Research ArticleA Path-Based Gradient Projection Algorithm for theCost-Based System Optimum Problem in Networks withContinuously Distributed Value of Time

Wen-Xiang Wu1 and Hai-Jun Huang2

1 Beijing Key Lab of Urban Intelligent Traffic Control Technology North China University of Technology Beijing 100144 China2 School of Economics and Management Beihang University Beijing 100191 China

Correspondence should be addressed to Wen-Xiang Wu wwx816gmailcom

Received 30 November 2013 Accepted 11 February 2014 Published 20 March 2014

Academic Editor Ching-Jong Liao

Copyright copy 2014 W-X Wu and H-J Huang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The cost-based system optimum problem in networks with continuously distributed value of time is formulated as a path-basedform which cannot be solved by the Frank-Wolfe algorithm In light of magnitude improvement in the availability of computermemory in recent years path-based algorithms have been regarded as a viable approach for traffic assignment problems withreasonably large network sizes We develop a path-based gradient projection algorithm for solving the cost-based system optimummodel based on Goldstein-Levitin-Polyak method which has been successfully applied to solve standard user equilibrium andsystem optimum problems The Sioux Falls network tested is used to verify the effectiveness of the algorithm

1 Introduction

The traffic assignment problem consists in determiningwhich routes to assign to the drivers who travel on atransportation network from some origins and some desti-nations Wardrop [1] stated two principles for determiningthe assignment The first principle leads to an equilibriumstate in which no user can reduce his travel time by usingan alternative route The second principle induces a systemoptimal state in which the total travel time of all users (or theaverage journey time in a network) isminimum regarding thebenefit of the society (ie whole traffic) It is well known thatin the standard traffic network equilibriummodel a so-calledmarginal-cost toll (equivalent to the negative externalitythat an additional individual imposes on other users of thesystem) can drive a user equilibrium flow pattern to a systemoptimum [2] This conventional traffic network equilibriummodel typically assumes that usersrsquo VOTs are identicalthat is homogeneous users However user heterogeneity ismanifested in the fact that some travelers take slower pathsto avoid tolls while others choose tolled roads to save timeSome studies pointed out that the VOT varies significantly

across individuals because of different socioeconomic char-acteristics trip purposes attitudes and inherent preferences[3ndash6]

The concept of value of time (VOT) plays a central role inroad pricing analysis as it describes how users make tradeoffsbetween money and time in response to road toll chargesThe network disutility can be measured in travel time ortravel cost It is obvious that different system optimal (SO)flow patterns will be obtained if we use different units (timeor money) to measure the system disutility Previous studiesthat address user heterogeneity can be classified into twocategories [7] The first is the multiclass approach in whichthe entire feasible VOT range is divided into several prede-termined intervals according to a discrete VOT distributionor some socioeconomic characteristics [2 8ndash10] The secondlets VOTs be continuously distributed across the populationof trips which is regarded to be more rational than others[11ndash15] In the these studies including those dealingwith two-route or highwaytransit two-mode problems [12 16ndash19] thecontinuously distributed VOT between each OD pair wastreated as a random variable following a probability densityfunction [13 15 20ndash22]

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 271358 9 pageshttpdxdoiorg1011552014271358

2 Journal of Applied Mathematics

For more realistically capturing the travelersrsquo path choicebehavior in response to toll charges Wu and Huang [23]assumed that the VOT varies significantly across individualsbecause of their different socioeconomic characteristics trippurposes attitudes and inherent preferences Furthermorethe VOT of each user is assumed to be deterministic and con-stant because the factors influencing VOT keep unchangedwithin a certain time period They extended the work ofYang andHuang [2] to the case with continuously distributedvalue of time across users for finding anonymous tolls torealize target flow pattern in networks with continuouslydistributed value of time To find anonymous tolls to realizesystemoptimum in cost unitsWu andHuang [23] proposed acost-based system optimum model in general networks withcontinuously distributed value of time but did not study itstraffic assignment algorithmsThe calculation of the model isvery difficult [24] It is formulated in a path-based form andcannot be solved by the Frank-Wolfe algorithmwhich is link-based and cannot provide path flows

In the past path-based algorithms even those not enu-merating all possible paths were traditionally discarded bytransportation researchers for solving large-scale networkproblems because of intensive memory requirements and thedifficulties in manipulating and storing paths [25] Howeverpath-based algorithms automatically provide not only thelink-flow solution but also the path-flow solution that maybe required in certain applications In light of magnitudeimprovement in the availability of computer memory inrecent years recent research on path-based algorithms hasdemonstrated and established that it is a viable approach fortraffic assignment problems with reasonably large networksizes [26ndash29] Chen et al [25] stated that much of theattention has been focused on two particular algorithms thedisaggregate simplicial decomposition (DSD) algorithm andthe gradient projection (GP) algorithm Application of GP tosolve the traffic assignment problem is relatively new but GPis as good as or better than DSD in direct comparisons

In this paper we develop a GP algorithm for the proposedcost-based system optimum model based on the Goldstein-Levitin-Polyak (GLP) GP method formulated by Bertsekas[30] for general nonlinear multicommodity problems whichis successfully used for solving traffic assignment problems[26] The proposed GLP algorithm includes five key opera-tions (1) assigning the flow on each path to the links along tofind the total link flows (2) computing link travel times andpath travel times and sorting path travel times in decreasingorder (3) computing the first derivative lengths (marginalsocial path travel costs) for all paths between each OD pairand finding the shortest first derivative lengths for eachOD pair (4) finding the second derivative lengths for eachOD pair and (5) updating the path flows using the secondderivative lengths as scaling

In the next section a system optimum problem in costunits is formulated in fixed demand networks with contin-uously distributed value of time In Section 3 we develop aGP algorithm for the cost-based system optimum model Anumerical example is presented in Section 4 for testing theGP algorithm Section 5 concludes the paper

Flow

x

120573w(x)

VOT

dw

Figure 1 Continuously distributed VOTs for users between eachOD pair 119908

2 Cost-Based System Optimum Problem

In this section we formulate the system optimum in costunits in a fixed demand network with heterogeneous users interms of a continuously distributed VOT Let 119866 = (119881 119860) bea directed network where 119881 is the set of nodes and 119860 the setof links Each link 119886 isin 119860 has an associated flow-dependenttravel time 119905

119886(V119886) which is assumed to be differentiable

convex andmonotonically increasing subject to the link flowV119886 Let 119882 be the set of all OD pairs 119877

119908the set of all paths

connecting OD pair 119908 isin 119882 and |119877119908| the number of paths

between each OD pair Let 119878119903119908be the set of drivers (users)

on path 119903 isin 119877119908and 119891

119903

119908the number of these drivers f =

( 119891119903

119908 ) We have V

119886= sum119908isin119882

sum119903isin119877119908

119891119903

119908120575119886119903

119908 119886 isin 119860 where

120575119886119903

119908equals 1 if path 119903 between OD pair 119908 contains link 119886

and 0 otherwise All users on link 119886 isin 119860 are charged by anexogenously given toll 120591

119886ge 0 Each path 119903 between each OD

pair 119908 is associated with a travel time 119905119903119908(f) and a travel cost

(dollar) 120591119903119908ge 0 119905119903119908(f) = sum

119886isin119860119905119886(V119886)120575119886119903

119908 and 120591

119903

119908= sum119886isin119860

120591119886120575119886119903

119908

Let 119889119908be the travel demand between each OD pair 119908

Instead of assuming a unified VOT for the whole popula-tion in this paper we set a unique specific VOT for eachtrip-maker Let the distribution of VOTs across individualsbetween each OD pair 119908 be characterized by a continuousfunction 120573

119908(119909) gt 0 Furthermore let the population between

each OD pair be ordered in decreasing order of their VOTsthat is 119889120573

119908(119909)119889119909 le 0 where 119909 is the 119909th user between each

OD pair 119908 as demonstrated in Figure 1 Note that here theVOT distribution is given In fact there exist many studieswhich address estimation of VOTs By observing choicesamong alternative combinations of cost and travel time basedon stated preference (SP) analysis due to the ability to controltime and cost variables [31] information about the relativeweighting of cost and time can be inferred and from this thedistribution of VOT can be derived In general literature onestimation of VOTs refers to two main models multinomiallogit model [32 33] and mixed logit model [34 35] Recentlythe mixed logit approach is popular because it does not have

Journal of Applied Mathematics 3

to assume the irrelevance of independent alternatives (IIA)property

Let 119895 = 1 2 |119877119908| denote the paths between each

OD pair 119908 satisfying 1199051

119908(f) ge 119905

2

119908(f) ge sdot sdot sdot ge 119905

|119877119908|

119908(f) The

system optimum naturally requires that the users with higherVOTs should choose faster paths and those with lower VOTschoose slower paths Otherwise the total travel cost can bereduced by switching a higher VOTuser on a slower path intoa faster path For the sake of notational consistency we definesum|119877119908|

119896=|119877119908|+1

119891119896

119908= 0 and sum

0

119896=1119891119896

119908= 0 The system optimum in

cost units can be formulated as the following minimizationproblem

minf

119885 (f) = sum

119908isin119882

|119877119908|

sum

119903=1

119905119903

119908(f) (int

119891119903

119908

0

120573119908(

|119877119908|

sum

119896=119903+1

119891119896

119908+ 119909)119889119909)

(1)

subject to

|119877119908|

sum

119896=1

119891119896

119908= 119889119908 (2)

119891119896

119908ge 0 (3)

where 119905119903

119908(f) = sum

119886isin119908119905119886(V119886)120575119886119903

119908 V119886

= sum119908isin119882

sum119903isin119877119908

119891119903

119908120575119886119903

119908

Recall that the users between each OD pair are arranged indecreasing order of their VOTs This determines the integralformulation appearing in (1)

The first-order optimality conditions of the above mini-mization problem are as follows

119905119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909 = 119892so119908cost

if 119891119903119908gt 0 119903 = 1 2

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(4)

119905119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909 ge 119892so119908cost

if 119891119903119908= 0 119903 = 1 2

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(5)

where 119892so119908cost is the minimal travel cost between each OD pair

119908

Note that the sum of the second and third terms on theleft hand side of (4) is the total externality caused by the119909th user of OD pair 119908 119909 = sum

|119877119908|

119896=119903119891119896

119908 or the 119891

119903

119908th user

of path 119903 The second term states that this new trip-makerimposes additional costs on all users who may belong toother OD pairs and other paths and have their own VOTs buttraverse the links of path 119903 This is the traditional congestionexternality reported in the literature The third term statesthat the costs of paths 1 to 119903 minus 1 are reduced since the 119891

119903

119908th

userrsquos path choice changes the VOTs of the users on thesepaths (this user chooses path 119903 rather than other longerpaths) This is another kind of externality attributed to theVOT distribution of OD trips Let

119888119903

119908(f) = 119905

119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

(6)Also for the remainder of the paper when we refer to the

first derivative lengths we mean the first derivatives of theobjective function which can also be regarded as marginalsocial travel cost corresponding to path flow on path 119903

Equations (4) and (5) can then be rewritten as119888119903

119908(f) = 119892

so119908cost if 119891119903

119908gt 0 119903 = 1 2 (

1003816100381610038161003816119877119908

1003816100381610038161003816minus 1)

119908 isin 119882

119888119903

119908(f) ge 119892

so119908cost if 119891119903

119908= 0 119903 = 1 2 (

1003816100381610038161003816119877119908

1003816100381610038161003816minus 1)

119908 isin 119882

(7)Therefore in a network where each user has a unique

VOT (7) state that at optimality the marginal social travelcosts on all the used paths connecting a given OD pair areequal and less than or equal to those on all the unused pathsThis is similar to a standard system optimum in which atoptimality the marginal social travel times on all the usedpaths between each given OD pair are equal and less than orequal to those on all the unused paths

Clearly the proposed model is more complicated thana standard system optimum According to the extremevalue theorem that a continuous function in the closed andbounded space attains itsmaximumandminimum the aboveminimization program must attain its minimum value Themost common algorithm used to solve traffic assignmentproblems is link-based Frank-Wolfe algorithm introduced byLeBlanc et al [36] However the objective function is ingeneral not convex since the path travel costs depending notonly on its own path but also on other paths are inseparableand asymmetric in terms of the path flows Fortunately for

4 Journal of Applied Mathematics

the differentiable and bounded objective the convexity of theconstraints allows the development of a gradient projectionalgorithm

3 Path-Based Traffic Assignment Algorithm

In this section we adopt the Goldstein-Levitin-Polyak algo-rithm to the traffic assignment problem In each iteration thetravel demand constraints (2) are eliminated by reformulatingthe path-flow variables in terms of nonshortest path flowsin terms of the first derivative lengths to make projectionoperation simpler This is implemented by partitioning 119891

119903

119908

into the shortest path flow 119891119903

119908and the nonshortest path flows

119891119903

119908and writing constraints (2) in terms of 119891119903

119908as follows

119891119903

119908= 119889119908minus

|119877119908|

sum

119903=1119903 = 119903

119891119903

119908 forall119908 isin 119882 (8)

It should be noted that 119903 is the shortest path in terms ofthe first derivative lengths but |119877

119908| is the shortest path in

terms of path travel times Putting constraints (8) into theobjective function we obtain a new formulation with just thenonnegativity constraints on the nonshortest path flows asthe decision variables Consider

min 119885(

f) (9)

119891119903

119908ge 0 forall119903 isin 119877

119908 119903 = 119903 119908 isin 119882 (10)

where 119885(

f) is the reformulated objective function onlyincluding the nonshortest path flows for all OD pairs Thisreformulation is a program with only nonnegativity con-straints The first and second derivatives of the new objectivefunction can be easily derived as follows

120597119885 (f)

120597119891119903

119908

=

120597119885 (f)120597119891119903

119908

minus

120597 (f)120597119891119903

119908

= 119888119903

119908(f) minus 119888

|119877119908|

119908(f)

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(11)

1205972

119885(f)

(120597119891119903

119908)2

=

1205972

119885 (f)(120597119891119903

119908)2minus 2

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

+

1205972

119885 (f)(120597119891119903

119908)

2

forall119903 isin 119877119908 119903 =

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(12)

where

1205972

119885 (f)(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)2

119889119909

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119896120575119908

119886119903

times int

119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908120597119891119903

119908

119889119909

1205972

119885 (f)(120597119891119903

119908)

2=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

times sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

Journal of Applied Mathematics 5

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

(13)

Putting (13) into (12) yields

1205972

119885(f)

(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891

119896

119908)

120597119891119903

119908

sum

119886isin119860

(119905119903

119908(f) minus 119905

119903

119908(f))

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ 2120573119908(

|119877119908|

sum

119896=119903

119891

119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

(120575119908

119886119903minus 120575119908

119886119903)2

120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

|119877119908|minus1

sum

119896=119903

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(14)

Note that the second derivative is in general not positiveThis is different from the transformed objective functionof the standard system optimum model [25] Thus in eachiteration the scaled GLP algorithm updates the path flowsaccording to the following iteration equations

119891119903

119908(119899 + 1) = [119891

119903

119908(119899) minus

120572 (119899)

119904119903

119908(119899)

(119888119903

119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899))]

+

if 119904119903119908(119899) gt 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899) 119908 isin 119882

50Demand

VOT

3

x

120573(x)

Figure 2 Continuously distributed VOTs of users

119891119903

119908(119899 + 1) = 119891

119903

119908(119899) if 119904119903

119908(119899) le 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899)

119908 isin 119882

119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus

|119877119908|

sum

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1)

(15)

where 119899 is the iteration number 120572(119899) is the step size119896119908(119899) is the shortest path in terms of the first derivative

lengths between each OD pair 119908 119904119903119908(119899) is a diagonal scaling

equivalent to the second derivative regarding path 119903 that is119904119903

119908(119899) = 120597

2

119885(f)(120597119891119903

119908)2 119888119903119908(f(119899)) and 119888

119896119908(119899)

119908(f(119899)) are the first

derivative lengths along path 119903 and path 119896119908(119899) between each

OD pair 119908 and []+ denotes the projection operation

Note that 119888119903119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899)) can be explained by (11)

and (15) certainly satisfies the flow conservation constraintsin (2) and nonnegativity conditions in (3)

With the above flow update equations the completealgorithmic steps can be summarized as follows

Step 1 (initialization) Set 119905119886(0) for all 119886 and perform all-or-

nothing assignments This yields path flows f119908(1) for all119908 isin

119882 and link flows 119909119886(1) for all 119886 Set iteration counter 119899 = 119897

Initialize the path-set 119877119908with the shortest path for each OD

pair 119908

Step 2 (update) Set 119905119886(119899) = 119905

119886(119909119886(119899)) for all 119886 Sort path

travel time in decreasing order that is 1199051119908(119899) ge 119905

2

119908(119899) ge sdot sdot sdot ge

119905|119877119908|

119908(119899) where 119905119896

119908(119899) = sum

119886isin119860119905119886(119909119886(119899))120575119908

119886119896 119896 = 1 2 |119877

119908|

Step 3 Update the first derivative lengths 119888119896119908(119899)of all the paths

119896 in 119877119908 for all119908 isin 119882

Step 4 (direction finding) Find the shortest paths 119896119908(119899) in

terms of marginal social travel cost between each OD pair 119908based on 119888

119896

119908(119899) If 119896

119908(119899) notin 119877

119908 then add it to 119877

119908and record

6 Journal of Applied Mathematics

1 2

3 4 5 6

9 8 7

12 11 10 16 18

17

14 15 19

23 22

13 24 21 20

22563

274

365

225

63

102947

171

418

66551

654

73

29077

665

51 29077

196

558

128

889

128889

840

09

16340

502

115

5517

21879

31218

584026

8839

61769

109

649

109649

16396

524

43

16396

555

17

105

888

137

106

128

889

153

502

167398

50211

577

49

52443

109649

Figure 3 The Sioux Falls test network

119888119896

119899

119908

119908(119899) Otherwise tag the shortest among the paths in 119877

119908as

119896119908(119899)

Step 5 (move) Set the new path flows as follows

119891119896

119908(119899 + 1) = max0 119891119896

119908(119899) minus

120572 (119899)

119904119896

119908(119899)

(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))

if 119904119903119908(119899) gt 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899) 119908 isin 119882

119891119896

119908(119899 + 1) = 119891

119896

119908(119899) if 119904119903

119908(119899) le 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899)

119908 isin 119882

(16)

where 119904119896

119908(119899) = 120597

2

119885(119891)(120597119891

119896

119908)

2

for all 119896 isin 119877119908 and 120572(119899) is a

scalar step-size modifierAlso 119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus sum|119877119908|

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1) for all

119896 isin 119877119908 119896 = 119896

119908(119899)

Update link flows 119909119886(119899 + 1) = sum

119908isin119882sum119896isin119877119908

119891119896

119908(119899 + 1)120575

119908

119896119886

Step 6 (convergence test) Determine the total deviation ofmarginal social travel costs between all OD pairs 119864 =

sum119908isin119882

sum119896isin119877119908

(119891119896

119908(119899)119889119908) sdot |(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))119888

119896

119908(119899)| If 119864 le 120576

then stop Otherwise set 119899 = 119899 + 119897 and go to Step 2

For convenience it is better to keep 120572(119899) constant (ie120572(119899) = 120572) since 119904

119896

119908(119899) is used for scaling [26] Given any

starting set of path flows there exists 120572 such that if 120572 isin [0 120572]

the sequence generated by this algorithm converges to theobjective function (1) [37]

4 Numerical Example

In the section a numerical example is presented to illustratethe effectiveness of the proposed model and algorithm Thetest network is the Sioux Falls network which is a mediumsized network with 24 nodes and 76 links as shown inFigure 3 This paper considers only one origin-destinationpair fromnode 1 to node 20 for conveniently providing a com-plete picture of the travel pattern which demonstrates all used

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

For more realistically capturing the travelersrsquo path choicebehavior in response to toll charges Wu and Huang [23]assumed that the VOT varies significantly across individualsbecause of their different socioeconomic characteristics trippurposes attitudes and inherent preferences Furthermorethe VOT of each user is assumed to be deterministic and con-stant because the factors influencing VOT keep unchangedwithin a certain time period They extended the work ofYang andHuang [2] to the case with continuously distributedvalue of time across users for finding anonymous tolls torealize target flow pattern in networks with continuouslydistributed value of time To find anonymous tolls to realizesystemoptimum in cost unitsWu andHuang [23] proposed acost-based system optimum model in general networks withcontinuously distributed value of time but did not study itstraffic assignment algorithmsThe calculation of the model isvery difficult [24] It is formulated in a path-based form andcannot be solved by the Frank-Wolfe algorithmwhich is link-based and cannot provide path flows

In the past path-based algorithms even those not enu-merating all possible paths were traditionally discarded bytransportation researchers for solving large-scale networkproblems because of intensive memory requirements and thedifficulties in manipulating and storing paths [25] Howeverpath-based algorithms automatically provide not only thelink-flow solution but also the path-flow solution that maybe required in certain applications In light of magnitudeimprovement in the availability of computer memory inrecent years recent research on path-based algorithms hasdemonstrated and established that it is a viable approach fortraffic assignment problems with reasonably large networksizes [26ndash29] Chen et al [25] stated that much of theattention has been focused on two particular algorithms thedisaggregate simplicial decomposition (DSD) algorithm andthe gradient projection (GP) algorithm Application of GP tosolve the traffic assignment problem is relatively new but GPis as good as or better than DSD in direct comparisons

In this paper we develop a GP algorithm for the proposedcost-based system optimum model based on the Goldstein-Levitin-Polyak (GLP) GP method formulated by Bertsekas[30] for general nonlinear multicommodity problems whichis successfully used for solving traffic assignment problems[26] The proposed GLP algorithm includes five key opera-tions (1) assigning the flow on each path to the links along tofind the total link flows (2) computing link travel times andpath travel times and sorting path travel times in decreasingorder (3) computing the first derivative lengths (marginalsocial path travel costs) for all paths between each OD pairand finding the shortest first derivative lengths for eachOD pair (4) finding the second derivative lengths for eachOD pair and (5) updating the path flows using the secondderivative lengths as scaling

In the next section a system optimum problem in costunits is formulated in fixed demand networks with contin-uously distributed value of time In Section 3 we develop aGP algorithm for the cost-based system optimum model Anumerical example is presented in Section 4 for testing theGP algorithm Section 5 concludes the paper

Flow

x

120573w(x)

VOT

dw

Figure 1 Continuously distributed VOTs for users between eachOD pair 119908

2 Cost-Based System Optimum Problem

In this section we formulate the system optimum in costunits in a fixed demand network with heterogeneous users interms of a continuously distributed VOT Let 119866 = (119881 119860) bea directed network where 119881 is the set of nodes and 119860 the setof links Each link 119886 isin 119860 has an associated flow-dependenttravel time 119905

119886(V119886) which is assumed to be differentiable

convex andmonotonically increasing subject to the link flowV119886 Let 119882 be the set of all OD pairs 119877

119908the set of all paths

connecting OD pair 119908 isin 119882 and |119877119908| the number of paths

between each OD pair Let 119878119903119908be the set of drivers (users)

on path 119903 isin 119877119908and 119891

119903

119908the number of these drivers f =

( 119891119903

119908 ) We have V

119886= sum119908isin119882

sum119903isin119877119908

119891119903

119908120575119886119903

119908 119886 isin 119860 where

120575119886119903

119908equals 1 if path 119903 between OD pair 119908 contains link 119886

and 0 otherwise All users on link 119886 isin 119860 are charged by anexogenously given toll 120591

119886ge 0 Each path 119903 between each OD

pair 119908 is associated with a travel time 119905119903119908(f) and a travel cost

(dollar) 120591119903119908ge 0 119905119903119908(f) = sum

119886isin119860119905119886(V119886)120575119886119903

119908 and 120591

119903

119908= sum119886isin119860

120591119886120575119886119903

119908

Let 119889119908be the travel demand between each OD pair 119908

Instead of assuming a unified VOT for the whole popula-tion in this paper we set a unique specific VOT for eachtrip-maker Let the distribution of VOTs across individualsbetween each OD pair 119908 be characterized by a continuousfunction 120573

119908(119909) gt 0 Furthermore let the population between

each OD pair be ordered in decreasing order of their VOTsthat is 119889120573

119908(119909)119889119909 le 0 where 119909 is the 119909th user between each

OD pair 119908 as demonstrated in Figure 1 Note that here theVOT distribution is given In fact there exist many studieswhich address estimation of VOTs By observing choicesamong alternative combinations of cost and travel time basedon stated preference (SP) analysis due to the ability to controltime and cost variables [31] information about the relativeweighting of cost and time can be inferred and from this thedistribution of VOT can be derived In general literature onestimation of VOTs refers to two main models multinomiallogit model [32 33] and mixed logit model [34 35] Recentlythe mixed logit approach is popular because it does not have

Journal of Applied Mathematics 3

to assume the irrelevance of independent alternatives (IIA)property

Let 119895 = 1 2 |119877119908| denote the paths between each

OD pair 119908 satisfying 1199051

119908(f) ge 119905

2

119908(f) ge sdot sdot sdot ge 119905

|119877119908|

119908(f) The

system optimum naturally requires that the users with higherVOTs should choose faster paths and those with lower VOTschoose slower paths Otherwise the total travel cost can bereduced by switching a higher VOTuser on a slower path intoa faster path For the sake of notational consistency we definesum|119877119908|

119896=|119877119908|+1

119891119896

119908= 0 and sum

0

119896=1119891119896

119908= 0 The system optimum in

cost units can be formulated as the following minimizationproblem

minf

119885 (f) = sum

119908isin119882

|119877119908|

sum

119903=1

119905119903

119908(f) (int

119891119903

119908

0

120573119908(

|119877119908|

sum

119896=119903+1

119891119896

119908+ 119909)119889119909)

(1)

subject to

|119877119908|

sum

119896=1

119891119896

119908= 119889119908 (2)

119891119896

119908ge 0 (3)

where 119905119903

119908(f) = sum

119886isin119908119905119886(V119886)120575119886119903

119908 V119886

= sum119908isin119882

sum119903isin119877119908

119891119903

119908120575119886119903

119908

Recall that the users between each OD pair are arranged indecreasing order of their VOTs This determines the integralformulation appearing in (1)

The first-order optimality conditions of the above mini-mization problem are as follows

119905119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909 = 119892so119908cost

if 119891119903119908gt 0 119903 = 1 2

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(4)

119905119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909 ge 119892so119908cost

if 119891119903119908= 0 119903 = 1 2

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(5)

where 119892so119908cost is the minimal travel cost between each OD pair

119908

Note that the sum of the second and third terms on theleft hand side of (4) is the total externality caused by the119909th user of OD pair 119908 119909 = sum

|119877119908|

119896=119903119891119896

119908 or the 119891

119903

119908th user

of path 119903 The second term states that this new trip-makerimposes additional costs on all users who may belong toother OD pairs and other paths and have their own VOTs buttraverse the links of path 119903 This is the traditional congestionexternality reported in the literature The third term statesthat the costs of paths 1 to 119903 minus 1 are reduced since the 119891

119903

119908th

userrsquos path choice changes the VOTs of the users on thesepaths (this user chooses path 119903 rather than other longerpaths) This is another kind of externality attributed to theVOT distribution of OD trips Let

119888119903

119908(f) = 119905

119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

(6)Also for the remainder of the paper when we refer to the

first derivative lengths we mean the first derivatives of theobjective function which can also be regarded as marginalsocial travel cost corresponding to path flow on path 119903

Equations (4) and (5) can then be rewritten as119888119903

119908(f) = 119892

so119908cost if 119891119903

119908gt 0 119903 = 1 2 (

1003816100381610038161003816119877119908

1003816100381610038161003816minus 1)

119908 isin 119882

119888119903

119908(f) ge 119892

so119908cost if 119891119903

119908= 0 119903 = 1 2 (

1003816100381610038161003816119877119908

1003816100381610038161003816minus 1)

119908 isin 119882

(7)Therefore in a network where each user has a unique

VOT (7) state that at optimality the marginal social travelcosts on all the used paths connecting a given OD pair areequal and less than or equal to those on all the unused pathsThis is similar to a standard system optimum in which atoptimality the marginal social travel times on all the usedpaths between each given OD pair are equal and less than orequal to those on all the unused paths

Clearly the proposed model is more complicated thana standard system optimum According to the extremevalue theorem that a continuous function in the closed andbounded space attains itsmaximumandminimum the aboveminimization program must attain its minimum value Themost common algorithm used to solve traffic assignmentproblems is link-based Frank-Wolfe algorithm introduced byLeBlanc et al [36] However the objective function is ingeneral not convex since the path travel costs depending notonly on its own path but also on other paths are inseparableand asymmetric in terms of the path flows Fortunately for

4 Journal of Applied Mathematics

the differentiable and bounded objective the convexity of theconstraints allows the development of a gradient projectionalgorithm

3 Path-Based Traffic Assignment Algorithm

In this section we adopt the Goldstein-Levitin-Polyak algo-rithm to the traffic assignment problem In each iteration thetravel demand constraints (2) are eliminated by reformulatingthe path-flow variables in terms of nonshortest path flowsin terms of the first derivative lengths to make projectionoperation simpler This is implemented by partitioning 119891

119903

119908

into the shortest path flow 119891119903

119908and the nonshortest path flows

119891119903

119908and writing constraints (2) in terms of 119891119903

119908as follows

119891119903

119908= 119889119908minus

|119877119908|

sum

119903=1119903 = 119903

119891119903

119908 forall119908 isin 119882 (8)

It should be noted that 119903 is the shortest path in terms ofthe first derivative lengths but |119877

119908| is the shortest path in

terms of path travel times Putting constraints (8) into theobjective function we obtain a new formulation with just thenonnegativity constraints on the nonshortest path flows asthe decision variables Consider

min 119885(

f) (9)

119891119903

119908ge 0 forall119903 isin 119877

119908 119903 = 119903 119908 isin 119882 (10)

where 119885(

f) is the reformulated objective function onlyincluding the nonshortest path flows for all OD pairs Thisreformulation is a program with only nonnegativity con-straints The first and second derivatives of the new objectivefunction can be easily derived as follows

120597119885 (f)

120597119891119903

119908

=

120597119885 (f)120597119891119903

119908

minus

120597 (f)120597119891119903

119908

= 119888119903

119908(f) minus 119888

|119877119908|

119908(f)

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(11)

1205972

119885(f)

(120597119891119903

119908)2

=

1205972

119885 (f)(120597119891119903

119908)2minus 2

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

+

1205972

119885 (f)(120597119891119903

119908)

2

forall119903 isin 119877119908 119903 =

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(12)

where

1205972

119885 (f)(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)2

119889119909

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119896120575119908

119886119903

times int

119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908120597119891119903

119908

119889119909

1205972

119885 (f)(120597119891119903

119908)

2=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

times sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

Journal of Applied Mathematics 5

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

(13)

Putting (13) into (12) yields

1205972

119885(f)

(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891

119896

119908)

120597119891119903

119908

sum

119886isin119860

(119905119903

119908(f) minus 119905

119903

119908(f))

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ 2120573119908(

|119877119908|

sum

119896=119903

119891

119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

(120575119908

119886119903minus 120575119908

119886119903)2

120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

|119877119908|minus1

sum

119896=119903

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(14)

Note that the second derivative is in general not positiveThis is different from the transformed objective functionof the standard system optimum model [25] Thus in eachiteration the scaled GLP algorithm updates the path flowsaccording to the following iteration equations

119891119903

119908(119899 + 1) = [119891

119903

119908(119899) minus

120572 (119899)

119904119903

119908(119899)

(119888119903

119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899))]

+

if 119904119903119908(119899) gt 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899) 119908 isin 119882

50Demand

VOT

3

x

120573(x)

Figure 2 Continuously distributed VOTs of users

119891119903

119908(119899 + 1) = 119891

119903

119908(119899) if 119904119903

119908(119899) le 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899)

119908 isin 119882

119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus

|119877119908|

sum

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1)

(15)

where 119899 is the iteration number 120572(119899) is the step size119896119908(119899) is the shortest path in terms of the first derivative

lengths between each OD pair 119908 119904119903119908(119899) is a diagonal scaling

equivalent to the second derivative regarding path 119903 that is119904119903

119908(119899) = 120597

2

119885(f)(120597119891119903

119908)2 119888119903119908(f(119899)) and 119888

119896119908(119899)

119908(f(119899)) are the first

derivative lengths along path 119903 and path 119896119908(119899) between each

OD pair 119908 and []+ denotes the projection operation

Note that 119888119903119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899)) can be explained by (11)

and (15) certainly satisfies the flow conservation constraintsin (2) and nonnegativity conditions in (3)

With the above flow update equations the completealgorithmic steps can be summarized as follows

Step 1 (initialization) Set 119905119886(0) for all 119886 and perform all-or-

nothing assignments This yields path flows f119908(1) for all119908 isin

119882 and link flows 119909119886(1) for all 119886 Set iteration counter 119899 = 119897

Initialize the path-set 119877119908with the shortest path for each OD

pair 119908

Step 2 (update) Set 119905119886(119899) = 119905

119886(119909119886(119899)) for all 119886 Sort path

travel time in decreasing order that is 1199051119908(119899) ge 119905

2

119908(119899) ge sdot sdot sdot ge

119905|119877119908|

119908(119899) where 119905119896

119908(119899) = sum

119886isin119860119905119886(119909119886(119899))120575119908

119886119896 119896 = 1 2 |119877

119908|

Step 3 Update the first derivative lengths 119888119896119908(119899)of all the paths

119896 in 119877119908 for all119908 isin 119882

Step 4 (direction finding) Find the shortest paths 119896119908(119899) in

terms of marginal social travel cost between each OD pair 119908based on 119888

119896

119908(119899) If 119896

119908(119899) notin 119877

119908 then add it to 119877

119908and record

6 Journal of Applied Mathematics

1 2

3 4 5 6

9 8 7

12 11 10 16 18

17

14 15 19

23 22

13 24 21 20

22563

274

365

225

63

102947

171

418

66551

654

73

29077

665

51 29077

196

558

128

889

128889

840

09

16340

502

115

5517

21879

31218

584026

8839

61769

109

649

109649

16396

524

43

16396

555

17

105

888

137

106

128

889

153

502

167398

50211

577

49

52443

109649

Figure 3 The Sioux Falls test network

119888119896

119899

119908

119908(119899) Otherwise tag the shortest among the paths in 119877

119908as

119896119908(119899)

Step 5 (move) Set the new path flows as follows

119891119896

119908(119899 + 1) = max0 119891119896

119908(119899) minus

120572 (119899)

119904119896

119908(119899)

(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))

if 119904119903119908(119899) gt 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899) 119908 isin 119882

119891119896

119908(119899 + 1) = 119891

119896

119908(119899) if 119904119903

119908(119899) le 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899)

119908 isin 119882

(16)

where 119904119896

119908(119899) = 120597

2

119885(119891)(120597119891

119896

119908)

2

for all 119896 isin 119877119908 and 120572(119899) is a

scalar step-size modifierAlso 119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus sum|119877119908|

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1) for all

119896 isin 119877119908 119896 = 119896

119908(119899)

Update link flows 119909119886(119899 + 1) = sum

119908isin119882sum119896isin119877119908

119891119896

119908(119899 + 1)120575

119908

119896119886

Step 6 (convergence test) Determine the total deviation ofmarginal social travel costs between all OD pairs 119864 =

sum119908isin119882

sum119896isin119877119908

(119891119896

119908(119899)119889119908) sdot |(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))119888

119896

119908(119899)| If 119864 le 120576

then stop Otherwise set 119899 = 119899 + 119897 and go to Step 2

For convenience it is better to keep 120572(119899) constant (ie120572(119899) = 120572) since 119904

119896

119908(119899) is used for scaling [26] Given any

starting set of path flows there exists 120572 such that if 120572 isin [0 120572]

the sequence generated by this algorithm converges to theobjective function (1) [37]

4 Numerical Example

In the section a numerical example is presented to illustratethe effectiveness of the proposed model and algorithm Thetest network is the Sioux Falls network which is a mediumsized network with 24 nodes and 76 links as shown inFigure 3 This paper considers only one origin-destinationpair fromnode 1 to node 20 for conveniently providing a com-plete picture of the travel pattern which demonstrates all used

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

to assume the irrelevance of independent alternatives (IIA)property

Let 119895 = 1 2 |119877119908| denote the paths between each

OD pair 119908 satisfying 1199051

119908(f) ge 119905

2

119908(f) ge sdot sdot sdot ge 119905

|119877119908|

119908(f) The

system optimum naturally requires that the users with higherVOTs should choose faster paths and those with lower VOTschoose slower paths Otherwise the total travel cost can bereduced by switching a higher VOTuser on a slower path intoa faster path For the sake of notational consistency we definesum|119877119908|

119896=|119877119908|+1

119891119896

119908= 0 and sum

0

119896=1119891119896

119908= 0 The system optimum in

cost units can be formulated as the following minimizationproblem

minf

119885 (f) = sum

119908isin119882

|119877119908|

sum

119903=1

119905119903

119908(f) (int

119891119903

119908

0

120573119908(

|119877119908|

sum

119896=119903+1

119891119896

119908+ 119909)119889119909)

(1)

subject to

|119877119908|

sum

119896=1

119891119896

119908= 119889119908 (2)

119891119896

119908ge 0 (3)

where 119905119903

119908(f) = sum

119886isin119908119905119886(V119886)120575119886119903

119908 V119886

= sum119908isin119882

sum119903isin119877119908

119891119903

119908120575119886119903

119908

Recall that the users between each OD pair are arranged indecreasing order of their VOTs This determines the integralformulation appearing in (1)

The first-order optimality conditions of the above mini-mization problem are as follows

119905119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909 = 119892so119908cost

if 119891119903119908gt 0 119903 = 1 2

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(4)

119905119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909 ge 119892so119908cost

if 119891119903119908= 0 119903 = 1 2

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(5)

where 119892so119908cost is the minimal travel cost between each OD pair

119908

Note that the sum of the second and third terms on theleft hand side of (4) is the total externality caused by the119909th user of OD pair 119908 119909 = sum

|119877119908|

119896=119903119891119896

119908 or the 119891

119903

119908th user

of path 119903 The second term states that this new trip-makerimposes additional costs on all users who may belong toother OD pairs and other paths and have their own VOTs buttraverse the links of path 119903 This is the traditional congestionexternality reported in the literature The third term statesthat the costs of paths 1 to 119903 minus 1 are reduced since the 119891

119903

119908th

userrsquos path choice changes the VOTs of the users on thesepaths (this user chooses path 119903 rather than other longerpaths) This is another kind of externality attributed to theVOT distribution of OD trips Let

119888119903

119908(f) = 119905

119903

119908(f) 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) + sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

(6)Also for the remainder of the paper when we refer to the

first derivative lengths we mean the first derivatives of theobjective function which can also be regarded as marginalsocial travel cost corresponding to path flow on path 119903

Equations (4) and (5) can then be rewritten as119888119903

119908(f) = 119892

so119908cost if 119891119903

119908gt 0 119903 = 1 2 (

1003816100381610038161003816119877119908

1003816100381610038161003816minus 1)

119908 isin 119882

119888119903

119908(f) ge 119892

so119908cost if 119891119903

119908= 0 119903 = 1 2 (

1003816100381610038161003816119877119908

1003816100381610038161003816minus 1)

119908 isin 119882

(7)Therefore in a network where each user has a unique

VOT (7) state that at optimality the marginal social travelcosts on all the used paths connecting a given OD pair areequal and less than or equal to those on all the unused pathsThis is similar to a standard system optimum in which atoptimality the marginal social travel times on all the usedpaths between each given OD pair are equal and less than orequal to those on all the unused paths

Clearly the proposed model is more complicated thana standard system optimum According to the extremevalue theorem that a continuous function in the closed andbounded space attains itsmaximumandminimum the aboveminimization program must attain its minimum value Themost common algorithm used to solve traffic assignmentproblems is link-based Frank-Wolfe algorithm introduced byLeBlanc et al [36] However the objective function is ingeneral not convex since the path travel costs depending notonly on its own path but also on other paths are inseparableand asymmetric in terms of the path flows Fortunately for

4 Journal of Applied Mathematics

the differentiable and bounded objective the convexity of theconstraints allows the development of a gradient projectionalgorithm

3 Path-Based Traffic Assignment Algorithm

In this section we adopt the Goldstein-Levitin-Polyak algo-rithm to the traffic assignment problem In each iteration thetravel demand constraints (2) are eliminated by reformulatingthe path-flow variables in terms of nonshortest path flowsin terms of the first derivative lengths to make projectionoperation simpler This is implemented by partitioning 119891

119903

119908

into the shortest path flow 119891119903

119908and the nonshortest path flows

119891119903

119908and writing constraints (2) in terms of 119891119903

119908as follows

119891119903

119908= 119889119908minus

|119877119908|

sum

119903=1119903 = 119903

119891119903

119908 forall119908 isin 119882 (8)

It should be noted that 119903 is the shortest path in terms ofthe first derivative lengths but |119877

119908| is the shortest path in

terms of path travel times Putting constraints (8) into theobjective function we obtain a new formulation with just thenonnegativity constraints on the nonshortest path flows asthe decision variables Consider

min 119885(

f) (9)

119891119903

119908ge 0 forall119903 isin 119877

119908 119903 = 119903 119908 isin 119882 (10)

where 119885(

f) is the reformulated objective function onlyincluding the nonshortest path flows for all OD pairs Thisreformulation is a program with only nonnegativity con-straints The first and second derivatives of the new objectivefunction can be easily derived as follows

120597119885 (f)

120597119891119903

119908

=

120597119885 (f)120597119891119903

119908

minus

120597 (f)120597119891119903

119908

= 119888119903

119908(f) minus 119888

|119877119908|

119908(f)

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(11)

1205972

119885(f)

(120597119891119903

119908)2

=

1205972

119885 (f)(120597119891119903

119908)2minus 2

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

+

1205972

119885 (f)(120597119891119903

119908)

2

forall119903 isin 119877119908 119903 =

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(12)

where

1205972

119885 (f)(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)2

119889119909

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119896120575119908

119886119903

times int

119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908120597119891119903

119908

119889119909

1205972

119885 (f)(120597119891119903

119908)

2=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

times sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

Journal of Applied Mathematics 5

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

(13)

Putting (13) into (12) yields

1205972

119885(f)

(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891

119896

119908)

120597119891119903

119908

sum

119886isin119860

(119905119903

119908(f) minus 119905

119903

119908(f))

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ 2120573119908(

|119877119908|

sum

119896=119903

119891

119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

(120575119908

119886119903minus 120575119908

119886119903)2

120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

|119877119908|minus1

sum

119896=119903

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(14)

Note that the second derivative is in general not positiveThis is different from the transformed objective functionof the standard system optimum model [25] Thus in eachiteration the scaled GLP algorithm updates the path flowsaccording to the following iteration equations

119891119903

119908(119899 + 1) = [119891

119903

119908(119899) minus

120572 (119899)

119904119903

119908(119899)

(119888119903

119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899))]

+

if 119904119903119908(119899) gt 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899) 119908 isin 119882

50Demand

VOT

3

x

120573(x)

Figure 2 Continuously distributed VOTs of users

119891119903

119908(119899 + 1) = 119891

119903

119908(119899) if 119904119903

119908(119899) le 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899)

119908 isin 119882

119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus

|119877119908|

sum

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1)

(15)

where 119899 is the iteration number 120572(119899) is the step size119896119908(119899) is the shortest path in terms of the first derivative

lengths between each OD pair 119908 119904119903119908(119899) is a diagonal scaling

equivalent to the second derivative regarding path 119903 that is119904119903

119908(119899) = 120597

2

119885(f)(120597119891119903

119908)2 119888119903119908(f(119899)) and 119888

119896119908(119899)

119908(f(119899)) are the first

derivative lengths along path 119903 and path 119896119908(119899) between each

OD pair 119908 and []+ denotes the projection operation

Note that 119888119903119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899)) can be explained by (11)

and (15) certainly satisfies the flow conservation constraintsin (2) and nonnegativity conditions in (3)

With the above flow update equations the completealgorithmic steps can be summarized as follows

Step 1 (initialization) Set 119905119886(0) for all 119886 and perform all-or-

nothing assignments This yields path flows f119908(1) for all119908 isin

119882 and link flows 119909119886(1) for all 119886 Set iteration counter 119899 = 119897

Initialize the path-set 119877119908with the shortest path for each OD

pair 119908

Step 2 (update) Set 119905119886(119899) = 119905

119886(119909119886(119899)) for all 119886 Sort path

travel time in decreasing order that is 1199051119908(119899) ge 119905

2

119908(119899) ge sdot sdot sdot ge

119905|119877119908|

119908(119899) where 119905119896

119908(119899) = sum

119886isin119860119905119886(119909119886(119899))120575119908

119886119896 119896 = 1 2 |119877

119908|

Step 3 Update the first derivative lengths 119888119896119908(119899)of all the paths

119896 in 119877119908 for all119908 isin 119882

Step 4 (direction finding) Find the shortest paths 119896119908(119899) in

terms of marginal social travel cost between each OD pair 119908based on 119888

119896

119908(119899) If 119896

119908(119899) notin 119877

119908 then add it to 119877

119908and record

6 Journal of Applied Mathematics

1 2

3 4 5 6

9 8 7

12 11 10 16 18

17

14 15 19

23 22

13 24 21 20

22563

274

365

225

63

102947

171

418

66551

654

73

29077

665

51 29077

196

558

128

889

128889

840

09

16340

502

115

5517

21879

31218

584026

8839

61769

109

649

109649

16396

524

43

16396

555

17

105

888

137

106

128

889

153

502

167398

50211

577

49

52443

109649

Figure 3 The Sioux Falls test network

119888119896

119899

119908

119908(119899) Otherwise tag the shortest among the paths in 119877

119908as

119896119908(119899)

Step 5 (move) Set the new path flows as follows

119891119896

119908(119899 + 1) = max0 119891119896

119908(119899) minus

120572 (119899)

119904119896

119908(119899)

(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))

if 119904119903119908(119899) gt 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899) 119908 isin 119882

119891119896

119908(119899 + 1) = 119891

119896

119908(119899) if 119904119903

119908(119899) le 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899)

119908 isin 119882

(16)

where 119904119896

119908(119899) = 120597

2

119885(119891)(120597119891

119896

119908)

2

for all 119896 isin 119877119908 and 120572(119899) is a

scalar step-size modifierAlso 119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus sum|119877119908|

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1) for all

119896 isin 119877119908 119896 = 119896

119908(119899)

Update link flows 119909119886(119899 + 1) = sum

119908isin119882sum119896isin119877119908

119891119896

119908(119899 + 1)120575

119908

119896119886

Step 6 (convergence test) Determine the total deviation ofmarginal social travel costs between all OD pairs 119864 =

sum119908isin119882

sum119896isin119877119908

(119891119896

119908(119899)119889119908) sdot |(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))119888

119896

119908(119899)| If 119864 le 120576

then stop Otherwise set 119899 = 119899 + 119897 and go to Step 2

For convenience it is better to keep 120572(119899) constant (ie120572(119899) = 120572) since 119904

119896

119908(119899) is used for scaling [26] Given any

starting set of path flows there exists 120572 such that if 120572 isin [0 120572]

the sequence generated by this algorithm converges to theobjective function (1) [37]

4 Numerical Example

In the section a numerical example is presented to illustratethe effectiveness of the proposed model and algorithm Thetest network is the Sioux Falls network which is a mediumsized network with 24 nodes and 76 links as shown inFigure 3 This paper considers only one origin-destinationpair fromnode 1 to node 20 for conveniently providing a com-plete picture of the travel pattern which demonstrates all used

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

the differentiable and bounded objective the convexity of theconstraints allows the development of a gradient projectionalgorithm

3 Path-Based Traffic Assignment Algorithm

In this section we adopt the Goldstein-Levitin-Polyak algo-rithm to the traffic assignment problem In each iteration thetravel demand constraints (2) are eliminated by reformulatingthe path-flow variables in terms of nonshortest path flowsin terms of the first derivative lengths to make projectionoperation simpler This is implemented by partitioning 119891

119903

119908

into the shortest path flow 119891119903

119908and the nonshortest path flows

119891119903

119908and writing constraints (2) in terms of 119891119903

119908as follows

119891119903

119908= 119889119908minus

|119877119908|

sum

119903=1119903 = 119903

119891119903

119908 forall119908 isin 119882 (8)

It should be noted that 119903 is the shortest path in terms ofthe first derivative lengths but |119877

119908| is the shortest path in

terms of path travel times Putting constraints (8) into theobjective function we obtain a new formulation with just thenonnegativity constraints on the nonshortest path flows asthe decision variables Consider

min 119885(

f) (9)

119891119903

119908ge 0 forall119903 isin 119877

119908 119903 = 119903 119908 isin 119882 (10)

where 119885(

f) is the reformulated objective function onlyincluding the nonshortest path flows for all OD pairs Thisreformulation is a program with only nonnegativity con-straints The first and second derivatives of the new objectivefunction can be easily derived as follows

120597119885 (f)

120597119891119903

119908

=

120597119885 (f)120597119891119903

119908

minus

120597 (f)120597119891119903

119908

= 119888119903

119908(f) minus 119888

|119877119908|

119908(f)

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(11)

1205972

119885(f)

(120597119891119903

119908)2

=

1205972

119885 (f)(120597119891119903

119908)2minus 2

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

+

1205972

119885 (f)(120597119891119903

119908)

2

forall119903 isin 119877119908 119903 =

1003816100381610038161003816119877119908

1003816100381610038161003816 119908 isin 119882

(12)

where

1205972

119885 (f)(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)2

119889119909

1205972

119885 (f)120597119891119903

119908120597119891119903

119908

=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ 120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886119903

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119896120575119908

119886119903

times int

119891119896

119908

0

120597120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

120597119891119903

119908120597119891119903

119908

119889119909

1205972

119885 (f)(120597119891119903

119908)

2=

120597120573119908(sum|119877119908|

119896=119903119891119896

119908)

120597119891119903

119908

sum

119886isin119860

119905119886(V119886) 120575119908

119886119903

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903

times sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

120575119908

119886119903120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903120575119908

119886ℎ

Journal of Applied Mathematics 5

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

(13)

Putting (13) into (12) yields

1205972

119885(f)

(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891

119896

119908)

120597119891119903

119908

sum

119886isin119860

(119905119903

119908(f) minus 119905

119903

119908(f))

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ 2120573119908(

|119877119908|

sum

119896=119903

119891

119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

(120575119908

119886119903minus 120575119908

119886119903)2

120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

|119877119908|minus1

sum

119896=119903

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(14)

Note that the second derivative is in general not positiveThis is different from the transformed objective functionof the standard system optimum model [25] Thus in eachiteration the scaled GLP algorithm updates the path flowsaccording to the following iteration equations

119891119903

119908(119899 + 1) = [119891

119903

119908(119899) minus

120572 (119899)

119904119903

119908(119899)

(119888119903

119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899))]

+

if 119904119903119908(119899) gt 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899) 119908 isin 119882

50Demand

VOT

3

x

120573(x)

Figure 2 Continuously distributed VOTs of users

119891119903

119908(119899 + 1) = 119891

119903

119908(119899) if 119904119903

119908(119899) le 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899)

119908 isin 119882

119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus

|119877119908|

sum

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1)

(15)

where 119899 is the iteration number 120572(119899) is the step size119896119908(119899) is the shortest path in terms of the first derivative

lengths between each OD pair 119908 119904119903119908(119899) is a diagonal scaling

equivalent to the second derivative regarding path 119903 that is119904119903

119908(119899) = 120597

2

119885(f)(120597119891119903

119908)2 119888119903119908(f(119899)) and 119888

119896119908(119899)

119908(f(119899)) are the first

derivative lengths along path 119903 and path 119896119908(119899) between each

OD pair 119908 and []+ denotes the projection operation

Note that 119888119903119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899)) can be explained by (11)

and (15) certainly satisfies the flow conservation constraintsin (2) and nonnegativity conditions in (3)

With the above flow update equations the completealgorithmic steps can be summarized as follows

Step 1 (initialization) Set 119905119886(0) for all 119886 and perform all-or-

nothing assignments This yields path flows f119908(1) for all119908 isin

119882 and link flows 119909119886(1) for all 119886 Set iteration counter 119899 = 119897

Initialize the path-set 119877119908with the shortest path for each OD

pair 119908

Step 2 (update) Set 119905119886(119899) = 119905

119886(119909119886(119899)) for all 119886 Sort path

travel time in decreasing order that is 1199051119908(119899) ge 119905

2

119908(119899) ge sdot sdot sdot ge

119905|119877119908|

119908(119899) where 119905119896

119908(119899) = sum

119886isin119860119905119886(119909119886(119899))120575119908

119886119896 119896 = 1 2 |119877

119908|

Step 3 Update the first derivative lengths 119888119896119908(119899)of all the paths

119896 in 119877119908 for all119908 isin 119882

Step 4 (direction finding) Find the shortest paths 119896119908(119899) in

terms of marginal social travel cost between each OD pair 119908based on 119888

119896

119908(119899) If 119896

119908(119899) notin 119877

119908 then add it to 119877

119908and record

6 Journal of Applied Mathematics

1 2

3 4 5 6

9 8 7

12 11 10 16 18

17

14 15 19

23 22

13 24 21 20

22563

274

365

225

63

102947

171

418

66551

654

73

29077

665

51 29077

196

558

128

889

128889

840

09

16340

502

115

5517

21879

31218

584026

8839

61769

109

649

109649

16396

524

43

16396

555

17

105

888

137

106

128

889

153

502

167398

50211

577

49

52443

109649

Figure 3 The Sioux Falls test network

119888119896

119899

119908

119908(119899) Otherwise tag the shortest among the paths in 119877

119908as

119896119908(119899)

Step 5 (move) Set the new path flows as follows

119891119896

119908(119899 + 1) = max0 119891119896

119908(119899) minus

120572 (119899)

119904119896

119908(119899)

(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))

if 119904119903119908(119899) gt 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899) 119908 isin 119882

119891119896

119908(119899 + 1) = 119891

119896

119908(119899) if 119904119903

119908(119899) le 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899)

119908 isin 119882

(16)

where 119904119896

119908(119899) = 120597

2

119885(119891)(120597119891

119896

119908)

2

for all 119896 isin 119877119908 and 120572(119899) is a

scalar step-size modifierAlso 119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus sum|119877119908|

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1) for all

119896 isin 119877119908 119896 = 119896

119908(119899)

Update link flows 119909119886(119899 + 1) = sum

119908isin119882sum119896isin119877119908

119891119896

119908(119899 + 1)120575

119908

119896119886

Step 6 (convergence test) Determine the total deviation ofmarginal social travel costs between all OD pairs 119864 =

sum119908isin119882

sum119896isin119877119908

(119891119896

119908(119899)119889119908) sdot |(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))119888

119896

119908(119899)| If 119864 le 120576

then stop Otherwise set 119899 = 119899 + 119897 and go to Step 2

For convenience it is better to keep 120572(119899) constant (ie120572(119899) = 120572) since 119904

119896

119908(119899) is used for scaling [26] Given any

starting set of path flows there exists 120572 such that if 120572 isin [0 120572]

the sequence generated by this algorithm converges to theobjective function (1) [37]

4 Numerical Example

In the section a numerical example is presented to illustratethe effectiveness of the proposed model and algorithm Thetest network is the Sioux Falls network which is a mediumsized network with 24 nodes and 76 links as shown inFigure 3 This paper considers only one origin-destinationpair fromnode 1 to node 20 for conveniently providing a com-plete picture of the travel pattern which demonstrates all used

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

119903minus1

sum

119896=1

119905119896

119908(f) int119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

(13)

Putting (13) into (12) yields

1205972

119885(f)

(120597119891119903

119908)2

=

120597120573119908(sum|119877119908|

119896=119903119891

119896

119908)

120597119891119903

119908

sum

119886isin119860

(119905119903

119908(f) minus 119905

119903

119908(f))

+ 2120573119908(

|119877119908|

sum

119896=119903

119891119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ 2120573119908(

|119877119908|

sum

119896=119903

119891

119896

119908) sum

119886isin119860

119889119905119886(V119886)

119889V119886

120575119908

119886119903(1 minus 120575

119908

119886119903)

+ sum

119908isin119882

|119877119908|

sum

ℎ=1

sum

119886isin119860

1198892

119905119886(V119886)

(119889V119886)2

(120575119908

119886119903minus 120575119908

119886119903)2

120575119908

119886ℎ

times int

119891ℎ

119908

0

120573119908(

|119877119908|

sum

119896=ℎ+1

119891119896

119908+ 119909)119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+ 2

119903minus1

sum

ℎ=1

sum

119886isin119860

119889119905119886(V119886)

119889V119886

(120575119908

119886119903minus 120575119908

119886119903) 120575119908

119886ℎ

times int

119891ℎ

119908

0

120597120573119908(sum|119877119908|

119896=ℎ+1119891119896

119908+ 119909)

120597119891119903

119908

119889119909

+

|119877119908|minus1

sum

119896=119903

119905119896

119908int

119891119896

119908

0

1205972

120573119908(sum|119877119908|

ℎ=119896+1119891ℎ

119908+ 119909)

(120597119891119903

119908)

2119889119909

forall119903 isin 119877119908 119903 = 119903 119908 isin 119882

(14)

Note that the second derivative is in general not positiveThis is different from the transformed objective functionof the standard system optimum model [25] Thus in eachiteration the scaled GLP algorithm updates the path flowsaccording to the following iteration equations

119891119903

119908(119899 + 1) = [119891

119903

119908(119899) minus

120572 (119899)

119904119903

119908(119899)

(119888119903

119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899))]

+

if 119904119903119908(119899) gt 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899) 119908 isin 119882

50Demand

VOT

3

x

120573(x)

Figure 2 Continuously distributed VOTs of users

119891119903

119908(119899 + 1) = 119891

119903

119908(119899) if 119904119903

119908(119899) le 0 forall119903 isin 119877

119908 119903 = 119896

119908(119899)

119908 isin 119882

119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus

|119877119908|

sum

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1)

(15)

where 119899 is the iteration number 120572(119899) is the step size119896119908(119899) is the shortest path in terms of the first derivative

lengths between each OD pair 119908 119904119903119908(119899) is a diagonal scaling

equivalent to the second derivative regarding path 119903 that is119904119903

119908(119899) = 120597

2

119885(f)(120597119891119903

119908)2 119888119903119908(f(119899)) and 119888

119896119908(119899)

119908(f(119899)) are the first

derivative lengths along path 119903 and path 119896119908(119899) between each

OD pair 119908 and []+ denotes the projection operation

Note that 119888119903119908(f(119899)) minus 119888

119896119908(119899)

119908(f(119899)) can be explained by (11)

and (15) certainly satisfies the flow conservation constraintsin (2) and nonnegativity conditions in (3)

With the above flow update equations the completealgorithmic steps can be summarized as follows

Step 1 (initialization) Set 119905119886(0) for all 119886 and perform all-or-

nothing assignments This yields path flows f119908(1) for all119908 isin

119882 and link flows 119909119886(1) for all 119886 Set iteration counter 119899 = 119897

Initialize the path-set 119877119908with the shortest path for each OD

pair 119908

Step 2 (update) Set 119905119886(119899) = 119905

119886(119909119886(119899)) for all 119886 Sort path

travel time in decreasing order that is 1199051119908(119899) ge 119905

2

119908(119899) ge sdot sdot sdot ge

119905|119877119908|

119908(119899) where 119905119896

119908(119899) = sum

119886isin119860119905119886(119909119886(119899))120575119908

119886119896 119896 = 1 2 |119877

119908|

Step 3 Update the first derivative lengths 119888119896119908(119899)of all the paths

119896 in 119877119908 for all119908 isin 119882

Step 4 (direction finding) Find the shortest paths 119896119908(119899) in

terms of marginal social travel cost between each OD pair 119908based on 119888

119896

119908(119899) If 119896

119908(119899) notin 119877

119908 then add it to 119877

119908and record

6 Journal of Applied Mathematics

1 2

3 4 5 6

9 8 7

12 11 10 16 18

17

14 15 19

23 22

13 24 21 20

22563

274

365

225

63

102947

171

418

66551

654

73

29077

665

51 29077

196

558

128

889

128889

840

09

16340

502

115

5517

21879

31218

584026

8839

61769

109

649

109649

16396

524

43

16396

555

17

105

888

137

106

128

889

153

502

167398

50211

577

49

52443

109649

Figure 3 The Sioux Falls test network

119888119896

119899

119908

119908(119899) Otherwise tag the shortest among the paths in 119877

119908as

119896119908(119899)

Step 5 (move) Set the new path flows as follows

119891119896

119908(119899 + 1) = max0 119891119896

119908(119899) minus

120572 (119899)

119904119896

119908(119899)

(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))

if 119904119903119908(119899) gt 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899) 119908 isin 119882

119891119896

119908(119899 + 1) = 119891

119896

119908(119899) if 119904119903

119908(119899) le 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899)

119908 isin 119882

(16)

where 119904119896

119908(119899) = 120597

2

119885(119891)(120597119891

119896

119908)

2

for all 119896 isin 119877119908 and 120572(119899) is a

scalar step-size modifierAlso 119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus sum|119877119908|

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1) for all

119896 isin 119877119908 119896 = 119896

119908(119899)

Update link flows 119909119886(119899 + 1) = sum

119908isin119882sum119896isin119877119908

119891119896

119908(119899 + 1)120575

119908

119896119886

Step 6 (convergence test) Determine the total deviation ofmarginal social travel costs between all OD pairs 119864 =

sum119908isin119882

sum119896isin119877119908

(119891119896

119908(119899)119889119908) sdot |(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))119888

119896

119908(119899)| If 119864 le 120576

then stop Otherwise set 119899 = 119899 + 119897 and go to Step 2

For convenience it is better to keep 120572(119899) constant (ie120572(119899) = 120572) since 119904

119896

119908(119899) is used for scaling [26] Given any

starting set of path flows there exists 120572 such that if 120572 isin [0 120572]

the sequence generated by this algorithm converges to theobjective function (1) [37]

4 Numerical Example

In the section a numerical example is presented to illustratethe effectiveness of the proposed model and algorithm Thetest network is the Sioux Falls network which is a mediumsized network with 24 nodes and 76 links as shown inFigure 3 This paper considers only one origin-destinationpair fromnode 1 to node 20 for conveniently providing a com-plete picture of the travel pattern which demonstrates all used

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

1 2

3 4 5 6

9 8 7

12 11 10 16 18

17

14 15 19

23 22

13 24 21 20

22563

274

365

225

63

102947

171

418

66551

654

73

29077

665

51 29077

196

558

128

889

128889

840

09

16340

502

115

5517

21879

31218

584026

8839

61769

109

649

109649

16396

524

43

16396

555

17

105

888

137

106

128

889

153

502

167398

50211

577

49

52443

109649

Figure 3 The Sioux Falls test network

119888119896

119899

119908

119908(119899) Otherwise tag the shortest among the paths in 119877

119908as

119896119908(119899)

Step 5 (move) Set the new path flows as follows

119891119896

119908(119899 + 1) = max0 119891119896

119908(119899) minus

120572 (119899)

119904119896

119908(119899)

(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))

if 119904119903119908(119899) gt 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899) 119908 isin 119882

119891119896

119908(119899 + 1) = 119891

119896

119908(119899) if 119904119903

119908(119899) le 0 forall119896 isin 119877

119908 119896 = 119896

119908(119899)

119908 isin 119882

(16)

where 119904119896

119908(119899) = 120597

2

119885(119891)(120597119891

119896

119908)

2

for all 119896 isin 119877119908 and 120572(119899) is a

scalar step-size modifierAlso 119891119896119908(119899)

119908(119899 + 1) = 119889

119908minus sum|119877119908|

119903=1119903 = 119896119908(119899)

119891119903

119908(119899 + 1) for all

119896 isin 119877119908 119896 = 119896

119908(119899)

Update link flows 119909119886(119899 + 1) = sum

119908isin119882sum119896isin119877119908

119891119896

119908(119899 + 1)120575

119908

119896119886

Step 6 (convergence test) Determine the total deviation ofmarginal social travel costs between all OD pairs 119864 =

sum119908isin119882

sum119896isin119877119908

(119891119896

119908(119899)119889119908) sdot |(119888119896

119908(119899) minus 119888

119896119908(119899)

119908(119899))119888

119896

119908(119899)| If 119864 le 120576

then stop Otherwise set 119899 = 119899 + 119897 and go to Step 2

For convenience it is better to keep 120572(119899) constant (ie120572(119899) = 120572) since 119904

119896

119908(119899) is used for scaling [26] Given any

starting set of path flows there exists 120572 such that if 120572 isin [0 120572]

the sequence generated by this algorithm converges to theobjective function (1) [37]

4 Numerical Example

In the section a numerical example is presented to illustratethe effectiveness of the proposed model and algorithm Thetest network is the Sioux Falls network which is a mediumsized network with 24 nodes and 76 links as shown inFigure 3 This paper considers only one origin-destinationpair fromnode 1 to node 20 for conveniently providing a com-plete picture of the travel pattern which demonstrates all used

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Table 1 Generated paths path costs and VOTs

Paths Link set Path flow Path cost VOTPath 1 1-2-6-5-4-11-14-23-22-21-20 29481 1667692 [10000 11179]Path 2 1-2-6-8-7-18-20 128851 1634423 [11179 19039]Path 3 1-2-6-8-16-17-19-20 67645 1512485 [16333 19039]Path 4 1-3-4-11-14-23-22-21-20 09821 1248157 [19039 19432]Path 5 1-3-4-5-9-8-16-17-19-20 16310 1229879 [19432 20084]Path 6 1-3-4-11-10-17-19-20 26448 1227562 [20084 21142]Path 7 1-3-12-11-14-23-22-21-20 13288 1216256 [21142 21674]Path 8 1-3-4-5-9-10-15-22-20 50010 1211957 [21674 23674]Path 9 1-3-12-11-14-15-19-20 16379 1209318 [23674 24329]Path 10 1-3-12-11-10-16-17-19-20 21797 1205415 [24329 25201]Path 11 1-3-12-11-10-17-19-20 04598 1195662 [25201 25385]Path 12 1-3-12-11-10-15-22-21-20 05293 1192367 [25385 25597]Path 13 1-3-12-13-24-21-20 110078 1178441 [25597 30000]

path flows all used link flows and path choice behaviorsWe here use the standard Bureau of Public Road (BPR) linkcost function for our numerical studyThe functional form isgiven by

119905119886(119909119886) = 119905119891(1 + 015(

119909119886

119862119886

)

4

) (17)

where 119905119891is the free-flow cost 119909

119886is the flow and119862

119886is the link

capacityAssume that the total demand 119889

12is 50 and all network

users which are differed by a continuously distributed VOTare given in Figure 2 as follows

120573 (119909) = 3 minus

119909

25

119909 isin [0 50] (18)

Note that in the numerical example we show that 120572 = 1

achieves very good convergence rateThe GP algorithm provides a complete picture of the

travel pattern and keeps track of the distribution of the ODflows among the different routes as shown in Table 1 andFigure 3 In the Sioux Falls network there are many pathsbetween OD pair 1ndash20 However the number of the usedpaths to define the system optimum keeps small that is thereare only thirteen used paths which are given in a decreasingorder in terms of path travel times It can be seen thatat optimality people with high VOTs would choose fasterpaths whereas people with low VOTs would choose slowerpathsThis is consistent with the requirements for the systemoptimum in Section 2

Figure 3 shows the optimal link-flow distribution Notethat the real lines refer to the used links and the dottedlines refer to the unused paths The link flows equal thevalues beside the corresponding links and are also graphicallydemonstrated by the widths of the corresponding lines It canbe easily seen that in the network most links in the forwarddirections are used butmost links in the backward directionsare unused

Table 2 compares the total travel times and total travelcosts under different types of traffic assignment respectively

Table 2 Total travel times and total travel costs under different typesof traffic assignment

Type of traffic assignment Total traveltime

Total travelcost

User equilibrium 6827 13655Time-based system optimum 6723 13343Cost-based system optimum 6912 13281

At the cost-based system optimum the total travel cost is thelowest whereas the total travel time is the highest At the time-based system optimum the total travel time is the lowestand the total travel cost is more than the cost-based systemoptimum but less than the user equilibrium

Figure 4 depicts the pattern of convergence toward theminimum This convergence pattern is demonstrated interms of the reduction in the value of the objective functionfrom iteration to iteration After the 21st iteration themarginal contribution of each successive iteration becomessmaller and smaller as the algorithm proceeds (this propertyis the basis for the convergence criterion used in the example)After the 40th iteration the value of the objective functionalmost keeps unchanged and is approximately equal to theminimum value 13281 verifying the effectiveness of theproposed GP algorithm

5 Conclusions

The VOT varies significantly across individuals because ofthe different socioeconomic characteristics trip purposesattitudes and inherent preferences Furthermore the VOT ofeach user is assumed to be deterministic and constant becausethe factors influencing VOT keep unchanged within a certaintime period We provide a theoretical investigation of thesystem optimum problem in fixed demand networks withcontinuous VOT distributionThis system optimumproblemis formulated as a minimization program in cost units whichis more complicated than standard system optimum This isbecause its objective function is in general nonconvex since

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 701

15

2

25

3

35

4

Iteration number

Tota

l tra

vel c

ost

times104

Figure 4 The objective function values versus the iterations of theGP algorithm

path travel costs depending not only on its own path but alsoon other paths are inseparable and asymmetric in terms of thepath flows Considering the convexity of the constraints wehave developed a path-based GLP algorithm for solving thismodel The test results in the Sioux Falls network show theeffectiveness of the algorithm

This study lays a solid foundation for road pricing to real-ize the cost-based system optimum in fixed demand generalnetworks with heterogeneous users For future research thisframework can be extended to the case of elastic demandand further investigate how to determine anonymous tolls torealize the system optimum in cost units

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by grants from the NationalBasic Research Program of China (2012CB725401) theNational Natural Science Foundation of China (71071004and 71271004) and the National High Technology Researchand Development Program of China (863 Program)(2012AA112401)

References

[1] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo Proceedings of the Institute of Civil Engineers II no1 pp 325ndash378 1952

[2] H Yang and H-J Huang ldquoThe multi-class multi-criteriatraffic network equilibrium and systems optimum problemrdquoTransportation Research B vol 38 no 1 pp 1ndash15 2004

[3] K A Small and J Yan ldquoThe value of ldquovalue pricingrdquo of roadssecond-best pricing and product differentiationrdquo Journal ofUrban Economics vol 49 no 2 pp 310ndash336 2001

[4] D Brownstone and K A Small ldquoValuing time and reliabilityassessing the evidence from road pricing demonstrationsrdquoTransportation Research A vol 39 no 4 pp 279ndash293 2005

[5] K A Small C Winston and J Yan ldquoUncovering the distri-bution of motoristsrsquo preferences for travel time and reliabilityrdquoEconometrica vol 73 no 4 pp 1367ndash1382 2005

[6] C Cirillo and KW Axhausen ldquoEvidence on the distribution ofvalues of travel time savings from a six-week diaryrdquo Transporta-tion Research A vol 40 no 5 pp 444ndash457 2006

[7] C-C Lu H S Mahmassani and X Zhou ldquoA bi-criteriondynamic user equilibrium traffic assignment model and solu-tion algorithm for evaluating dynamic road pricing strategiesrdquoTransportation Research C vol 16 no 4 pp 371ndash389 2008

[8] R Lindsey ldquoExistence uniqueness and trip cost functionproperties of user equilibrium in the bottleneck model withmultiple user classesrdquo Transportation Science vol 38 no 3 pp293ndash314 2004

[9] D Han and H Yang ldquoThe multi-class multi-criterion trafficequilibrium and the efficiency of congestion pricingrdquo Trans-portation Research E vol 44 no 5 pp 753ndash773 2008

[10] A Clark A Sumalee S Shepherd and R Connors ldquoOn theexistence and uniqueness of first best tolls in networks withmultiple user classes and elastic demandrdquoTransportmetrica vol5 no 2 pp 141ndash157 2009

[11] T C Lam and K A Small ldquoThe value of time and reliabilitymeasurement from a value pricing experimentrdquo TransportationResearch E vol 37 no 2-3 pp 231ndash251 2001

[12] E T Verhoef andK A Small ldquoProduct differentiation on roadsconstrained congestion pricingwith heterogeneous usersrdquo Jour-nal of Transport Economics and Policy vol 38 no 1 pp 127ndash1562004

[13] P Marcotte and D L Zhu ldquoExistence and computation ofoptimal tolls in multiclass network equilibrium problemsrdquoOperations Research Letters vol 37 no 3 pp 211ndash214 2009

[14] V van den Berg and E T Verhoef ldquoCongestion tolling inthe bottleneck model with heterogeneous values of timerdquoTransportation Research B vol 45 no 1 pp 60ndash78 2011

[15] D Zhu C Li and G Chen ldquoExistence of strongly valid tolls formulticlass network equilibrium problemsrdquo Acta MathematicaScientia B vol 32 no 3 pp 1093ndash1101 2012

[16] JMayet andMHansen ldquoCongestion pricingwith continuouslydistributed values of timerdquo Journal of Transport Economics andPolicy vol 34 no 3 pp 359ndash369 2000

[17] F Xiao and H Yang ldquoEfficiency loss of private road withcontinuously distributed value-of-timerdquo Transportmetrica vol4 no 1 pp 19ndash32 2008

[18] F Xiao and H M Zhang ldquoPareto-improving and self-sustainable pricing for themorning commute with nonidenticalcommutersrdquo Transportation Science pp 1ndash11 2013

[19] L J Tian H J Huang and H Yang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[20] R Cole Y Dodis and T Roughgarden ldquoPricing network edgesfor heterogeneous selfish usersrdquo in Proceedings of the 35 AnnualACM Symposium on Theory of Computing pp 521ndash530 ACMSan Diego Calif USA 2003

[21] L Fleischer K Jain and M Mahdian ldquoTolls for heterogeneousselfish users in multicommodity networks and generalizedcongestion gamesrdquo in Proceedings of the 45th Annual IEEESymposium on Foundations of Computer Science (FOCS rsquo04) pp277ndash285 Rome Italy October 2004

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

[22] G Karakostas and S G Kolliopoulos ldquoEdge pricing of mul-ticommodity networks for heterogeneous selfish usersrdquo inProceedings of the 45th Annual IEEE Symposium on Foundationsof Computer Science (FOCS rsquo04) pp 268ndash276 Rome ItalyOctober 2004

[23] W X Wu and H J Huang ldquoFinding anonymous tolls to realizetarget flow pattern in networks with continuously distributedvalue of timerdquo submitted to Transportation Research B 2013

[24] R B Dial ldquoNetwork-optimized road pricing part II algorithmsand examplesrdquo Operations Research vol 47 no 2 pp 327ndash3361999

[25] A Chen D-H Lee and R Jayakrishnan ldquoComputational studyof state-of-the-art path-based traffic assignment algorithmsrdquoMathematics and Computers in Simulation vol 59 no 6 pp509ndash518 2002

[26] R JayakrishnanW K Tsai J N Prashker and S RajadhyakshaldquoFaster path-based algorithm for traffic assignmentrdquo Trans-portation Research Record no 1443 pp 75ndash83 1994

[27] T Larsson and M Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment prob-lemrdquo Transportation Science vol 26 no 1 pp 4ndash17 1992

[28] C Sun R Jayakrishnan and W K Tsai ldquoComputational studyof a path-based algorithm and its variants for static trafficassignmentrdquo Transportation Research Record no 1537 pp 106ndash115 1996

[29] M Tatineni H Edwards and D Boyce ldquoComparison of disag-gregate simplicial decomposition and Frank-Wolfe algorithmsfor user-optimal route choicerdquo Transportation Research Recordno 1617 pp 157ndash162 1998

[30] D P Bertsekas ldquoOn the Goldstein-Levitin-Polyak gradientprojection methodrdquo IEEE Transactions on Automatic Controlvol 21 no 2 pp 174ndash184 1976

[31] H Gunn ldquoAn introduction to the valuation of travel-timesavings and losserdquo inHandbook of TransportModelling ElsevierScience 2000

[32] D Brownstone andK Train ldquoForecasting newproduct penetra-tionwith flexible substitution patternsrdquo Journal of Econometricsvol 89 no 1-2 pp 109ndash129 1998

[33] K E Train ldquoRecreation demand models with taste differencesover peoplerdquo Land Economics vol 74 no 2 pp 230ndash239 1998

[34] S Algers P Bergstrom M Dahlberg and J L Dillen ldquoMixedlogit estimation of the value of travel timerdquo Uppsala-WorkingPaper Series 199815 1998

[35] S Hess M Bierlaire and J W Polak ldquoEstimation of value oftravel-time savings using mixed logit modelsrdquo TransportationResearch A vol 39 no 2-3 pp 221ndash236 2005

[36] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium traffic assign-ment problemrdquo Transportation Research vol 9 no 5 pp 309ndash318 1975

[37] D Bertsekas and R Gallager Data Networks Prentice HallEnglewood Cliffs NJ USA 2nd edition 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of