) and (13), except that now the feasible set is different in that (2) must also be satisfied. Moreover, in their multimodal framework, monotonicity conditions are imposed on the multimodal link cost functions in order to derive qualitative properties. This approach, as will become more evident in Section 3, cannot be used in the case of our multicriteria model, since the multiclass generalized link cost functions are constructed from the travel time and travel cost functions according to (5) and, hence, monotonicity conditions, if applied, need to be imposed on the time (cf. (3)) and cost (cf. (4)) functions, rather than on the generalized link cost functions (6). This creates challenges, from a qualitative, as well as from a computational, perspective, as will

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become more apparent in the subsequent sections, where such issues are addressed. Indeed, the imposition of monotonicity conditions on (3) and (4) does not guarantee the same monotonicity properties of (6). Remark 3. In the case that the travel demands are fixed and the weights are not link-dependent but are class-dependent, that is, wi1a ¼ wi1 and wi2a ¼ wi2 for all links a 2 L and classes i, then the above model reduces to the fixed demand multiclass, multicriteria model of Nagurney (2000). Our elastic demand model is defined on a feasible set that is not compact (as are the fixed demand models cited in Table 1) and, thus, existence of an equilibrium solution is no longer guaranteed. We address this issue in the following section. We note that if one is provided with the O/D pair travel disutility functions kðdÞ, rather than the demand functions, and ki x > 0 8i; x, then we can write down immediately the following variational inequality formulation for the multiclass, multicriteria model with given O/D pair travel disutility functions: Corollary 1. The variational inequality formulation of the multiclass, multicriteria traffic network model with known O/D pair travel disutility (inverse demand) functions kðdÞ satisfying equilibrium i i condition (10) with ki x replaced by kx ðd Þ and, under the assumption that kx > 0 for all i and x, in which case equilibrium condition (11) collapses to the feasibility condition (8), is given by: Determine ðf~; dÞ 2 K1 , satisfying k X X i¼1 a2L

k X   X   uia ðf~ Þ  fai  fai  kix ðd Þ  dxi  dxi P 0

8ðf ; dÞ 2 K1 ;

ð23Þ

i¼1 x2W

where K1 fðf~; dÞ j 9~x P 0; satisfying, (1), (2) and (8)}. Proof. The first term in variational inequality (12) remains whereas ki x in the second term is replaced by kix ðd Þ, and the third term is identically equal to zero under the assumption of positive equilibrium O/D pair travel disutilities. Hence, the result is variational inequality (23).  Observe that if there is only a single class of traveler in that all travelers perceive their disutility associated with selecting routes in an identical fashion but the travelers are, nevertheless, multicriteria decision-makers in that they perceive their disutility associated with selecting routes as a weighting of the travel times and travel costs, then the multicriteria model with known O/D pair travel disutility functions collapses to the model of Dafermos (1982) in the case of only a single mode of transportation and in which the ‘‘generalized’’ cost denoted by c1a on a link a 2 L therein coincides with the generalized link cost functions u1a for all links. Remark 4. In the case of fixed travel demands, in which case the class O/D demands in (8) are assumed known and fixed, and in which case equilibrium condition (10) holds and the first part of (11) also does, with dxi substituted for dxi ðk Þ, and with ki x assumed positive for all classes i and O/D pairs x, the variational inequality formulation collapses to the first term in variational

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inequality (23), with the feasible set redefined accordingly. We emphasize that such a fixed demand model generalizes the fixed demand models in Table 1 in the sense that the weights are now also link-dependent, in addition to being class-dependent.

3. Qualitative properties In this section, we derive some qualitative properties of the solution to variational inequality (12), in particular, an existence result. We then obtain some uniqueness results. Subsequently, we investigate properties of the function F that enters the variational inequality formulation of the governing equilibrium conditions for the multiclass, multicriteria traffic network model. Recall that the feasible set K underlying the variational inequality (12) is not a compact set as is the case in the fixed demand model (see Nagurney, 2000). Moreover, imposing strong monotonicity conditions on the travel cost and travel time functions will not guarantee strong monotonicity of the multiclass generalized link cost functions (6) since the former are the functions of the total link flows. Furthermore, it is generally not reasonable to assume strong monotonicity for the travel demand functions. Consequently, one cannot appeal to the standard theory of variational inequalities (see Kinderlehrer and Stampacchia, 1980) to utilize a condition such as strong monotonicity to guarantee existence (as well as uniqueness) of the solution pattern. In this section, hence, we propose a weaker condition under which the existence of a solution to variational inequality (12) is guaranteed. The condition is an adaptation of a condition imposed to establish existence by Dafermos (1986) for a single-class, single-criterion elastic demand model, which is a special case of the model of Dafermos and Nagurney (1984). Let c and t be given continuous functions with the following properties: There exist positive numbers c^; ^t; and k2 such that cia ðf Þ P c^ tai ðf Þ P ^t dxi ðkÞ < k2

8a; i; 8a; i; 8i; x

and f 2 K;

ð24aÞ

and f 2 K;

ð24bÞ

with kix P k1 ;

ð25Þ

where k1 ¼ min wi1a c^ þ wi2a^t P 0: i;a

Thus, uia ðf~Þ P k1

8a; i; and f 2 K:

Conditions (24a) and (24b) assume only that the uncongested parts of the travel costs and the travel times are not zero, with the common belief that congested travel cost and time always exceed the uncongested ones. Condition (25) assumes that travel demands would not be too large (i.e., infinity) if the travel disutility is greater than the minimum weighted uncongested travel cost and travel time. A bounded vector d implies that f is bounded. This implies that cðf Þ and tðf Þ are bounded and, therefore, u and k are bounded. Let V be sufficiently large such that

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X

V > fbi

max uia ðf~Þ:

455

ð26Þ

6 k2 W

Define the compact, convex set n o S ðf~; d; kÞ j 0 6 k 6 V ; 0 6 d 6 k2 ; 9~x P 0 j ð1Þ; ð2Þ; and ð8Þ hold :

ð27Þ

Referring to Dafermos (1986) (see also Nagurney, 1999), we now present the existence result with two lemmas. Consider the variational inequality problem: Determine X 2 S such that hF ðX ÞT ; y  X i P 0

8y 2 S:

ð28Þ

Since F is continuous and S is compact, according to the standard variational inequality theory (cf. Kinderlehrer and Stampacchia, 1980), there exists at least one solution, X ¼ ðf~ ; d ; k Þ to variational inequality (28). Now we claim that X is also a solution to the variational inequality (12). Lemma 1. If X ¼ ðf~ ; d ; k Þ is any solution of variational inequality (28), then dxi < k2

8i; x;

ð29Þ

ki x

8i; x:

ð30Þ


Proof. Analogous to the proof of Lemma 1 in Dafermos (1986).  Lemma 2. Let X ¼ ðf~ ; d ; k Þ be a solution of variational inequality (28). Suppose that dxi < k2

8i; x

ð31Þ

ki x

8i; x:

ð32Þ


Then X is a solution to the original variational inequality (12). Proof. Analogous to the proof of Lemma 2 in Dafermos (1986).  Based on Lemmas 1 and 2, we can immediately obtain the following result: Theorem 2 (Existence of a solution to variational inequality (12)). Let c and t be given continuous functions satisfying conditions (24a), (24b) and (25). Then variational inequality (12) has at least one solution. Using similar arguments one may establish existence of a solution to the model with known O/ D pair travel disutility (or inverse demand) functions. Thus, we have the following: Corollary 2 (Existence of a solution to variational inequality (23)). Let c and t be given continuous functions satisfying conditions (24a) and (24b). Assume that the inverse demand function vector k is known and given and is greater than zero. Then variational inequality (23) has at least one solution.

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We now turn to examining uniqueness. Recall from the standard theory of variational inequalities that strict monotonicity of the function F (cf. (13)) guarantees uniqueness of the solution X , provided that a solution exists. However, although we cannot expect the generalized link cost functions in the multiclass, multicriteria to be strictly monotone they, nevertheless, may be monotone, that is, we may have that: hðuðf~1 Þ  uðf~2 ÞÞT ; f~1  f~2 i P 0

8f~1 ; f~2 2 K:

We now consider a special case of the above model in which we establish uniqueness not of the vector of class link loads f~ but, rather, the uniqueness of the total link loads f . Specifically, consider a generalized link cost function of the form: uia ¼ wia ta þ ð1  wia Þca

8a; i;

ð33Þ

where ta ¼ ga ðf Þ þ aa ;

ca ¼ ga ðf Þ þ ba

8a 2 L:

ð34Þ

Hence, the generalized link cost function uia for each class and link is a weighted average of the travel time and travel cost on a link. Moreover, the variable term is identical for both the travel time and the travel cost function on a given link. wia is the link-dependent weight for class i travelers. Assume now that t and c are each strictly monotone in f, that is, hðtðf 1 Þ  tðf 2 ÞÞT ; f 1  f 2 i > 0

8f 1 ; f 2 2 K;

f 1 6¼ f 2 ;

ð35Þ

8f 1 ; f 2 2 K;

f 1 6¼ f 2 :

ð36Þ

and T

hðcðf 1 Þ  cðf 2 ÞÞ ; f 1  f 2 i > 0 Then we have the following:

Theorem 3 (Uniqueness of the total link load pattern in the special case). The total link load pattern f induced by a solution f~ to variational inequality (12) in the case of link generalized cost functions which are a weighted average of the travel time and travel cost functions on a link and these are functions of the total link load f and differ only by the fixed terms, is guaranteed to be unique if the travel time and travel cost functions are each strictly monotone in f. Proof. Assume that there are two solutions to variational inequality (12) given by ðf~0 ; d 0 ; k0 Þ and ðf~00 ; d 00 ; k00 Þ. Denote the total link load patterns induced by these class patterns through (2) by f 0 and f 00 , respectively. Then substituting d ¼ d 0 , k ¼ k0 into (12), we must have that k X  X i

 0 wa ðga ðf 0 Þ þ aa Þ þ ð1  wia Þðga ðf 0 Þ þ ba Þ  fai  fai P 0

8f~ 2 K:

ð37Þ

i¼1 a2L

Substituting then d ¼ d 00 , k ¼ k00 into (12), we must have, in turn, that k X  X i      00 wa ga ðf 00 Þ þ aa þ 1  wia ga ðf 00 Þ þ ba  fai  fai P 0 i¼1 a2L

8f~ 2 K:

ð38Þ

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457

Let f~ ¼ f~00 and substitute into (37) and let f~ ¼ f~0 and substitute into (38). Adding the resultants yields, after algebraic simplification: X    ð39Þ ga ðf 0 Þ  ga ðf 00 Þ  fa0  fa00 6 0; a2L

which is in contradiction to the assumption of strict monotonicity and, hence, we must have that fa0 ¼ fa00 for all a 2 L and, thus, uniqueness of the total link load pattern induced by the equilibrium class link load pattern has been established.  Here, hence, we have extended the uniqueness result obtained in Nagurney (2000) where it was assumed that travel cost functions and travel time functions are separable functions of the total link load on the link, that is, ca ¼ ga ðfa Þ þ aa ;

ta ¼ ga ðfa Þ þ ba :

Moreover, here the link travel disutilities are also permitted to be link-dependent and not only class-dependent. For some related results, see (Marcotte and Zhu, 1994). In addition, we can establish in a manner similar to that of Theorem 3, the following result for variational inequality (23). Corollary 3 (Uniqueness of the total link load pattern with given O/D pair travel disutility (inverse demand) functions). The total link load pattern f induced by a solution f~ to variational inequality (23) in the case of link generalized cost functions u which are a weighted average of the travel time and travel cost functions on a link and these are functions of the total link load f and differ only by the fixed terms as in (33) and (34), is guaranteed to be unique if the travel time and travel cost functions are each strictly monotone in f. Theorem 4 (Uniqueness of the equilibrium O/D pair travel disutility pattern). Assume that the travel demands dðkÞ are strictly monotone decreasing and that the generalized link cost functions are monotone increasing. Then the equilibrium O/D pair travel disutility pattern k is unique. Proof. Assume that there are two equilibrium solutions ðf~0 ; d 0 ; k0 Þ and ðf~00 ; d 00 ; k00 Þ to variational inequality (12), where k0 6¼ k00 . 00 00 Substitution of ðf~0 ; d 0 ; k0 Þ into (12) as the solution along with fai ¼ fai 8a; i, kix ¼ kix 8x; i, and 00 dxi ¼ dxi 8i; x into variational inequality (12) yields k X k X    00  X  00  X 0 0 0 uia f~0  fai  fai  kix  dxi  dxi i¼1 x2W

i¼1 a2L

þ

k X

X





 00 0 0 dxi  dxi ðk0 Þ  kix  kix P 0:

ð40Þ

i¼1 x2W 0

0 kix

Substituting now ðf~00 ; d 00 ; k00 Þ into (12) along with the feasible pattern: fai ¼ fai 8a; i, kix ¼ 0 8i; x, and dxi ¼ dxi 8i; x into (12) yields

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A. Nagurney, J. Dong / Transportation Research Part B 36 (2002) 445–469 k X X

k X    0  X  0  00 00 00 uia f~00  fai  fai  kix  dxi  dxi i¼1 x2W

i¼1 a2L

þ

k X X



 0  00 00 dxi  dxi ðk00 Þ  kix  kix P 0:

ð41Þ

i¼1 x2W

Summing (40) and (41) yields k X X

dxi ðk00 Þ





dxi ðk0 Þ

k X   0  00   X 00 i i0  kx  kx P uia ðf~0 Þ  uia ðf~00 Þ  fai  fai P 0;

i¼1 x2W

i¼1 a2L

ð42Þ since u is assumed to be monotone. But (42) is in contradiction to the assumption that dðkÞ is strictly monotone decreasing. Hence, we must have that k0 ¼ k00 .  We now prove that the equilibrium pattern is unique under analogous conditions for variational inequality (23). Corollary 4 (Uniqueness of the equilibrium travel demand pattern with given O/D pair travel disutility functions). Assume that the O/D pair travel disutility (inverse demand) functions k ¼ kðdÞ in (23) are strictly monotone decreasing and that the generalized user link cost functions are monotone increasing. Then the equilibrium travel demand pattern d is unique. Proof. Assume that there are two equilibrium solutions ðf~0 ; d 0 Þ and ðf~00 ; d 00 Þ to variational inequality (23). 00 00 Substitution of ðf~0 ; d 0 Þ into (23) as the solution along with fai ¼ fai 8a; i, dxi ¼ dxi 8x; i into variational inequality (12) yields k X X

00 0 uia ðf~0 Þ  ðfai  fai Þ 

k X X

 00  0 kix ðd 0 Þ  dxi  dxi P 0:

ð43Þ

i¼1 x2W

i¼1 a2L

0

0

Substituting now ðf~00 ; d 00 Þ into (23) along with the feasible pattern: fai ¼ fai 8a; i, dxi ¼ dxi 8i; x into (23) yields k X X

k X  0  X  0  00 00 uia ðf~00 Þ  fai  fai  kix ðd 00 Þ  dxi  dxi P 0:

i¼1 a2L

ð44Þ

i¼1 x2W

Summing (43) and (44) yields k X k X   0  X  X  i 00   00 0 00 kx ðd Þ  kix ðd 0 Þ  dxi  dxi P uia ðf~0 Þ  uia ðf~00 Þ  fai  fai P 0; i¼1 x2W

i¼1 a2L

ð45Þ since u is assumed to be monotone. But (45) is in contradiction to the assumption that kðdÞ is strictly monotone decreasing. Hence, we must have that d 0 ¼ d 00 . 

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459

In addition, we show in the subsequent theorem that the function F is monotone, provided that the underlying travel time and travel cost functions and travel demand functions are all monotone functions in X and the generalized user link cost functions take on a special form. The monotonicity property is used in establishing convergence of the algorithm in the following section. Theorem 5 (Monotonicity in the special case). Assume that the link generalized cost functions are as in (33) with both the travel time and travel cost functions differing on a given link only by the fixed cost terms as in (34). Assume also that these functions are monotone in f and the demand functions are monotone decreasing in k. Then the function F that enters the variational inequality problem (12) governing the multiclass, multicriteria traffic network equilibrium model with elastic demand functions is monotone. Proof. Recall the definition of monotonicity, that is, hðF ðX 1 Þ  F ðX 2 ÞÞT ; X 1  X 2 i P 0

8X 1 ; X 2 2 K:

ð46Þ

In view of (16), evaluating (46) for the model is equivalent to evaluating the following: k X X h   i X i2 i ~2 i2 vip ðf~1 Þ  ki1 ð f Þ  k  v  ½xi1 p p  xp  x x i¼1 x2W p2Px k X X



" dxi ðk1 Þ

i¼1 x2W 1

2

8ð~x ; ~x Þ 2



X

! dxi ðk2 Þ

xi1 p



2

RkJ þ:

p2Px P Rkn þ ;

1



X

!# xi2 p

i2  ½ki1 x  kx 

p2Px

8ðk ; k Þ 2

ð47Þ

Due to (1), (2), (7) and (8), (47) can be simplified to: k X k X X X i2 i 1 i 2 i1 i2 ðua ðf Þ  ua ðf ÞÞ  ðfa  fa Þ  ðdxi ðk1 Þ  dxi ðk2 ÞÞ  ðki1 w  kw Þ: i¼1 a2L

ð48Þ

i¼1 x2W

Substituting (33) and (34) into (48), one has the following: k X k X X X i2 ðga ðf 1 Þ  ga ðf 2 ÞÞ  ðfai1  fai2 Þ  ðdxi ðk1 Þ  dxi ðk2 ÞÞ  ðki1 x  kx Þ: i¼1 a2L

ð49Þ

i¼1 x2W

Using (2) and (49) can be expressed as: X a2L

k X    i 1    X  i2 ga ðf 1 Þ  ga ðf 2 Þ  fa1  fa2  dx ðk Þ  dxi ðk2 Þ  ki1 x  kx :

ð50Þ

i¼1 x2W

Since c and t are monotone increasing, therefore, g is monotone increasing in f. Hence, the first term of (50) is nonnegative. Since d is monotone decreasing, the second term of (50) is nonpositive. Therefore, (50) must be nonnegative, that is, (46) must hold.  Corollary 5 (Monotonicity with given O/D pair travel disutility (inverse demand) functions). Assume that the generalized link cost functions u are as in (33) with both the travel time and travel cost functions differing on a given link only by the fixed cost terms as in (34). Assume also that these

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functions are monotone in f and that the O/D pair travel disutility functions k are monotone in d. Then the function that enters the variational inequality problem (23) governing the multiclass, multicriteria traffic network equilibrium model with known O/D pair disutility functions is monotone. The following example illustrates that, given strictly monotone t and c in f, the function u, although not strictly monotone in f~, may, nevertheless, be monotone in f~. Moreover, the function u is not of the special form (33) and (34). 3.1. An example Consider the example depicted in Fig. 1 consisting of two nodes x and y, two links a and b, and a single O/D pair x ¼ ðx; yÞ. Let path p1 ¼ a and p2 ¼ b. Assume that there are two classes of users and that the travel cost functions are given by: ta ðf Þ ¼ 4fa þ 2fb þ 4;

tb ðf Þ ¼ 6fb þ 4fa þ 4;

whereas the travel cost functions are given by: ca ðf Þ ¼ 2fa þ 5;

cb ðf Þ ¼ 2fb þ 10:

Assume that the weights for class 1 are: 1 w11a ¼ w12a ¼ ; 2

1 w11b ¼ ; 4

3 w12b ¼ : 4

whereas the weights for class 2 are: 1 w21a ¼ ; 4

3 w22a ¼ ; 4

1 w21b ¼ w22b ¼ : 2

The travel demand function for class 1 is dx1 ðkÞ ¼ 5k1x  2k2x þ 50 and the travel demand function for class 2 is dx2 ðkÞ ¼ 7k2x  3k1x þ 70. The generalized link cost functions are, hence, for class 1: 1 1 1 u1a ¼ ta þ ca ¼ 3fa þ fb þ 4 ; 2 2 2

1 3 1 u1b ¼ tb þ cb ¼ 3fb þ fa þ 8 ; 4 4 2

and for class 2:

Fig. 1. Network topology for a multicriteria example.

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461

1 3 1 1 3 1 1 u2a ¼ ta þ ca ¼ 2 fa þ fb þ 4 ; u2b ¼ tb þ cb ¼ 4fb þ 2fa þ 7: 4 4 2 2 4 2 2 Clearly, both t and c are each separately strictly monotone in f since their Jacobians given, respectively, by:     4 2 2 0 ; 4 6 0 2 are positive definite. However, the above-constructed class generalized link cost functions are not strictly monotone in f~ since the Jacobian matrix of the generalized link cost functions, given by: 1 0 3 1 3 1 B 1 3 1 3C C; B 1 1 @2 2 12 12 A 2 2 2 4 2 4 has eigenvalues not all positive and given by 0, 8 12, 4, and 0. The generalized link cost functions are, nevertheless, monotone (see Nagurney, 1999) since the Jacobian matrix is positive semidefinite. Consequently, we cannot guarantee uniqueness of an equilibrium multiclass link load pattern. In the following theorem we provide another property of F (in addition to monotonicity) which is needed for convergence of the algorithmic scheme in Section 4. Theorem 6 (Lipschitz continuity). If the generalized link cost functions and the travel demand functions have bounded first-order derivatives, then the function F ðX Þ is Lipschitz continuous, that is, there exists a positive constant L such that kF ðX 1 Þ  F ðX 2 Þk 6 LkX 1  X 2 k

8X 1 ; X 2 2 K:

ð51Þ

Proof. Denote F ðX Þ ¼ ðF1 ðX Þ; . . . ; FkLþ2kW ðX ÞÞT . Since Fl ðX Þ : RkLþ2kW 7! R1 is a smooth function, from the Taylor Theorem, we have that, for any X 1 ; X 2 2 K, there exist nl 2 RkLþ2kW ; l ¼ 1; . . . ; kL þ 2kW , such that Fl ðX 1 Þ  Fl ðX 2 Þ ¼ rFl ðnl ÞðX 1  X 2 Þ; Let

0 B B B B B B rF B B B B B B @

l ¼ 1; . . . ; kL þ 2kW :

ð52Þ 1

rF1 ðn1 Þ

C C C C C C C: C C C C C A

   rFl ðnl Þ   

ð53Þ

rFkLþ2kW ðnkLþ2kW Þ Since the link travel disutility functions and the travel demand functions have bounded first-order derivatives, so there exists L P 0, such that

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krF k 6 L:

ð54Þ

Therefore, using (53) and the basic properties of the linear norm operator, we have the following: 0 1   rF1 ðn1 ÞðX 1  X 2 Þ   B C .. B C . B C B C 1 2 1 2  C B kF ðX Þ  F ðX Þk ¼ B rFl ðnl ÞðX  X Þ C B C .. B C @ A .     rF 1 2 kLþ2kW ðnkLþ2kW ÞðX  X Þ ¼ krF ðX 1  X 2 Þk 6 krF k  kX 1  X 2 k:

ð55Þ

In view of (54), one can conclude that kF ðX 1 Þ  F ðX 2 Þk 6 LkX 1  X 2 k

8X 1 ; X 2 2 K: 

ð56Þ

The following result may be obtained using similar arguments as those above. Corollary 6 (Lipschitz continuity with known O/D pair travel disutility functions). If u and the O/D pair disutility functions k have bounded first-order derivatives, then the function that enters the variational inequality (23) is Lipschitz continuous.

4. The algorithm In this section, an algorithm is presented which can be applied to solve any variational inequality problem in standard form (13) and is guaranteed to converge provided that the function F that enters the variational inequality is monotone and Lipschitz continuous (and that a solution exists). The algorithm is the modified projection method of Korpelevich (1977). The statement of the modified projection method is as follows, where T denotes an iteration counter: Modified projection method Step 0: Initialization Set X 0 2 K. Let T ¼ 1 and let c be a scalar such that 0 < c < 1=L, where L is the Lipschitz continuity constant (cf. Korpelevich, 1977). Step 1: Computation Compute X T by solving the variational inequality subproblem: hðX T þ cF ðX T1 ÞT  X T1 ÞT ; X  X T i P 0 8X 2 K:

ð57Þ

A. Nagurney, J. Dong / Transportation Research Part B 36 (2002) 445–469

463

Step 2: Adaptation Compute X T by solving the variational inequality subproblem: hðX T þ cF ðX T ÞT  X T1 ÞT ; X  X T i P 0;

8X 2 K:

ð58Þ

Step 3: Convergence Verification If maxjXlT  XlT1 j 6 , for all l, with  > 0, a pre-specified tolerance, then stop; else, set T ¼: T þ 1, and go to Step 1. We now discuss the modified projection method more fully. Recall the definition of the projection of X on the closed convex set K, with respect to the Euclidean norm, and denoted by PK X , as y ¼ PK X ¼ arg min kX  zk:

ð59Þ

z2K

In particular, note that X T generated by the modified projection method as the solution to the T variational inequality subproblem (57) is actually the projection of ðX T1  cF ðX T1 Þ Þ on the closed convex set K. In other words, h i T X T ¼ PK X T1  cF ðX T1 Þ : ð60Þ Similarly, X T generated by the solution to variational inequality subproblem (58) is the projection of X T1  cF ðX T ÞT on K, that is, h i T T1 T T  X ¼ PK X  cF ðX Þ : ð61Þ We now give an explicit statement of the modified projection method for the solution of variational inequality problem (12) for the multiclass, multicriteria traffic network equilibrium model. Modified projection method for the solution of variational inequality (12) Step 0: Initialization Set ðf~0 ; d 0 ; k0 Þ 2 K. Let T ¼ 1 and set c such that 0 < c < 1=L, where L is the Lipschitz constant for the problem. Step 1: Computation T Compute ðf~ ; dT ; kT Þ 2 K by solving the variational inequality subproblem: k X X

T T1 fai þ cðuia ðf~T1 ÞÞ  fai

i¼1 a2L

þ

k X X i¼1 x2W



k X   X    T1 T1 dxiT  ckix  dxi  fai  faiT þ  dxi  dxiT i¼1 x2W

 T1 iT1  dxi ðkT1 ÞÞ  kix  ðkix  kiT kiT x þ cðdx x ÞP0

8ðf~; d; kÞ 2 K:

ð62Þ

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A. Nagurney, J. Dong / Transportation Research Part B 36 (2002) 445–469

Step 2: Adaptation Compute ðf~T ; d T ; kT Þ 2 K by solving the variational inequality subproblem: k X X

k X  X  T T T1 T T T iT1 fai þ cðuia ðf~ Þ  fai Þ  ðfai  fai Þ þ dxi  ckiT  d  ðdxi  dxi Þ x x i¼1 x2W

i¼1 a2L

þ

k X X

T

T1

kix þ cðdxiT  dxi ðkT ÞÞ  kix



T

 ðkix  kix Þ P 0

8ðf~; d; kÞ 2 K:

ð63Þ

i¼1 x2W T

T1

T

T1

T

T1

Step 3: Convergence Verification If jfai  fai j 6 , jdxi  dxi j 6 , and jkix  kix j 6 , for all i ¼ 1; . . . ; k; x 2 W , and all a 2 L, with  > 0, a pre-specified tolerance, then stop; otherwise, set T :¼ T þ 1, and go to Step 1. We now state the convergence result for the modified projection method for this model. We then show the realization of the modified projection method for the solution of variational inequality (23), where the O/D pair travel disutility functions are assumed given. Theorem 7 (Convergence in the special case). Assume that the link generalized cost functions u take the form of (33) and (34) and are monotone increasing, and also satisfy conditions (24a), (24b) and (25). Assume that the travel demand functions d are monotone decreasing, and also that u and d have bounded first-order derivatives. Then the modified projection method described above converges to the solution of the variational inequality (12). Proof. According to Korpelevich (1977), the modified projection method converges to the solution of the variational inequality problem of the form (13) provided that F is monotone and Lipschitz continuous and that a solution exists. Existence of a solution follows from Theorem 2. Monotonicity follows from Theorem 5 whereas Lipschitz continuity, in turn, follows from Theorem 6 under the assumption that the travel disutility functions have bounded first-order derivatives.  Modified projection method for the solution of variational inequality (23) Step 0: Initialization Set ðf~0 ; d 0 Þ 2 K1 . Let T ¼ 1 and set c such that 0 < c < 1=L, where L is the Lipschitz constant for the problem. Step 1: Computation T Compute ðf~ ; dT Þ 2 K1 by solving the variational inequality subproblem: k X  X i i ~T1 iT1  ÞÞ  fa fa T þ cðua ðf  ðfai  faiT Þ i¼1 a2L

þ

k X X i¼1 x2W

 T1  ðdxi  dxiT Þ P 0 8ðf~; dÞ 2 K1 : dxiT  ckix ðd T1 Þ  dxi

ð64Þ

A. Nagurney, J. Dong / Transportation Research Part B 36 (2002) 445–469

465

Step 2: Adaptation Compute ðf~T ; d T Þ 2 K1 by solving the variational inequality subproblem: k X  X T T T1 T fai þ cðuia ðf~ Þ  fai Þ  ðfai  fai Þ i¼1 a2L

þ

k X X

T

T1

dxi  ckix ðdT Þ  dxi



T

 ðdxi  dxi Þ P 0

8ðf~; dÞ 2 K1 :

ð65Þ

i¼1 x2W

Step 3: Convergence Verification T

T1

T

T1

If jfai  fai j 6 , jdxi  dxi j 6 , for all i ¼ 1; . . . ; k; x 2 W , and all a 2 L, with  > 0, a prespecified tolerance, then stop; otherwise, set T :¼ T þ 1, and go to Step 1. Corollary 7 (Convergence with known O/D pair travel disutility functions). Assume that the generalized link cost functions u take the form of (33) and (34) and are monotone increasing, and also satisfy conditions (24a) and (24b). Assume that the disutility functions k are known and are positive and monotone decreasing. Also assume that u and k have bounded first-order derivatives. Then the modified projection method described above converges to the solution of the variational inequality (23). Remark 5. Note that although Theorem 7 and Corollary 7 establish convergence of the modified projection method in the case of generalized link cost functions of a special form, for which we were able to establish the monotonicity property, the method may, nevertheless, converge in the case of other functions as the numerical example in the following section demonstrates. Also, it is worth noting that the variational inequality formulations derived and computed here are link-based, rather than path-based. We discuss what algorithm we use for the embedded subproblems (62) and (63) (or (64) and (65)) in the following section.

5. Numerical example In this section, we provide a numerical example. Specifically, we utilize the modified projection method for the solution of variational inequality (12) discussed in the preceding section in order to compute the equilibrium multiclass link load (and path), demand, and O/D pair travel disutility patterns. The traffic network example consists of two classes of users. We then, for completeness, invert the demand functions, and solve the example using, again, the modified projection method but for the solution of variational inequality (23), which assumes that travel disutility functions are given for each O/D pair, rather than the travel demand functions. The convergence criterion was that the absolute value of the path flows and the O/D pair travel disutilities at two successive iterations was less than or equal to  with  set to 103 . In the case of known inverse demand functions, we just had the former test. The c parameter used in the modified projection method (see Section 4) was set to 0.1 for both implementations of the modified projection method. Since for the modified projection method to converge one must have

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that 0 < c < 1=L, where L is the Lipschitz constant in (51), clearly c will be dependent on the generalized link cost functions and demand (or inverse demand) functions since L is. The travel demand for each class and O/D pair was equally distributed among the paths connecting the O/D pair to construct the initial feasible path flow pattern and induced link load pattern and the initial demand was set to one hundred for each class and each O/D pair. The O/D pair travel disutilities were initialized to 1 for all classes and all O/D pairs. In the case of known O/D pair travel disutility functions, the demands were initialized similarly. For the solution of the variational inequality subproblems (62) and (63), and, also, (64) and (65), we utilized the Euler method (see Nagurney and Zhang, 1996, 1997). Specifically, the Euler method is a special case of the general iterative scheme of Dupuis and Nagurney (1993) proposed for the solution of projected dynamical systems. The Euler method can be interpreted as a projection method with a varying step-size. It was utilized for the solution of the variational inequality subproblems since it results (due to the underlying feasible sets) in closed form expressions for the variables at each iteration of the method. We report the CPU time (exclusive of input and output) and the number of iterations required for convergence of the modified projection method for the numerical example. 5.1. A numerical example The network example had the topology depicted in Fig. 2. The network consisted of 10 nodes, 13 links, and two O/D pairs where x1 ¼ ð1; 8Þ and x2 ¼ ð2; 10Þ with travel demand functions by class given by: dx1 1 ¼ k1x1 þ 200; dx1 2 ¼ k1x2 þ 200; dx2 1 ¼  12 k2x1 þ 100, and dx2 2 ¼ k2x2 þ 100. The paths connecting the O/D pairs were: for O/D pair x1 : p1 ¼ ð1; 2; 7Þ, p2 ¼ ð1; 6; 11Þ, p3 ¼ ð5; 10; 11Þ and for O/D pair x2 : p4 ¼ ð2; 3; 4; 9Þ, p5 ¼ ð2; 3; 8; 13Þ, p6 ¼ ð2; 7; 12; 13Þ, and p7 ¼ ð6; 11; 12; 13Þ. The weights were constructed as follows: For each class i and for each link a in a class we let wi2a ¼ ð1  wi1a Þ. Hence, we only report the wi1a terms. For class 1, the weights were: w111 ¼ :25;

w112 ¼ :25;

w113 ¼ :5;

w118 ¼ :75;

w119 ¼ :25;

w1110 ¼ :1;

w114 ¼ :5;

w115 ¼ :25;

w116 ¼ :4;

w1111 ¼ :25;

w1112 ¼ :5;

w214 ¼ :25;

w215 ¼ :5;

w117 ¼ :4;

w1113 ¼ :5:

whereas for class 2, the weights were: w211 ¼ :25; w218 ¼ :3;

w212 ¼ :75; w219 ¼ :4;

w213 ¼ :25; w2110 ¼ :5;

w2111 ¼ :2;

w2112 ¼ :2;

w216 ¼ :5;

w217 ¼ :1;

w2113 ¼ :6:

The generalized link cost functions were constructed according to (5) with the travel time and travel cost functions as given in Table 2.

Fig. 2. Network topology for numerical example.

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467

Table 2 The travel time and travel cost functions for the links Link a

ta ðf Þ

1 2 3 4 5 6 7 8 9 10 11 12 13

0:00005f14 0:00003f24 0:00005f34 0:00003f44

ca ðf Þ 0:00005f14 þ 5f1 þ 1 0:00003f24 þ 4f2 þ 2f3 þ 2 0:00005f34 þ 3f3 þ f1 þ 1 0:00003f44 þ 6f4 þ 2f6 þ 4 4f5 þ 8 0:00007f64 þ 7f6 þ 2f2 þ 6 8f7 þ 7 0:00001f84 þ 7f8 þ 3f5 þ 6 8f9 þ 5 0:00003f104 þ 6f10 þ 2f8 þ 3 0:00004f114 þ 4f11 þ 3f10 þ 4 0:00002f124 þ 6f12 þ 2f9 þ 5 0:00003f134 þ 9f13 þ 3f8 þ 3

þ 4f1 þ 2f3 þ 2 þ 2f2 þ f5 þ 1 þ f3 þ :5f2 þ 3 þ 7f4 þ 3f1 þ 1

5f5 þ 2 0:00007f64 þ 3f6 þ f9 þ 4 4f7 þ 6 0:00001f84 þ 4f8 þ 2f10 þ 1 2f9 þ 8 0:00003f104 þ 4f10 þ f12 þ 7 0:00004f114 þ 6f11 þ 2f13 þ 2 0:00002f12 f 4 þ 4f12 þ 2f5 þ 1 0:00003f134 þ 7f13 þ 4f10 þ 8

The modified projection method converged in 2.89 CPU seconds and required 340 iterations for convergence. It yielded the equilibrium multiclass link load and total link load pattern given in Table 3. The equilibrium path flow pattern induced by the equilibrium multiclass link load pattern is reported in Table 4. 1 The computed equilibrium O/D pair travel disutilities were: k1 x1 ¼ 189:7131, kx2 ¼ 191:8373, 2 2 kx1 ¼ 182:1076, and kx2 ¼ 99:9997. The computed travel demands were: dx1 1 ¼ 10:2716, dx1 2 ¼ 8:1645, dx1 1 ¼ 8:9972, and dx2 2 ¼ 0:000. The incurred path travel disutilities were: for class 1, O/D pair x1 : v1p1 ¼ 192:5735, v1p2 ¼ 189:7080, 1 vp3 ¼ 189:6710, and for class 1, O/D pair x2 : v1p4 ¼ 191:8376, v1p5 ¼ 191:8364, v1p7 ¼ 195:6023, and v1p6 ¼ 192:7368; for class 2, O/D pair x1 : v2p1 ¼ 182:1136, v2p2 ¼ 185:7855, v2p3 ¼ 182:1528, and for class 2, O/D pair x2 : v2p4 ¼ 166:3723, v2p5 ¼ 190:3493, v2p6 ¼ 186:4531, and v2p7 ¼ 190:1250. Table 3 The equilibrium link flows for the example Link a

Class 1 – fa1

Class 2 – fa2

Total flow – fa

1 2 3 4 5 6 7 8 9 10 11 12 13

2.8000 8.1645 8.1645 5.0393 7.4718 2.8002 0.0000 3.1252 5.0393 7.4718 10.2721 0.0000 3.1252

7.2174 7.2174 0.0000 0.0000 1.7793 0.0000 7.2174 0.0000 0.0000 1.7793 1.7793 0.0000 0.0000

10.0177 15.3819 8.1645 5.0393 9.2511 2.8002 7.2174 3.1252 5.0393 9.2511 12.0513 0.0000 3.1252

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Table 4 The equilibrium path flows for the example Path p

Class 1– x1 p

Class 2 – x2 p

p1 p2 p3 p4 p5 p6 p7

0.0000 2.8003 7.4722 5.0393 3.1252 0.0000 0.0000

7.2174 0.0000 1.7788 0.0000 0.0000 0.0000 0.0000

1 1 The incurred O/D pair travel demands were: dx1 1 ðk1 x1 Þ ¼ 10:2869, dx2 ðkx2 Þ ¼ 8:1622, 2 2 ¼ 8:9462, dx2 ðkx2 Þ ¼ 0:0000. The combination of the modified projection method embedded with the Euler method for the solution of the traffic network subproblems of Steps 1 and 2 yielded accurate solutions in a timely manner. Indeed, both equilibrium conditions (10) and (11) were satisfied with good accuracy. In addition, as described previously, we then inverted the travel demand functions and constructed the following travel disutility functions: k1x1 ðdx1 1 Þ ¼ dx1 1 þ 200, k1x2 ðdx1 2 Þ ¼ dx1 2 þ 200, k2x1 ðk2x1 Þ ¼ 2dx2 1 þ 200, and k2x2 ðdx2 2 Þ ¼ k2x2 þ 100. The modified projection method converged in .79 CPU seconds and 82 iterations and yielded a class path flow pattern (and, consequently, a class link load and total link load pattern) which was identical to the one computed above to within two decimal points. Clearly, these networks are small-scale in nature. The effectiveness of the modified projection method for the solution of larger scale examples, along with the possible incorporation of columngeneration techniques, is a topic for future research.

dx2 1 ðk2 x1 Þ

Acknowledgements This research was supported, in part, by NSF Grant No. IIS-0002647. The first author’s research was also supported, in part, by NSF Grant No. INT-0000309 and the John F. Smith Memorial Fund at the Isenberg School of Management at the University of Massachusetts at Amherst. All the research support is gratefully acknowledged. The authors are indebted to the two anonymous referees for their careful reading of the manuscript and for their useful suggestions which improved the presentation of this work.

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