Research ArticleTransportation Network Design considering Morning andEvening Peak-Hour Demands

Hua Wang1 Gui-Yuan Xiao2 Li-Ye Zhang3 and Yangbeibei Ji4

1 School of Economics and Management Tongji University Shanghai 200096 China2 College of Civil and Architectural Engineering Guilin University of Technology Guilin 541004 China3 School of Traffic and Transportation Engineering Changsha University of Science and Technology Changsha 410076 China4 School of Management Shanghai University Shanghai 200444 China

Correspondence should be addressed to Li-Ye Zhang liyezhangufgmailcom

Received 18 December 2013 Accepted 17 January 2014 Published 10 March 2014

Academic Editor X Zhang

Copyright copy 2014 Hua Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Previous studies of transportation network design problem (NDP) always consider one peak-hour origin-destination (O-D)demand distribution However the NDP based on one peak-hour O-D demand matrix might be unable to model the real trafficpatterns due to diverse traffic characteristics in themorning and evening peaks and impacts of network structure and link sensitivityThis paper proposes anNDPmodel simultaneously considering bothmorning and evening peak-hour demandsTheNDP problemis formulated as a bilevel programming model where the upper level is to minimize the weighted sum of total travel time fornetwork users travelling in both morning and evening commute peaks and the lower level is to characterize user equilibriumchoice behaviors of the travelers in two peaks The proposed NDP model is transformed into an equivalent mixed integer linearprogramming (MILP) which can be efficiently solved by optimization solvers Numerical examples are finally performed todemonstrate the effectiveness of the developed model It is shown that the proposed NDP model has more promising design effectof improving network efficiency than the traditional NDP model considering one peak-hour demand and avoids the misleadingselection of improved links

1 Introduction

The transportation network design problem (NDP) is char-acterized as one of the most important and challengingoptimization problems in the transportation system [1] Itaims to improve the network system efficiency by expandinglink capacities on existing roads or building new roadslanesfor the network In general the NDP problem can be wellformulated by a bilevel programming model in which theStackelberg behavior between network planner and networkusers is characterizedThe upper level problem is to optimizea set of design objectives (eg minimizing total travel time)for the urban transportation system by setting link capacityexpansion scheme with some necessary constraints (egfinancial budget constraint) The lower level problem is todescribe the userrsquos behavior in terms of pathmode choicedeparture time choice and origindestination choice

After a pioneering work of Abdulaal and Leblanc [2]more and more researchers have paid attention to the NDPproblem with focus on advanced model formulation andeffective algorithm design The achievements made beforelast century on the NDP studies can be found in twocomprehensive reviews Magnanti and Wong [3] and Yangand Bell [4] In recent decade we have also witnessed a largenumber of emerging advances on theNDP studiesThese newadvances are pertinent to uncertain parameters (eg stochas-tic demand and capacity variation) sustainability constraintsand distinct time dimension Uncertainty is one underlyingand important feature of travel activity and plays a critical rolein network design and planning policy To avoid unnecessaryrisks and possibly misleading outcomes demand uncertaintyandor capacity variation have been taken into account inthe NDP problem (eg [5 6]) by introducing expectedvalue model and robust optimization model Two typical

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 806916 10 pageshttpdxdoiorg1011552014806916

2 Mathematical Problems in Engineering

approaches are used to model the stochastic traffic flowsunder uncertain traffic conditions One is to develop atwo-stage formulation by means of scenario-constructionapproach where a number of finite scenarios of uncertainparameters with known probability distribution will be gen-erated (eg [7 8]) Another way called probability-analyticalmethod is to derive a reliability-based traffic assignmentwhere the users follow probabilistic user equilibrium underuncertain traffic conditions (eg [9ndash11])

Time dimension as another important factor has alsoattracted the researchersrsquo attention in recent NDP studiesThree scales of time dimension can be considered in theNDP problem short-termreal-time (seconds) medium-term (days) and long-term (years) The NDP consideringtimedimension is defined as time-dependentNDPwhich canbe specifically classified into multiperiod NDP and dynamicNDP according to different scales of time dimension Theformer is generally to analyze the NDP problem taking intoaccount a long-term spanning construction andmaintenanceof transport infrastructure [12ndash15] The latter is to preciselycharacterize the real-time trafficdynamics andunsteady-stateconditions in the NDP by introducing dynamic traffic assign-ment Hopefully it can be used to estimate microinteractionamong adjacent links and thereby identify and examine thepossible bottleneck [6 16 17] A comprehensive review ofrecent developments on the NDP studies can be referred toMeng et al [18]

By retrieving and reviewing theNDP literature especiallythe valuable reviews across several decades [3 4 18 19] it canbe found that almost all NDP studies assume that an NDPmodel based on one peak-hour O-D demand distribution iscapable and effective to explain the real traffic patterns Forurban traffic there always are two peaks of a daily trafficflow corresponding tomorning commute and even commuterespectively onweekday [20] For commuting travel trips therepeatability of hourly fluctuation of traffic flow gives rise to acomparative stable peak-hour demand But such repeatabilityof hourly variation of traffic flow does not mean the same O-D demand distributions in the morning and evening peaks

Although the daily commuters have high repetitive travelactivities the peak-hour O-D demand distribution on thenetwork will be different in the morning and evening peaksThis fact has been recognized and emphasized in the trans-portation network modeling problem such as commutingpattern analysis [21] and road tolling and parking fee opti-mization [22 23] It is always assumed that the O-D demandmatrix is asymmetric diagonal for the morning and evenpeaks namely 119902

119900119889= 119902119889119900 if only daily regular commuters

are consideredThe O-D demandmatrix used in the networkplanning and design comes up with the peak-hour trafficsurvey data (eg link traffic volumes) in either morning orevening peak In practice the traffic volumes in the morningpeak may largely differ from that of evening peak commutewhich has been demonstrated by some empirical studies(eg [20 24]) The O-D demand matrices thus may also bedifferent for two peaks because they are estimated based onthe corresponding peak-hour traffic volumes In other wordsthe travelers in two commuting peaks would make differenttravel decision in terms of both path choice and departure

1

1

2

2

Figure 1 An illustrative example

time choice [25] Therefore the NDP considering one peak-hour O-D demand might be misleading and even incorrectOn the one hand the O-D demand distribution on a realnetwork is asymmetric due to different traffic patterns inthe morning and evening peaks though the morning andevening round commuting trips can be assumed to be equalWe should overlook demand uncertainty and diverse spatial-temporal characteristic of travel activities in two peaks Forexample the noncommuting trips in morning peak (egshopping going to hospital and tourism) may not returnback to the origin places in evening peak Similarly partof activities for the travelers who travel in evening peak donot commence in the morning and evening peaks but othertraveling periods Therefore which peak-hour commutepattern should be used to estimate the O-D demand matrixis a controversial and intractable issue

On the other hand network structure and link sensitivitycould have important impact on the network performanceevaluation for two peaks Even though the O-D pairs andtheir demands can be assumed to be symmetric in two peaksnamely 119902

119900119889= 119902119889119900 the path choices in two commute peaks

may be still different which largely depends on the networktopological structure If different pathslinks are chosen intwo peaks network situation in terms of traffic congestionwould show large difference between two peaks In thisrespect the NDP considering one peak-hour O-D demandmay generate misleading and even wrong outcomes Moreimportantly link sensitivity of the NDP effect should not beunderestimated and overlooked under different peaks Takea simple network comprising of two links shown in Figure 1as an example Assume that 119902

119900119889in the morning peak is

equal to 119902119889119900

in the evening peak Consider two O-D pairs11990212= 30 119902

21= 10 in the morning peak and 119902

12= 10

11990221= 30 in the evening peak Links 1 and 2 have equal

capacities of 20 vehh and free flow travel time of 1 minuteThe available budget invested on link capacity expansion isset as 20 equivalently BPR function is used to measure thelink travel time performance

It is not difficult to see that link 1 will be improved byadding 20 units in the NDP considering morning peak-hourdemand and link 2 will be improved by expanding 20 unitsin the NDP considering evening peak-hour demand Thetotal travel times on the network for these two NDPs areequal namely 104305 minutes In the NDP simultaneouslyconsidering morning and evening peak-hour demands bothlinks 1 and 2 will be improved by adding 10 units respectivelyand thereby the total travel time on the network will bereduced to 89037 minutes Evidently the NDP consideringone peak-hour demand matrix is not the promising scheme

Mathematical Problems in Engineering 3

to improve transportation system performance Therefore itis necessary to develop a new network design model in orderto avoid misleading decision-making

To address the above problems we develop a networkdesign model simultaneously taking into account morningand evening peak-hour demandsThe traditionalNDPmodelconsidering one peak-hour O-D demand has been regardedas a special case of the proposed model The proposed modelis formulated as a bilevel programming model with objectiveto minimize the weighted sum of total travel times of thetraffic patterns in the morning and evening peaks In thelower level network users are assumed to follow the userequilibrium principle of their route choice decision Theproposed model is finally transformed into an equivalentmixed integer linear programming (MILP) so that a globaloptimum can be obtained by means of MIP solvers (egCPLEX)

The rest of the paper is organized as follows In the nextsection the new NDP model is formulated by simultane-ously considering the morning and evening peal-hour O-Ddemand distributions Then an equivalent MILP model isintroduced Numerical examples are discussed in Section 4Finally conclusions and further studies are given in Section 5

2 The Model

This section builds a network design model simultaneouslyconsidering morning and evening peak-hour demand matri-ces Consider a directed transportation network 119866(119873119860)comprising of a set119873 of nodes and a set 119860 of directed linksSince traffic flow patterns in two peaks will be involved thefollowing notations are defined for two peaks respectivelywhich are summarized in the Appendix (Notations)

The link travel time in the morning or evening peak isassumed to be continuous convex and strictly increasingfunction of its own link flow such as the widely-applied BRPfunction Meanwhile our focus is put on the deterministicNDP problem with fixed demand

21 Model Formulation Similar to the traditional NDPmodel taking into account one peak-hour demand matrix abilevel program is formulated to model the proposed NDPproblem considering two peak-hour demand matrices inthat the Stackelberg behavior between network planner andnetwork users can be well characterized In the upper levelthe network design objective is to minimize the weightedsum of total travel times (TTC) of the users traveling inthe morning and evening peaks The lower level is theuser equilibrium traffic assignment problem which is usedto characterize the userrsquos route choice behavior The NDPformulation is given below

Upper Level

minf119898 f119890 x

TTC = 120572 sum119908isin119882119898

120583119898

119908(f119898 x) 119902119898

119908+ 120573 sum

119908isin119882119890

120583119890

119908(f119890 x) 119902119890

119908 (1)

subject to119909119886ge 0 119886 isin 119860 (2)

sum

119886isin119860

119887119886(x) le 119861 (3)

Constraint (2) means a nonnegative link capacity expan-sion and (3) is the total invested budget constraint Theconstruction cost for each improved link 119887

119886(119909119886) can be

approximately estimated by a linear function119887119886(119909119886) = 120574119886119909119886 119886 isin 119860 (4)

The traffic flow patterns 119891119898(x) 119891119890(x) 120583119898119908(x) and 120583119890

119908(x)

can be obtained by solving the lower level traffic assignmentproblems in the morning and evening peaks

Lower Level(1) User equilibrium model in the morning peak is as

below

minx sum

119886isin119860

int

V119898119886

0

119905119898

119886(x 120596) 119889120596 (5)

subject to

V119898119886= sum

119908isin119882119898

sum

119903isin119877119898

119908

119891119898

119903119908120575119898

119886119903119908 119886 isin 119860

sum

119903isin119877119898

119908

119891119898

119903119908= 119902119898

119908 119908 isin 119882

119898

119891119898

119903119908ge 0 119903 isin 119877

119898

119908 119908 isin 119882

119898

(6)

(2) User equilibrium model in the evening peak is asbelow

minx sum

119886isin119860

int

V119890119886

0

119905119890

119886(x 120596) 119889120596 (7)

subject to

V119890119886= sum

119908isin119882119890

sum

119903isin119877119890

119908

119891119890

119903119908120575119890

119886119903119908 119886 isin 119860

sum

119903isin119877119890

119908

119891119890

119903119908= 119902119890

119908 119908 isin 119882

119890

119891119890

119903119908ge 0 119903 isin 119877

119890

119908 119908 isin 119882

119890

(8)

It should be noted that all travelers in the morning andevening peaks will complete their travel journeys on the sametransportation network that is the link capacity expansionschemewould be designed and served for the travelers in bothmorning and evening peaks Hopefully it is intuitive that thetravelers in both morning and evening peaks will all benefitfrom the NDP scheme

We can easily see that if 120572 = 1 120573 = 0 the proposedmodel is equivalent to the traditionalNDPmodel consideringmorning peak-hour demand distribution in that the eveningpeak-hour traffic patterns are not considered in the objectivefunction Similarly if 120572 = 0 120573 = 1 the proposed model isequivalent to the traditional NDPmodel considering eveningpeak-hour demand distribution since themorning commutepattern has no impact on the objective function

4 Mathematical Problems in Engineering

22 Extensions of considering Elastic Demand and UncertainDemand The proposed model can be also extended toaccount for the elastic demand andor uncertain demandWehere give a brief discussion of these extensions In realitywhether a potential traveler decides to finish herhis traveljourney in great extent depends on the traffic congestion onthe network no matter what commute peak is consideredTheO-D demand thus can be assumed to be a function of theO-D travel cost Take the O-D travel demand in the morningpeak as an example

119902119898

119908= 119889119908(120583119898

119908) 119908 isin 119882

119898

(9)

where 119889119908(sdot) is the demand function between O-D pair 119908 isin

119882119898 and its inverse function is represented by 119889minus1

119908(sdot) In

general 119889119908(sdot) is assumed to be a positive continuously

differentiable and strictly decreasing functionwith respect toshortest path cost 120583119898

119908 such as negative exponential function

It is not difficult to deal with the proposed NDP modelwith elastic demand since the traffic assignments for twopeaks with elastic demand can be easily solved by a super-network method or an improved Frank-Wolfe algorithm inSheffi [26]

Another interesting extension is to take into account ofthe uncertain travel demands in two peaks The importanceof considering uncertainties in the NDP is to avoid theunnecessary risks and misleading policies The O-D demandin either morning commute peak or evening commute peakwill fluctuate in future To capture these impacts we candevelop a general stochastic NDP according to the belowframework

minVx119865 (V xΞ) (10)

subject to

V isin Ω (xΞ)

(Vx) isin Θ(11)

The bold notations V Ξ are random variables whichdenote stochastic link flows and uncertain demands in themorning and evening peaks Notation Θ denotes the set ofadditional constraints of variablesV and x andΩ(119909Ξ) is theset of feasible link flows on stochastic network

Two modeling methods can be used to characterizethe morning and evening peak-hour demand uncertaintiesin the NDP problem One is to develop the probabilisticor reliability-based user equilibrium model of deriving thestochastic traffic flow patterns in the morning and eveningpeaks and then embed them into the upper level optimizationof the NDP problem For such probabilistic user equilibriummodel it can be referred to Lo et al [9] Shao et al [10]and Wang et al [11] Another way is to formulate a two-stage NDP model where the uncertainties of two peak-hour demands can be captured by the random samplesgenerated by some scenario-construction methods such assample average approximation [7 8] The main differencebetween two modeling approaches and the detailed formu-lation framework can be found in Meng et al [18]

Via

ViHa

ViHminus1a

Vika

Vi2a

Vi1a

Vi0aX0

a X1a X2

a Xja

Xa

XNminus1a XN

a

(Xlowasta Vilowast

a )

Figure 2 Discretize binary space into feasible regions

3 The Equivalent MILP Model

This study focuses on the NDP problem with two peak-hour demand impacts and on examining the design effectsbetween the proposed model and the conventional one withone peak-hour demandmatrix In order to precisely comparethe network design effects it would be better to solve theNDP model by a global solution algorithm We here use aglobal solution algorithm proposed by Wang and Lo [27] bytransforming the bilevel NDP model into single-level MILPThe transformation of MILP includes linearization of designobjective function link travel time function and other sideconstraints

For the sake of simplicity the fixed demand is consideredand the BRP function is used tomeasure the congestion effectof the link travel time namely

119905119894

119886(V119894119886 119909119886) = 1199050

119886(1 + 015(

V119894119886

1198880119886+ 119909119886

)

4

) 119894 = 119898 119890 (12)

where 119905119894119886and V119894119886are used to represent the link travel time and

link flow for the morning peak if 119894 = 119898 or for the eveningpeak if 119894 = 119890 Hereafter we briefly revisit the transformationof MILP

31 Linearization of Link Travel Time Function It can be seenin (12) that the link travel time function for each commutepeak is a function of link flow in the peak and design variablenamely link capacity expansion Similar to Wang and Lo[27] Luathep et al [28] and Zhang and Van [29] the binaryspace in terms of link flow and link capacity expansion canbe divided into119867 times 119873 feasible regions as shown in Figure 2Let V119894119897119886 V119894119906119886

be the lower and upper bounds of V119894119886 and let 119909119897

119886

119909119906

119886be the lower and upper bounds of 119909

119886 For all discretized

intervals we have V119894119897119886lt 119870119894

119886ℎlt 119870119894

119886ℎ+1lt V119894119906119886

and 119909119897119886lt

119871119886119899lt 119871119886119899+1

lt 119909119906

119886 ℎ isin 1 2 119867 119899 isin 1 2 119873 and

119894 = 119898 119890

Mathematical Problems in Engineering 5

For any feasible region [ℎ 119899] the link travel time function(12) can be approximated as a linear function by Taylorexpansion

119905119894

119886(V119894119886 119909119886) = 119886119894ℎ119899

119886V119894119886+ 119887119894ℎ119899

119886119909119886+ 119888119894ℎ119899

119886

if 119870119894119886ℎle V119894119886le 119870119894

119886ℎ+1 119871119886119899le 119909119886le 119871119886119899+1

(13)

where coefficients 119886119894ℎ119899119886

119887119894ℎ119899119886

119888119894ℎ119899119886

can be obtained by deter-mining the partial derivatives of the BPR link performancefunction

119886119894ℎ119899

119886=120597119905119894

119886

120597V119894119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119887119894ℎ119899

119886=120597119905119894

119886

120597119909119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119888119894ℎ119899

119886= 119905119894

119886(119870119894

119886ℎ 119871119886119899) minus 119870119894

119886ℎsdot 119886119894ℎ119899

119886minus 119871119886119899sdot 119887119894ℎ119899

119886

(14)

It is clear that link travel time can be precisely estimatedby (13) if very large 119867 119873 are set to guarantee sufficientnumber of binary space splits The link travel time functionthus can be replaced by an equivalent set of mixed integerlinear constraints For link 119886 119886 isin 119860 (ℎ = 1 119867 119899 =1 119873 119894 = 119898 119890) we have

119871 sdot 120585119894

119886ℎle V119894119886minus 119870119894

119886ℎle 119880 sdot (1 minus 120585

119894

119886ℎ) minus 120576

120581119894

119886ℎ= 120585119894

119886ℎ+1minus 120585119894

119886ℎ

119871 sdot 120591119886119899le 119909119886minus 119871119886119899le 119880 sdot (1 minus 120591

119886119899) minus 120576

120582119886119899= 120591119886119899+1

minus 120591119886119899

120595119894ℎ119899

119886= 120581119894

119886ℎ+ 120582119886119899

119871 sdot (2 minus 120595119894ℎ119899

119886) le 119905119886minus (119886119894ℎ119899

119886sdot V119894119886+ 119887119894ℎ119899

119886sdot 119909119886+ 119888119894ℎ119899

119886)

le 119880 sdot (2 minus 120595119894ℎ119899

119886)

integer 120585119894119886ℎ 120591119886119899isin 0 1

(15)

where 119871 119880 are respectively a very large negative constantand a very large positive constant 120576 is a very small positiveconstant It is not difficult to prove that (15) is equivalent to thelinear approximation function (13) We here do not attemptto present the proof again because the detailed proof can befound in [27]

32 Linearization of the Constraints The proposed NDPmodel includes three kinds of constraints deterministic userequilibrium constraint definitional constaints (eg demandconservation) and other side constraints (invested budgetconstaint and boundary constraints of design variables)

(1) Deterministic User Equilibrium Constraint Recall thatthe deterministic user equilibrium principle can also beexpressed by complementary constraint which is derived

from the first-order condition of the lower level trafficassignment problem

119891119894

119903119908sdot (119888119894

119903119908minus 120583119894

119908) = 0 119888

119894

119903119908minus 120583119894

119908= 0 forall119903 119908 119894 (16)

The ldquoif-thenrdquo complementary constraint can be trans-formed into an equivalent set of constraints by introducinga set of binary variables shown as

119871 sdot 120590119894

119903119908+ 120576 le 119891

119894

119903119908le 119880 sdot (1 minus 120590

119894

119903119908)

119871 sdot 120590119894

119903119908le 119888119894

119903119908minus 120583119894

119908le 119880120590119894

119903119908

119888119894

119903119908minus 120583119894

119908ge 0

integer 120590119894119903119908isin 0 1 119894 = 119898 119890

(17)

Evidently in (17) if 120590119894119903119908= 0 we have 119891119894

119903119908gt 0 and 119888119894

119903119908minus

120583119894

119908= 0 otherwise120590119894

119903119908= 1 we have119891119894

119903119908= 0 and 119888119894

119903119908minus120583119894

119908ge 0

That is the user equilibrium condition holds

(2) Definitional ConstaintsWe have

sum

119903isin119877119894

119908

119891119894

119903119908= 119902119894

119908

V119894119886= sum

119908isin119882119894

sum

119903isin119877119894

119908

119891119894

119903119908120575119894

119886119903119908

119888119894

119903119908= sum

119886isin119860

119905119894

119886(V119894119886 119909119886) 120575119894

119886119903119908

V119894119886ge 0 119891

119894

119903119908ge 0

(18)

119905119894

119886ge 1199050

119886 119894 isin 119898 119890 (19)

The definitional constraints in (18) are all linear con-straints due to their additive properties

(3) Other Side ConstraintsWe have

0 le 119909119886le 119909119906

119886 0 le sum

119886isin119860

120574119886119909119886le 119861 (20)

So far we have completely transformed the lower leveluser equilibrium traffic assignment problem into an equiv-alent set of mixed integer linear constraints Since the designobjective function (1) is also linear for the NDP with fixeddemand the bilevel NDP model simultaneously consideringmorning and evening peak-hour demands can be perfectlytransformed into the equivalent MILP

33 Solution Algorithm Comparing to a nonlinear andnonconvex bilevel NDP it is simple and effective to solvethe transformed MILP problem A more attractive meritof solving the NDP problem by transforming into MILPis that a global solution can be guaranteed The globaloptimal solution is helpful and convincing for exploring

6 Mathematical Problems in Engineering

Table 1 O-D demands in two peaks

O-D pair 1rarr 3 3rarr 1 2rarr 3 3rarr 2Commute demandin the morning peak (119902119898

119908) 25 10 20 10

Commute demandin the evening peak (119902119890

119908) 10 25 10 20

Table 2 Parameters used for numerical examples

Link number 1 2 3 41199050

11988650 100 150 100

1198880

11988615 20 20 25

120574119886

50 100 150 100

2

2

31

1

3

4

Figure 3 The transportation network used in numerical examples

the design effect of the proposed NDP model The MILPmodel can easily be solved by off-the-shelf MIP solversincluding IBM ILOG CPLEX LINGO and GUROBI It hasbeen demonstrated that the bilevel optimization problem canbe fully transformed into equivalent MILP and solved by theMIP solvers efficiently and precisely (eg [27]) [29] In thispaper the proposed bilevel NDP model will be solved as theequivalent MILP by the CPLEX solver

4 Numerical Examples

41 Preliminary The numerical examples are used to illus-trate the difference between the proposed NDP model andthe traditional one considering traffic patterns in one peakIn the numerical study a small network shown in Figure 3 isused to demonstrate the property of the proposedmodelThistransportation network comprises of 3 nodes 4 links and 4O-D pairs in each commute peak All 4 links are consideredin the candidate set of the capacity improvement schemenamely 119860 = 119860 The travel demands for each O-D pairin morning and evening peaks are given in Table 1 Table 2provides the link performance parameters 1199050

119886and 1198880119886 and

the link capacity expansion cost coefficient 120574119886 The weighted

parameters in design objective function are set as 120572 = 05120573 = 05 The total budget invested on the link capacityimprovement scheme is 300 The commercial optimizationpackage CPLEX-125 is used to solve the MILP model witha gap tolerance of 01 All experiments run on Windows7 system with the following attributes Intel Core i5-252025 GHz times 2 and 4GB RAM

42 Comparison of the NDP Schemes We investigate threeNDP schemes and make a comparison of them in terms ofnetwork design effect These NDP schemes are the proposedNDP simultaneously consideringmorning and evening peak-hour commuting demands the traditional NDP only con-sidering morning peak-hour demand matrix and the tra-ditional NDP only considering evening peak-hour demandmatrix The outcomes of three NDP schemes are provided inTables 3 4 and 5 respectively

In the NDP considering morning peak-hour demandthe total travel time in the morning peak is 860551 Thesefour links will be improved by adding capacities of 974918202 2998 and 2427 respectively Once this NDP schemeis implemented the total travel time for the network userscommuting in the evening peak is 1296560 It can be foundthat travelers of O-D pairs 1-3 and 2-3 will largely benefit fromSchemeA in themorning and evening peaks But the travelersof another two O-D pairs obtain little benefit from the NDPScheme A

In the NDP considering evening peak-hour demand thetotal travel time in the evening peak is 851822 Links 1 and4 will be expanded by adding capacities of 7637 and 26181respectively and links 2 and 3maintain their initial capacitiesOnce the NDP Scheme B is performed the total travel timefor the network users commuting in the morning peak is958753 We can see that travelers of O-D pairs 1-3 3-1 and3-2 will benefit from Scheme A in two commute peaks in thattheir path travel times will be reduced by the NDP schemeBut Scheme B does not bring any benefit for the travelers ofO-D pair 2-3

In the NDP simultaneously considering two peak-hourdemands the total travel time in themorning peak is 902092and the total travel time in the evening peak is 877038Links 1 2 and 4 will be expanded by adding capacities of6152 6201 and 20725 respectively and link 3 will not beconsidered to be improved It is easy to see that travelers ofall O-D pairs will benefit from the NDP Scheme C in twocommute peaks

By comparing the NDPs considering one peak-hourdemand namely Schemes A and B it is shown that SchemeB is better than Scheme A in terms of reducing total traveltime on the network Specifically the sum of total travel timeof two peaks in Scheme B is far less than that in SchemeA in the sense that the travelers commuting in the eveningpeak largely reduce their travel costs in Scheme B Thisclearly indicates that the effects of the NDPs considering onepeak-hour demand will be affected by the network structureand link sensitivity which bring some troubles in choosingappropriate peak-hour demand matrix The fact that SchemeA is far inferior to Scheme B on design effect also gives us areminder that it should be careful to select the commute peakin performing traffic data collection

We also compare the performance of the NDP simul-taneously considering two peak-hour demands (Scheme C)with the NDP considering one peak-hour demand (SchemeB) As expected the sum of total travel time of two peaksin Scheme C is less than that in Scheme B although totaltravel time in the evening peak of Scheme C is slightly morethan that of Scheme BThe reason is that Scheme B overlooks

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

approaches are used to model the stochastic traffic flowsunder uncertain traffic conditions One is to develop atwo-stage formulation by means of scenario-constructionapproach where a number of finite scenarios of uncertainparameters with known probability distribution will be gen-erated (eg [7 8]) Another way called probability-analyticalmethod is to derive a reliability-based traffic assignmentwhere the users follow probabilistic user equilibrium underuncertain traffic conditions (eg [9ndash11])

Time dimension as another important factor has alsoattracted the researchersrsquo attention in recent NDP studiesThree scales of time dimension can be considered in theNDP problem short-termreal-time (seconds) medium-term (days) and long-term (years) The NDP consideringtimedimension is defined as time-dependentNDPwhich canbe specifically classified into multiperiod NDP and dynamicNDP according to different scales of time dimension Theformer is generally to analyze the NDP problem taking intoaccount a long-term spanning construction andmaintenanceof transport infrastructure [12ndash15] The latter is to preciselycharacterize the real-time trafficdynamics andunsteady-stateconditions in the NDP by introducing dynamic traffic assign-ment Hopefully it can be used to estimate microinteractionamong adjacent links and thereby identify and examine thepossible bottleneck [6 16 17] A comprehensive review ofrecent developments on the NDP studies can be referred toMeng et al [18]

By retrieving and reviewing theNDP literature especiallythe valuable reviews across several decades [3 4 18 19] it canbe found that almost all NDP studies assume that an NDPmodel based on one peak-hour O-D demand distribution iscapable and effective to explain the real traffic patterns Forurban traffic there always are two peaks of a daily trafficflow corresponding tomorning commute and even commuterespectively onweekday [20] For commuting travel trips therepeatability of hourly fluctuation of traffic flow gives rise to acomparative stable peak-hour demand But such repeatabilityof hourly variation of traffic flow does not mean the same O-D demand distributions in the morning and evening peaks

Although the daily commuters have high repetitive travelactivities the peak-hour O-D demand distribution on thenetwork will be different in the morning and evening peaksThis fact has been recognized and emphasized in the trans-portation network modeling problem such as commutingpattern analysis [21] and road tolling and parking fee opti-mization [22 23] It is always assumed that the O-D demandmatrix is asymmetric diagonal for the morning and evenpeaks namely 119902

119900119889= 119902119889119900 if only daily regular commuters

are consideredThe O-D demandmatrix used in the networkplanning and design comes up with the peak-hour trafficsurvey data (eg link traffic volumes) in either morning orevening peak In practice the traffic volumes in the morningpeak may largely differ from that of evening peak commutewhich has been demonstrated by some empirical studies(eg [20 24]) The O-D demand matrices thus may also bedifferent for two peaks because they are estimated based onthe corresponding peak-hour traffic volumes In other wordsthe travelers in two commuting peaks would make differenttravel decision in terms of both path choice and departure

1

1

2

2

Figure 1 An illustrative example

time choice [25] Therefore the NDP considering one peak-hour O-D demand might be misleading and even incorrectOn the one hand the O-D demand distribution on a realnetwork is asymmetric due to different traffic patterns inthe morning and evening peaks though the morning andevening round commuting trips can be assumed to be equalWe should overlook demand uncertainty and diverse spatial-temporal characteristic of travel activities in two peaks Forexample the noncommuting trips in morning peak (egshopping going to hospital and tourism) may not returnback to the origin places in evening peak Similarly partof activities for the travelers who travel in evening peak donot commence in the morning and evening peaks but othertraveling periods Therefore which peak-hour commutepattern should be used to estimate the O-D demand matrixis a controversial and intractable issue

On the other hand network structure and link sensitivitycould have important impact on the network performanceevaluation for two peaks Even though the O-D pairs andtheir demands can be assumed to be symmetric in two peaksnamely 119902

119900119889= 119902119889119900 the path choices in two commute peaks

may be still different which largely depends on the networktopological structure If different pathslinks are chosen intwo peaks network situation in terms of traffic congestionwould show large difference between two peaks In thisrespect the NDP considering one peak-hour O-D demandmay generate misleading and even wrong outcomes Moreimportantly link sensitivity of the NDP effect should not beunderestimated and overlooked under different peaks Takea simple network comprising of two links shown in Figure 1as an example Assume that 119902

119900119889in the morning peak is

equal to 119902119889119900

in the evening peak Consider two O-D pairs11990212= 30 119902

21= 10 in the morning peak and 119902

12= 10

11990221= 30 in the evening peak Links 1 and 2 have equal

capacities of 20 vehh and free flow travel time of 1 minuteThe available budget invested on link capacity expansion isset as 20 equivalently BPR function is used to measure thelink travel time performance

It is not difficult to see that link 1 will be improved byadding 20 units in the NDP considering morning peak-hourdemand and link 2 will be improved by expanding 20 unitsin the NDP considering evening peak-hour demand Thetotal travel times on the network for these two NDPs areequal namely 104305 minutes In the NDP simultaneouslyconsidering morning and evening peak-hour demands bothlinks 1 and 2 will be improved by adding 10 units respectivelyand thereby the total travel time on the network will bereduced to 89037 minutes Evidently the NDP consideringone peak-hour demand matrix is not the promising scheme

Mathematical Problems in Engineering 3

to improve transportation system performance Therefore itis necessary to develop a new network design model in orderto avoid misleading decision-making

To address the above problems we develop a networkdesign model simultaneously taking into account morningand evening peak-hour demandsThe traditionalNDPmodelconsidering one peak-hour O-D demand has been regardedas a special case of the proposed model The proposed modelis formulated as a bilevel programming model with objectiveto minimize the weighted sum of total travel times of thetraffic patterns in the morning and evening peaks In thelower level network users are assumed to follow the userequilibrium principle of their route choice decision Theproposed model is finally transformed into an equivalentmixed integer linear programming (MILP) so that a globaloptimum can be obtained by means of MIP solvers (egCPLEX)

The rest of the paper is organized as follows In the nextsection the new NDP model is formulated by simultane-ously considering the morning and evening peal-hour O-Ddemand distributions Then an equivalent MILP model isintroduced Numerical examples are discussed in Section 4Finally conclusions and further studies are given in Section 5

2 The Model

This section builds a network design model simultaneouslyconsidering morning and evening peak-hour demand matri-ces Consider a directed transportation network 119866(119873119860)comprising of a set119873 of nodes and a set 119860 of directed linksSince traffic flow patterns in two peaks will be involved thefollowing notations are defined for two peaks respectivelywhich are summarized in the Appendix (Notations)

The link travel time in the morning or evening peak isassumed to be continuous convex and strictly increasingfunction of its own link flow such as the widely-applied BRPfunction Meanwhile our focus is put on the deterministicNDP problem with fixed demand

21 Model Formulation Similar to the traditional NDPmodel taking into account one peak-hour demand matrix abilevel program is formulated to model the proposed NDPproblem considering two peak-hour demand matrices inthat the Stackelberg behavior between network planner andnetwork users can be well characterized In the upper levelthe network design objective is to minimize the weightedsum of total travel times (TTC) of the users traveling inthe morning and evening peaks The lower level is theuser equilibrium traffic assignment problem which is usedto characterize the userrsquos route choice behavior The NDPformulation is given below

Upper Level

minf119898 f119890 x

TTC = 120572 sum119908isin119882119898

120583119898

119908(f119898 x) 119902119898

119908+ 120573 sum

119908isin119882119890

120583119890

119908(f119890 x) 119902119890

119908 (1)

subject to119909119886ge 0 119886 isin 119860 (2)

sum

119886isin119860

119887119886(x) le 119861 (3)

Constraint (2) means a nonnegative link capacity expan-sion and (3) is the total invested budget constraint Theconstruction cost for each improved link 119887

119886(119909119886) can be

approximately estimated by a linear function119887119886(119909119886) = 120574119886119909119886 119886 isin 119860 (4)

The traffic flow patterns 119891119898(x) 119891119890(x) 120583119898119908(x) and 120583119890

119908(x)

can be obtained by solving the lower level traffic assignmentproblems in the morning and evening peaks

Lower Level(1) User equilibrium model in the morning peak is as

below

minx sum

119886isin119860

int

V119898119886

0

119905119898

119886(x 120596) 119889120596 (5)

subject to

V119898119886= sum

119908isin119882119898

sum

119903isin119877119898

119908

119891119898

119903119908120575119898

119886119903119908 119886 isin 119860

sum

119903isin119877119898

119908

119891119898

119903119908= 119902119898

119908 119908 isin 119882

119898

119891119898

119903119908ge 0 119903 isin 119877

119898

119908 119908 isin 119882

119898

(6)

(2) User equilibrium model in the evening peak is asbelow

minx sum

119886isin119860

int

V119890119886

0

119905119890

119886(x 120596) 119889120596 (7)

subject to

V119890119886= sum

119908isin119882119890

sum

119903isin119877119890

119908

119891119890

119903119908120575119890

119886119903119908 119886 isin 119860

sum

119903isin119877119890

119908

119891119890

119903119908= 119902119890

119908 119908 isin 119882

119890

119891119890

119903119908ge 0 119903 isin 119877

119890

119908 119908 isin 119882

119890

(8)

It should be noted that all travelers in the morning andevening peaks will complete their travel journeys on the sametransportation network that is the link capacity expansionschemewould be designed and served for the travelers in bothmorning and evening peaks Hopefully it is intuitive that thetravelers in both morning and evening peaks will all benefitfrom the NDP scheme

We can easily see that if 120572 = 1 120573 = 0 the proposedmodel is equivalent to the traditionalNDPmodel consideringmorning peak-hour demand distribution in that the eveningpeak-hour traffic patterns are not considered in the objectivefunction Similarly if 120572 = 0 120573 = 1 the proposed model isequivalent to the traditional NDPmodel considering eveningpeak-hour demand distribution since themorning commutepattern has no impact on the objective function

4 Mathematical Problems in Engineering

22 Extensions of considering Elastic Demand and UncertainDemand The proposed model can be also extended toaccount for the elastic demand andor uncertain demandWehere give a brief discussion of these extensions In realitywhether a potential traveler decides to finish herhis traveljourney in great extent depends on the traffic congestion onthe network no matter what commute peak is consideredTheO-D demand thus can be assumed to be a function of theO-D travel cost Take the O-D travel demand in the morningpeak as an example

119902119898

119908= 119889119908(120583119898

119908) 119908 isin 119882

119898

(9)

where 119889119908(sdot) is the demand function between O-D pair 119908 isin

119882119898 and its inverse function is represented by 119889minus1

119908(sdot) In

general 119889119908(sdot) is assumed to be a positive continuously

differentiable and strictly decreasing functionwith respect toshortest path cost 120583119898

119908 such as negative exponential function

It is not difficult to deal with the proposed NDP modelwith elastic demand since the traffic assignments for twopeaks with elastic demand can be easily solved by a super-network method or an improved Frank-Wolfe algorithm inSheffi [26]

Another interesting extension is to take into account ofthe uncertain travel demands in two peaks The importanceof considering uncertainties in the NDP is to avoid theunnecessary risks and misleading policies The O-D demandin either morning commute peak or evening commute peakwill fluctuate in future To capture these impacts we candevelop a general stochastic NDP according to the belowframework

minVx119865 (V xΞ) (10)

subject to

V isin Ω (xΞ)

(Vx) isin Θ(11)

The bold notations V Ξ are random variables whichdenote stochastic link flows and uncertain demands in themorning and evening peaks Notation Θ denotes the set ofadditional constraints of variablesV and x andΩ(119909Ξ) is theset of feasible link flows on stochastic network

Two modeling methods can be used to characterizethe morning and evening peak-hour demand uncertaintiesin the NDP problem One is to develop the probabilisticor reliability-based user equilibrium model of deriving thestochastic traffic flow patterns in the morning and eveningpeaks and then embed them into the upper level optimizationof the NDP problem For such probabilistic user equilibriummodel it can be referred to Lo et al [9] Shao et al [10]and Wang et al [11] Another way is to formulate a two-stage NDP model where the uncertainties of two peak-hour demands can be captured by the random samplesgenerated by some scenario-construction methods such assample average approximation [7 8] The main differencebetween two modeling approaches and the detailed formu-lation framework can be found in Meng et al [18]

Via

ViHa

ViHminus1a

Vika

Vi2a

Vi1a

Vi0aX0

a X1a X2

a Xja

Xa

XNminus1a XN

a

(Xlowasta Vilowast

a )

Figure 2 Discretize binary space into feasible regions

3 The Equivalent MILP Model

This study focuses on the NDP problem with two peak-hour demand impacts and on examining the design effectsbetween the proposed model and the conventional one withone peak-hour demandmatrix In order to precisely comparethe network design effects it would be better to solve theNDP model by a global solution algorithm We here use aglobal solution algorithm proposed by Wang and Lo [27] bytransforming the bilevel NDP model into single-level MILPThe transformation of MILP includes linearization of designobjective function link travel time function and other sideconstraints

For the sake of simplicity the fixed demand is consideredand the BRP function is used tomeasure the congestion effectof the link travel time namely

119905119894

119886(V119894119886 119909119886) = 1199050

119886(1 + 015(

V119894119886

1198880119886+ 119909119886

)

4

) 119894 = 119898 119890 (12)

where 119905119894119886and V119894119886are used to represent the link travel time and

link flow for the morning peak if 119894 = 119898 or for the eveningpeak if 119894 = 119890 Hereafter we briefly revisit the transformationof MILP

31 Linearization of Link Travel Time Function It can be seenin (12) that the link travel time function for each commutepeak is a function of link flow in the peak and design variablenamely link capacity expansion Similar to Wang and Lo[27] Luathep et al [28] and Zhang and Van [29] the binaryspace in terms of link flow and link capacity expansion canbe divided into119867 times 119873 feasible regions as shown in Figure 2Let V119894119897119886 V119894119906119886

be the lower and upper bounds of V119894119886 and let 119909119897

119886

119909119906

119886be the lower and upper bounds of 119909

119886 For all discretized

intervals we have V119894119897119886lt 119870119894

119886ℎlt 119870119894

119886ℎ+1lt V119894119906119886

and 119909119897119886lt

119871119886119899lt 119871119886119899+1

lt 119909119906

119886 ℎ isin 1 2 119867 119899 isin 1 2 119873 and

119894 = 119898 119890

Mathematical Problems in Engineering 5

For any feasible region [ℎ 119899] the link travel time function(12) can be approximated as a linear function by Taylorexpansion

119905119894

119886(V119894119886 119909119886) = 119886119894ℎ119899

119886V119894119886+ 119887119894ℎ119899

119886119909119886+ 119888119894ℎ119899

119886

if 119870119894119886ℎle V119894119886le 119870119894

119886ℎ+1 119871119886119899le 119909119886le 119871119886119899+1

(13)

where coefficients 119886119894ℎ119899119886

119887119894ℎ119899119886

119888119894ℎ119899119886

can be obtained by deter-mining the partial derivatives of the BPR link performancefunction

119886119894ℎ119899

119886=120597119905119894

119886

120597V119894119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119887119894ℎ119899

119886=120597119905119894

119886

120597119909119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119888119894ℎ119899

119886= 119905119894

119886(119870119894

119886ℎ 119871119886119899) minus 119870119894

119886ℎsdot 119886119894ℎ119899

119886minus 119871119886119899sdot 119887119894ℎ119899

119886

(14)

It is clear that link travel time can be precisely estimatedby (13) if very large 119867 119873 are set to guarantee sufficientnumber of binary space splits The link travel time functionthus can be replaced by an equivalent set of mixed integerlinear constraints For link 119886 119886 isin 119860 (ℎ = 1 119867 119899 =1 119873 119894 = 119898 119890) we have

119871 sdot 120585119894

119886ℎle V119894119886minus 119870119894

119886ℎle 119880 sdot (1 minus 120585

119894

119886ℎ) minus 120576

120581119894

119886ℎ= 120585119894

119886ℎ+1minus 120585119894

119886ℎ

119871 sdot 120591119886119899le 119909119886minus 119871119886119899le 119880 sdot (1 minus 120591

119886119899) minus 120576

120582119886119899= 120591119886119899+1

minus 120591119886119899

120595119894ℎ119899

119886= 120581119894

119886ℎ+ 120582119886119899

119871 sdot (2 minus 120595119894ℎ119899

119886) le 119905119886minus (119886119894ℎ119899

119886sdot V119894119886+ 119887119894ℎ119899

119886sdot 119909119886+ 119888119894ℎ119899

119886)

le 119880 sdot (2 minus 120595119894ℎ119899

119886)

integer 120585119894119886ℎ 120591119886119899isin 0 1

(15)

where 119871 119880 are respectively a very large negative constantand a very large positive constant 120576 is a very small positiveconstant It is not difficult to prove that (15) is equivalent to thelinear approximation function (13) We here do not attemptto present the proof again because the detailed proof can befound in [27]

32 Linearization of the Constraints The proposed NDPmodel includes three kinds of constraints deterministic userequilibrium constraint definitional constaints (eg demandconservation) and other side constraints (invested budgetconstaint and boundary constraints of design variables)

(1) Deterministic User Equilibrium Constraint Recall thatthe deterministic user equilibrium principle can also beexpressed by complementary constraint which is derived

from the first-order condition of the lower level trafficassignment problem

119891119894

119903119908sdot (119888119894

119903119908minus 120583119894

119908) = 0 119888

119894

119903119908minus 120583119894

119908= 0 forall119903 119908 119894 (16)

The ldquoif-thenrdquo complementary constraint can be trans-formed into an equivalent set of constraints by introducinga set of binary variables shown as

119871 sdot 120590119894

119903119908+ 120576 le 119891

119894

119903119908le 119880 sdot (1 minus 120590

119894

119903119908)

119871 sdot 120590119894

119903119908le 119888119894

119903119908minus 120583119894

119908le 119880120590119894

119903119908

119888119894

119903119908minus 120583119894

119908ge 0

integer 120590119894119903119908isin 0 1 119894 = 119898 119890

(17)

Evidently in (17) if 120590119894119903119908= 0 we have 119891119894

119903119908gt 0 and 119888119894

119903119908minus

120583119894

119908= 0 otherwise120590119894

119903119908= 1 we have119891119894

119903119908= 0 and 119888119894

119903119908minus120583119894

119908ge 0

That is the user equilibrium condition holds

(2) Definitional ConstaintsWe have

sum

119903isin119877119894

119908

119891119894

119903119908= 119902119894

119908

V119894119886= sum

119908isin119882119894

sum

119903isin119877119894

119908

119891119894

119903119908120575119894

119886119903119908

119888119894

119903119908= sum

119886isin119860

119905119894

119886(V119894119886 119909119886) 120575119894

119886119903119908

V119894119886ge 0 119891

119894

119903119908ge 0

(18)

119905119894

119886ge 1199050

119886 119894 isin 119898 119890 (19)

The definitional constraints in (18) are all linear con-straints due to their additive properties

(3) Other Side ConstraintsWe have

0 le 119909119886le 119909119906

119886 0 le sum

119886isin119860

120574119886119909119886le 119861 (20)

So far we have completely transformed the lower leveluser equilibrium traffic assignment problem into an equiv-alent set of mixed integer linear constraints Since the designobjective function (1) is also linear for the NDP with fixeddemand the bilevel NDP model simultaneously consideringmorning and evening peak-hour demands can be perfectlytransformed into the equivalent MILP

33 Solution Algorithm Comparing to a nonlinear andnonconvex bilevel NDP it is simple and effective to solvethe transformed MILP problem A more attractive meritof solving the NDP problem by transforming into MILPis that a global solution can be guaranteed The globaloptimal solution is helpful and convincing for exploring

6 Mathematical Problems in Engineering

Table 1 O-D demands in two peaks

O-D pair 1rarr 3 3rarr 1 2rarr 3 3rarr 2Commute demandin the morning peak (119902119898

119908) 25 10 20 10

Commute demandin the evening peak (119902119890

119908) 10 25 10 20

Table 2 Parameters used for numerical examples

Link number 1 2 3 41199050

11988650 100 150 100

1198880

11988615 20 20 25

120574119886

50 100 150 100

2

2

31

1

3

4

Figure 3 The transportation network used in numerical examples

the design effect of the proposed NDP model The MILPmodel can easily be solved by off-the-shelf MIP solversincluding IBM ILOG CPLEX LINGO and GUROBI It hasbeen demonstrated that the bilevel optimization problem canbe fully transformed into equivalent MILP and solved by theMIP solvers efficiently and precisely (eg [27]) [29] In thispaper the proposed bilevel NDP model will be solved as theequivalent MILP by the CPLEX solver

4 Numerical Examples

41 Preliminary The numerical examples are used to illus-trate the difference between the proposed NDP model andthe traditional one considering traffic patterns in one peakIn the numerical study a small network shown in Figure 3 isused to demonstrate the property of the proposedmodelThistransportation network comprises of 3 nodes 4 links and 4O-D pairs in each commute peak All 4 links are consideredin the candidate set of the capacity improvement schemenamely 119860 = 119860 The travel demands for each O-D pairin morning and evening peaks are given in Table 1 Table 2provides the link performance parameters 1199050

119886and 1198880119886 and

the link capacity expansion cost coefficient 120574119886 The weighted

parameters in design objective function are set as 120572 = 05120573 = 05 The total budget invested on the link capacityimprovement scheme is 300 The commercial optimizationpackage CPLEX-125 is used to solve the MILP model witha gap tolerance of 01 All experiments run on Windows7 system with the following attributes Intel Core i5-252025 GHz times 2 and 4GB RAM

42 Comparison of the NDP Schemes We investigate threeNDP schemes and make a comparison of them in terms ofnetwork design effect These NDP schemes are the proposedNDP simultaneously consideringmorning and evening peak-hour commuting demands the traditional NDP only con-sidering morning peak-hour demand matrix and the tra-ditional NDP only considering evening peak-hour demandmatrix The outcomes of three NDP schemes are provided inTables 3 4 and 5 respectively

In the NDP considering morning peak-hour demandthe total travel time in the morning peak is 860551 Thesefour links will be improved by adding capacities of 974918202 2998 and 2427 respectively Once this NDP schemeis implemented the total travel time for the network userscommuting in the evening peak is 1296560 It can be foundthat travelers of O-D pairs 1-3 and 2-3 will largely benefit fromSchemeA in themorning and evening peaks But the travelersof another two O-D pairs obtain little benefit from the NDPScheme A

In the NDP considering evening peak-hour demand thetotal travel time in the evening peak is 851822 Links 1 and4 will be expanded by adding capacities of 7637 and 26181respectively and links 2 and 3maintain their initial capacitiesOnce the NDP Scheme B is performed the total travel timefor the network users commuting in the morning peak is958753 We can see that travelers of O-D pairs 1-3 3-1 and3-2 will benefit from Scheme A in two commute peaks in thattheir path travel times will be reduced by the NDP schemeBut Scheme B does not bring any benefit for the travelers ofO-D pair 2-3

In the NDP simultaneously considering two peak-hourdemands the total travel time in themorning peak is 902092and the total travel time in the evening peak is 877038Links 1 2 and 4 will be expanded by adding capacities of6152 6201 and 20725 respectively and link 3 will not beconsidered to be improved It is easy to see that travelers ofall O-D pairs will benefit from the NDP Scheme C in twocommute peaks

By comparing the NDPs considering one peak-hourdemand namely Schemes A and B it is shown that SchemeB is better than Scheme A in terms of reducing total traveltime on the network Specifically the sum of total travel timeof two peaks in Scheme B is far less than that in SchemeA in the sense that the travelers commuting in the eveningpeak largely reduce their travel costs in Scheme B Thisclearly indicates that the effects of the NDPs considering onepeak-hour demand will be affected by the network structureand link sensitivity which bring some troubles in choosingappropriate peak-hour demand matrix The fact that SchemeA is far inferior to Scheme B on design effect also gives us areminder that it should be careful to select the commute peakin performing traffic data collection

We also compare the performance of the NDP simul-taneously considering two peak-hour demands (Scheme C)with the NDP considering one peak-hour demand (SchemeB) As expected the sum of total travel time of two peaksin Scheme C is less than that in Scheme B although totaltravel time in the evening peak of Scheme C is slightly morethan that of Scheme BThe reason is that Scheme B overlooks

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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Mathematical Problems in Engineering

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Mathematical Problems in Engineering 3

to improve transportation system performance Therefore itis necessary to develop a new network design model in orderto avoid misleading decision-making

To address the above problems we develop a networkdesign model simultaneously taking into account morningand evening peak-hour demandsThe traditionalNDPmodelconsidering one peak-hour O-D demand has been regardedas a special case of the proposed model The proposed modelis formulated as a bilevel programming model with objectiveto minimize the weighted sum of total travel times of thetraffic patterns in the morning and evening peaks In thelower level network users are assumed to follow the userequilibrium principle of their route choice decision Theproposed model is finally transformed into an equivalentmixed integer linear programming (MILP) so that a globaloptimum can be obtained by means of MIP solvers (egCPLEX)

The rest of the paper is organized as follows In the nextsection the new NDP model is formulated by simultane-ously considering the morning and evening peal-hour O-Ddemand distributions Then an equivalent MILP model isintroduced Numerical examples are discussed in Section 4Finally conclusions and further studies are given in Section 5

2 The Model

This section builds a network design model simultaneouslyconsidering morning and evening peak-hour demand matri-ces Consider a directed transportation network 119866(119873119860)comprising of a set119873 of nodes and a set 119860 of directed linksSince traffic flow patterns in two peaks will be involved thefollowing notations are defined for two peaks respectivelywhich are summarized in the Appendix (Notations)

The link travel time in the morning or evening peak isassumed to be continuous convex and strictly increasingfunction of its own link flow such as the widely-applied BRPfunction Meanwhile our focus is put on the deterministicNDP problem with fixed demand

21 Model Formulation Similar to the traditional NDPmodel taking into account one peak-hour demand matrix abilevel program is formulated to model the proposed NDPproblem considering two peak-hour demand matrices inthat the Stackelberg behavior between network planner andnetwork users can be well characterized In the upper levelthe network design objective is to minimize the weightedsum of total travel times (TTC) of the users traveling inthe morning and evening peaks The lower level is theuser equilibrium traffic assignment problem which is usedto characterize the userrsquos route choice behavior The NDPformulation is given below

Upper Level

minf119898 f119890 x

TTC = 120572 sum119908isin119882119898

120583119898

119908(f119898 x) 119902119898

119908+ 120573 sum

119908isin119882119890

120583119890

119908(f119890 x) 119902119890

119908 (1)

subject to119909119886ge 0 119886 isin 119860 (2)

sum

119886isin119860

119887119886(x) le 119861 (3)

Constraint (2) means a nonnegative link capacity expan-sion and (3) is the total invested budget constraint Theconstruction cost for each improved link 119887

119886(119909119886) can be

approximately estimated by a linear function119887119886(119909119886) = 120574119886119909119886 119886 isin 119860 (4)

The traffic flow patterns 119891119898(x) 119891119890(x) 120583119898119908(x) and 120583119890

119908(x)

can be obtained by solving the lower level traffic assignmentproblems in the morning and evening peaks

Lower Level(1) User equilibrium model in the morning peak is as

below

minx sum

119886isin119860

int

V119898119886

0

119905119898

119886(x 120596) 119889120596 (5)

subject to

V119898119886= sum

119908isin119882119898

sum

119903isin119877119898

119908

119891119898

119903119908120575119898

119886119903119908 119886 isin 119860

sum

119903isin119877119898

119908

119891119898

119903119908= 119902119898

119908 119908 isin 119882

119898

119891119898

119903119908ge 0 119903 isin 119877

119898

119908 119908 isin 119882

119898

(6)

(2) User equilibrium model in the evening peak is asbelow

minx sum

119886isin119860

int

V119890119886

0

119905119890

119886(x 120596) 119889120596 (7)

subject to

V119890119886= sum

119908isin119882119890

sum

119903isin119877119890

119908

119891119890

119903119908120575119890

119886119903119908 119886 isin 119860

sum

119903isin119877119890

119908

119891119890

119903119908= 119902119890

119908 119908 isin 119882

119890

119891119890

119903119908ge 0 119903 isin 119877

119890

119908 119908 isin 119882

119890

(8)

It should be noted that all travelers in the morning andevening peaks will complete their travel journeys on the sametransportation network that is the link capacity expansionschemewould be designed and served for the travelers in bothmorning and evening peaks Hopefully it is intuitive that thetravelers in both morning and evening peaks will all benefitfrom the NDP scheme

We can easily see that if 120572 = 1 120573 = 0 the proposedmodel is equivalent to the traditionalNDPmodel consideringmorning peak-hour demand distribution in that the eveningpeak-hour traffic patterns are not considered in the objectivefunction Similarly if 120572 = 0 120573 = 1 the proposed model isequivalent to the traditional NDPmodel considering eveningpeak-hour demand distribution since themorning commutepattern has no impact on the objective function

4 Mathematical Problems in Engineering

22 Extensions of considering Elastic Demand and UncertainDemand The proposed model can be also extended toaccount for the elastic demand andor uncertain demandWehere give a brief discussion of these extensions In realitywhether a potential traveler decides to finish herhis traveljourney in great extent depends on the traffic congestion onthe network no matter what commute peak is consideredTheO-D demand thus can be assumed to be a function of theO-D travel cost Take the O-D travel demand in the morningpeak as an example

119902119898

119908= 119889119908(120583119898

119908) 119908 isin 119882

119898

(9)

where 119889119908(sdot) is the demand function between O-D pair 119908 isin

119882119898 and its inverse function is represented by 119889minus1

119908(sdot) In

general 119889119908(sdot) is assumed to be a positive continuously

differentiable and strictly decreasing functionwith respect toshortest path cost 120583119898

119908 such as negative exponential function

It is not difficult to deal with the proposed NDP modelwith elastic demand since the traffic assignments for twopeaks with elastic demand can be easily solved by a super-network method or an improved Frank-Wolfe algorithm inSheffi [26]

Another interesting extension is to take into account ofthe uncertain travel demands in two peaks The importanceof considering uncertainties in the NDP is to avoid theunnecessary risks and misleading policies The O-D demandin either morning commute peak or evening commute peakwill fluctuate in future To capture these impacts we candevelop a general stochastic NDP according to the belowframework

minVx119865 (V xΞ) (10)

subject to

V isin Ω (xΞ)

(Vx) isin Θ(11)

The bold notations V Ξ are random variables whichdenote stochastic link flows and uncertain demands in themorning and evening peaks Notation Θ denotes the set ofadditional constraints of variablesV and x andΩ(119909Ξ) is theset of feasible link flows on stochastic network

Two modeling methods can be used to characterizethe morning and evening peak-hour demand uncertaintiesin the NDP problem One is to develop the probabilisticor reliability-based user equilibrium model of deriving thestochastic traffic flow patterns in the morning and eveningpeaks and then embed them into the upper level optimizationof the NDP problem For such probabilistic user equilibriummodel it can be referred to Lo et al [9] Shao et al [10]and Wang et al [11] Another way is to formulate a two-stage NDP model where the uncertainties of two peak-hour demands can be captured by the random samplesgenerated by some scenario-construction methods such assample average approximation [7 8] The main differencebetween two modeling approaches and the detailed formu-lation framework can be found in Meng et al [18]

Via

ViHa

ViHminus1a

Vika

Vi2a

Vi1a

Vi0aX0

a X1a X2

a Xja

Xa

XNminus1a XN

a

(Xlowasta Vilowast

a )

Figure 2 Discretize binary space into feasible regions

3 The Equivalent MILP Model

This study focuses on the NDP problem with two peak-hour demand impacts and on examining the design effectsbetween the proposed model and the conventional one withone peak-hour demandmatrix In order to precisely comparethe network design effects it would be better to solve theNDP model by a global solution algorithm We here use aglobal solution algorithm proposed by Wang and Lo [27] bytransforming the bilevel NDP model into single-level MILPThe transformation of MILP includes linearization of designobjective function link travel time function and other sideconstraints

For the sake of simplicity the fixed demand is consideredand the BRP function is used tomeasure the congestion effectof the link travel time namely

119905119894

119886(V119894119886 119909119886) = 1199050

119886(1 + 015(

V119894119886

1198880119886+ 119909119886

)

4

) 119894 = 119898 119890 (12)

where 119905119894119886and V119894119886are used to represent the link travel time and

link flow for the morning peak if 119894 = 119898 or for the eveningpeak if 119894 = 119890 Hereafter we briefly revisit the transformationof MILP

31 Linearization of Link Travel Time Function It can be seenin (12) that the link travel time function for each commutepeak is a function of link flow in the peak and design variablenamely link capacity expansion Similar to Wang and Lo[27] Luathep et al [28] and Zhang and Van [29] the binaryspace in terms of link flow and link capacity expansion canbe divided into119867 times 119873 feasible regions as shown in Figure 2Let V119894119897119886 V119894119906119886

be the lower and upper bounds of V119894119886 and let 119909119897

119886

119909119906

119886be the lower and upper bounds of 119909

119886 For all discretized

intervals we have V119894119897119886lt 119870119894

119886ℎlt 119870119894

119886ℎ+1lt V119894119906119886

and 119909119897119886lt

119871119886119899lt 119871119886119899+1

lt 119909119906

119886 ℎ isin 1 2 119867 119899 isin 1 2 119873 and

119894 = 119898 119890

Mathematical Problems in Engineering 5

For any feasible region [ℎ 119899] the link travel time function(12) can be approximated as a linear function by Taylorexpansion

119905119894

119886(V119894119886 119909119886) = 119886119894ℎ119899

119886V119894119886+ 119887119894ℎ119899

119886119909119886+ 119888119894ℎ119899

119886

if 119870119894119886ℎle V119894119886le 119870119894

119886ℎ+1 119871119886119899le 119909119886le 119871119886119899+1

(13)

where coefficients 119886119894ℎ119899119886

119887119894ℎ119899119886

119888119894ℎ119899119886

can be obtained by deter-mining the partial derivatives of the BPR link performancefunction

119886119894ℎ119899

119886=120597119905119894

119886

120597V119894119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119887119894ℎ119899

119886=120597119905119894

119886

120597119909119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119888119894ℎ119899

119886= 119905119894

119886(119870119894

119886ℎ 119871119886119899) minus 119870119894

119886ℎsdot 119886119894ℎ119899

119886minus 119871119886119899sdot 119887119894ℎ119899

119886

(14)

It is clear that link travel time can be precisely estimatedby (13) if very large 119867 119873 are set to guarantee sufficientnumber of binary space splits The link travel time functionthus can be replaced by an equivalent set of mixed integerlinear constraints For link 119886 119886 isin 119860 (ℎ = 1 119867 119899 =1 119873 119894 = 119898 119890) we have

119871 sdot 120585119894

119886ℎle V119894119886minus 119870119894

119886ℎle 119880 sdot (1 minus 120585

119894

119886ℎ) minus 120576

120581119894

119886ℎ= 120585119894

119886ℎ+1minus 120585119894

119886ℎ

119871 sdot 120591119886119899le 119909119886minus 119871119886119899le 119880 sdot (1 minus 120591

119886119899) minus 120576

120582119886119899= 120591119886119899+1

minus 120591119886119899

120595119894ℎ119899

119886= 120581119894

119886ℎ+ 120582119886119899

119871 sdot (2 minus 120595119894ℎ119899

119886) le 119905119886minus (119886119894ℎ119899

119886sdot V119894119886+ 119887119894ℎ119899

119886sdot 119909119886+ 119888119894ℎ119899

119886)

le 119880 sdot (2 minus 120595119894ℎ119899

119886)

integer 120585119894119886ℎ 120591119886119899isin 0 1

(15)

where 119871 119880 are respectively a very large negative constantand a very large positive constant 120576 is a very small positiveconstant It is not difficult to prove that (15) is equivalent to thelinear approximation function (13) We here do not attemptto present the proof again because the detailed proof can befound in [27]

32 Linearization of the Constraints The proposed NDPmodel includes three kinds of constraints deterministic userequilibrium constraint definitional constaints (eg demandconservation) and other side constraints (invested budgetconstaint and boundary constraints of design variables)

(1) Deterministic User Equilibrium Constraint Recall thatthe deterministic user equilibrium principle can also beexpressed by complementary constraint which is derived

from the first-order condition of the lower level trafficassignment problem

119891119894

119903119908sdot (119888119894

119903119908minus 120583119894

119908) = 0 119888

119894

119903119908minus 120583119894

119908= 0 forall119903 119908 119894 (16)

The ldquoif-thenrdquo complementary constraint can be trans-formed into an equivalent set of constraints by introducinga set of binary variables shown as

119871 sdot 120590119894

119903119908+ 120576 le 119891

119894

119903119908le 119880 sdot (1 minus 120590

119894

119903119908)

119871 sdot 120590119894

119903119908le 119888119894

119903119908minus 120583119894

119908le 119880120590119894

119903119908

119888119894

119903119908minus 120583119894

119908ge 0

integer 120590119894119903119908isin 0 1 119894 = 119898 119890

(17)

Evidently in (17) if 120590119894119903119908= 0 we have 119891119894

119903119908gt 0 and 119888119894

119903119908minus

120583119894

119908= 0 otherwise120590119894

119903119908= 1 we have119891119894

119903119908= 0 and 119888119894

119903119908minus120583119894

119908ge 0

That is the user equilibrium condition holds

(2) Definitional ConstaintsWe have

sum

119903isin119877119894

119908

119891119894

119903119908= 119902119894

119908

V119894119886= sum

119908isin119882119894

sum

119903isin119877119894

119908

119891119894

119903119908120575119894

119886119903119908

119888119894

119903119908= sum

119886isin119860

119905119894

119886(V119894119886 119909119886) 120575119894

119886119903119908

V119894119886ge 0 119891

119894

119903119908ge 0

(18)

119905119894

119886ge 1199050

119886 119894 isin 119898 119890 (19)

The definitional constraints in (18) are all linear con-straints due to their additive properties

(3) Other Side ConstraintsWe have

0 le 119909119886le 119909119906

119886 0 le sum

119886isin119860

120574119886119909119886le 119861 (20)

So far we have completely transformed the lower leveluser equilibrium traffic assignment problem into an equiv-alent set of mixed integer linear constraints Since the designobjective function (1) is also linear for the NDP with fixeddemand the bilevel NDP model simultaneously consideringmorning and evening peak-hour demands can be perfectlytransformed into the equivalent MILP

33 Solution Algorithm Comparing to a nonlinear andnonconvex bilevel NDP it is simple and effective to solvethe transformed MILP problem A more attractive meritof solving the NDP problem by transforming into MILPis that a global solution can be guaranteed The globaloptimal solution is helpful and convincing for exploring

6 Mathematical Problems in Engineering

Table 1 O-D demands in two peaks

O-D pair 1rarr 3 3rarr 1 2rarr 3 3rarr 2Commute demandin the morning peak (119902119898

119908) 25 10 20 10

Commute demandin the evening peak (119902119890

119908) 10 25 10 20

Table 2 Parameters used for numerical examples

Link number 1 2 3 41199050

11988650 100 150 100

1198880

11988615 20 20 25

120574119886

50 100 150 100

2

2

31

1

3

4

Figure 3 The transportation network used in numerical examples

the design effect of the proposed NDP model The MILPmodel can easily be solved by off-the-shelf MIP solversincluding IBM ILOG CPLEX LINGO and GUROBI It hasbeen demonstrated that the bilevel optimization problem canbe fully transformed into equivalent MILP and solved by theMIP solvers efficiently and precisely (eg [27]) [29] In thispaper the proposed bilevel NDP model will be solved as theequivalent MILP by the CPLEX solver

4 Numerical Examples

41 Preliminary The numerical examples are used to illus-trate the difference between the proposed NDP model andthe traditional one considering traffic patterns in one peakIn the numerical study a small network shown in Figure 3 isused to demonstrate the property of the proposedmodelThistransportation network comprises of 3 nodes 4 links and 4O-D pairs in each commute peak All 4 links are consideredin the candidate set of the capacity improvement schemenamely 119860 = 119860 The travel demands for each O-D pairin morning and evening peaks are given in Table 1 Table 2provides the link performance parameters 1199050

119886and 1198880119886 and

the link capacity expansion cost coefficient 120574119886 The weighted

parameters in design objective function are set as 120572 = 05120573 = 05 The total budget invested on the link capacityimprovement scheme is 300 The commercial optimizationpackage CPLEX-125 is used to solve the MILP model witha gap tolerance of 01 All experiments run on Windows7 system with the following attributes Intel Core i5-252025 GHz times 2 and 4GB RAM

42 Comparison of the NDP Schemes We investigate threeNDP schemes and make a comparison of them in terms ofnetwork design effect These NDP schemes are the proposedNDP simultaneously consideringmorning and evening peak-hour commuting demands the traditional NDP only con-sidering morning peak-hour demand matrix and the tra-ditional NDP only considering evening peak-hour demandmatrix The outcomes of three NDP schemes are provided inTables 3 4 and 5 respectively

In the NDP considering morning peak-hour demandthe total travel time in the morning peak is 860551 Thesefour links will be improved by adding capacities of 974918202 2998 and 2427 respectively Once this NDP schemeis implemented the total travel time for the network userscommuting in the evening peak is 1296560 It can be foundthat travelers of O-D pairs 1-3 and 2-3 will largely benefit fromSchemeA in themorning and evening peaks But the travelersof another two O-D pairs obtain little benefit from the NDPScheme A

In the NDP considering evening peak-hour demand thetotal travel time in the evening peak is 851822 Links 1 and4 will be expanded by adding capacities of 7637 and 26181respectively and links 2 and 3maintain their initial capacitiesOnce the NDP Scheme B is performed the total travel timefor the network users commuting in the morning peak is958753 We can see that travelers of O-D pairs 1-3 3-1 and3-2 will benefit from Scheme A in two commute peaks in thattheir path travel times will be reduced by the NDP schemeBut Scheme B does not bring any benefit for the travelers ofO-D pair 2-3

In the NDP simultaneously considering two peak-hourdemands the total travel time in themorning peak is 902092and the total travel time in the evening peak is 877038Links 1 2 and 4 will be expanded by adding capacities of6152 6201 and 20725 respectively and link 3 will not beconsidered to be improved It is easy to see that travelers ofall O-D pairs will benefit from the NDP Scheme C in twocommute peaks

By comparing the NDPs considering one peak-hourdemand namely Schemes A and B it is shown that SchemeB is better than Scheme A in terms of reducing total traveltime on the network Specifically the sum of total travel timeof two peaks in Scheme B is far less than that in SchemeA in the sense that the travelers commuting in the eveningpeak largely reduce their travel costs in Scheme B Thisclearly indicates that the effects of the NDPs considering onepeak-hour demand will be affected by the network structureand link sensitivity which bring some troubles in choosingappropriate peak-hour demand matrix The fact that SchemeA is far inferior to Scheme B on design effect also gives us areminder that it should be careful to select the commute peakin performing traffic data collection

We also compare the performance of the NDP simul-taneously considering two peak-hour demands (Scheme C)with the NDP considering one peak-hour demand (SchemeB) As expected the sum of total travel time of two peaksin Scheme C is less than that in Scheme B although totaltravel time in the evening peak of Scheme C is slightly morethan that of Scheme BThe reason is that Scheme B overlooks

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

22 Extensions of considering Elastic Demand and UncertainDemand The proposed model can be also extended toaccount for the elastic demand andor uncertain demandWehere give a brief discussion of these extensions In realitywhether a potential traveler decides to finish herhis traveljourney in great extent depends on the traffic congestion onthe network no matter what commute peak is consideredTheO-D demand thus can be assumed to be a function of theO-D travel cost Take the O-D travel demand in the morningpeak as an example

119902119898

119908= 119889119908(120583119898

119908) 119908 isin 119882

119898

(9)

where 119889119908(sdot) is the demand function between O-D pair 119908 isin

119882119898 and its inverse function is represented by 119889minus1

119908(sdot) In

general 119889119908(sdot) is assumed to be a positive continuously

differentiable and strictly decreasing functionwith respect toshortest path cost 120583119898

119908 such as negative exponential function

It is not difficult to deal with the proposed NDP modelwith elastic demand since the traffic assignments for twopeaks with elastic demand can be easily solved by a super-network method or an improved Frank-Wolfe algorithm inSheffi [26]

Another interesting extension is to take into account ofthe uncertain travel demands in two peaks The importanceof considering uncertainties in the NDP is to avoid theunnecessary risks and misleading policies The O-D demandin either morning commute peak or evening commute peakwill fluctuate in future To capture these impacts we candevelop a general stochastic NDP according to the belowframework

minVx119865 (V xΞ) (10)

subject to

V isin Ω (xΞ)

(Vx) isin Θ(11)

The bold notations V Ξ are random variables whichdenote stochastic link flows and uncertain demands in themorning and evening peaks Notation Θ denotes the set ofadditional constraints of variablesV and x andΩ(119909Ξ) is theset of feasible link flows on stochastic network

Two modeling methods can be used to characterizethe morning and evening peak-hour demand uncertaintiesin the NDP problem One is to develop the probabilisticor reliability-based user equilibrium model of deriving thestochastic traffic flow patterns in the morning and eveningpeaks and then embed them into the upper level optimizationof the NDP problem For such probabilistic user equilibriummodel it can be referred to Lo et al [9] Shao et al [10]and Wang et al [11] Another way is to formulate a two-stage NDP model where the uncertainties of two peak-hour demands can be captured by the random samplesgenerated by some scenario-construction methods such assample average approximation [7 8] The main differencebetween two modeling approaches and the detailed formu-lation framework can be found in Meng et al [18]

Via

ViHa

ViHminus1a

Vika

Vi2a

Vi1a

Vi0aX0

a X1a X2

a Xja

Xa

XNminus1a XN

a

(Xlowasta Vilowast

a )

Figure 2 Discretize binary space into feasible regions

3 The Equivalent MILP Model

This study focuses on the NDP problem with two peak-hour demand impacts and on examining the design effectsbetween the proposed model and the conventional one withone peak-hour demandmatrix In order to precisely comparethe network design effects it would be better to solve theNDP model by a global solution algorithm We here use aglobal solution algorithm proposed by Wang and Lo [27] bytransforming the bilevel NDP model into single-level MILPThe transformation of MILP includes linearization of designobjective function link travel time function and other sideconstraints

For the sake of simplicity the fixed demand is consideredand the BRP function is used tomeasure the congestion effectof the link travel time namely

119905119894

119886(V119894119886 119909119886) = 1199050

119886(1 + 015(

V119894119886

1198880119886+ 119909119886

)

4

) 119894 = 119898 119890 (12)

where 119905119894119886and V119894119886are used to represent the link travel time and

link flow for the morning peak if 119894 = 119898 or for the eveningpeak if 119894 = 119890 Hereafter we briefly revisit the transformationof MILP

31 Linearization of Link Travel Time Function It can be seenin (12) that the link travel time function for each commutepeak is a function of link flow in the peak and design variablenamely link capacity expansion Similar to Wang and Lo[27] Luathep et al [28] and Zhang and Van [29] the binaryspace in terms of link flow and link capacity expansion canbe divided into119867 times 119873 feasible regions as shown in Figure 2Let V119894119897119886 V119894119906119886

be the lower and upper bounds of V119894119886 and let 119909119897

119886

119909119906

119886be the lower and upper bounds of 119909

119886 For all discretized

intervals we have V119894119897119886lt 119870119894

119886ℎlt 119870119894

119886ℎ+1lt V119894119906119886

and 119909119897119886lt

119871119886119899lt 119871119886119899+1

lt 119909119906

119886 ℎ isin 1 2 119867 119899 isin 1 2 119873 and

119894 = 119898 119890

Mathematical Problems in Engineering 5

For any feasible region [ℎ 119899] the link travel time function(12) can be approximated as a linear function by Taylorexpansion

119905119894

119886(V119894119886 119909119886) = 119886119894ℎ119899

119886V119894119886+ 119887119894ℎ119899

119886119909119886+ 119888119894ℎ119899

119886

if 119870119894119886ℎle V119894119886le 119870119894

119886ℎ+1 119871119886119899le 119909119886le 119871119886119899+1

(13)

where coefficients 119886119894ℎ119899119886

119887119894ℎ119899119886

119888119894ℎ119899119886

can be obtained by deter-mining the partial derivatives of the BPR link performancefunction

119886119894ℎ119899

119886=120597119905119894

119886

120597V119894119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119887119894ℎ119899

119886=120597119905119894

119886

120597119909119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119888119894ℎ119899

119886= 119905119894

119886(119870119894

119886ℎ 119871119886119899) minus 119870119894

119886ℎsdot 119886119894ℎ119899

119886minus 119871119886119899sdot 119887119894ℎ119899

119886

(14)

It is clear that link travel time can be precisely estimatedby (13) if very large 119867 119873 are set to guarantee sufficientnumber of binary space splits The link travel time functionthus can be replaced by an equivalent set of mixed integerlinear constraints For link 119886 119886 isin 119860 (ℎ = 1 119867 119899 =1 119873 119894 = 119898 119890) we have

119871 sdot 120585119894

119886ℎle V119894119886minus 119870119894

119886ℎle 119880 sdot (1 minus 120585

119894

119886ℎ) minus 120576

120581119894

119886ℎ= 120585119894

119886ℎ+1minus 120585119894

119886ℎ

119871 sdot 120591119886119899le 119909119886minus 119871119886119899le 119880 sdot (1 minus 120591

119886119899) minus 120576

120582119886119899= 120591119886119899+1

minus 120591119886119899

120595119894ℎ119899

119886= 120581119894

119886ℎ+ 120582119886119899

119871 sdot (2 minus 120595119894ℎ119899

119886) le 119905119886minus (119886119894ℎ119899

119886sdot V119894119886+ 119887119894ℎ119899

119886sdot 119909119886+ 119888119894ℎ119899

119886)

le 119880 sdot (2 minus 120595119894ℎ119899

119886)

integer 120585119894119886ℎ 120591119886119899isin 0 1

(15)

where 119871 119880 are respectively a very large negative constantand a very large positive constant 120576 is a very small positiveconstant It is not difficult to prove that (15) is equivalent to thelinear approximation function (13) We here do not attemptto present the proof again because the detailed proof can befound in [27]

32 Linearization of the Constraints The proposed NDPmodel includes three kinds of constraints deterministic userequilibrium constraint definitional constaints (eg demandconservation) and other side constraints (invested budgetconstaint and boundary constraints of design variables)

(1) Deterministic User Equilibrium Constraint Recall thatthe deterministic user equilibrium principle can also beexpressed by complementary constraint which is derived

from the first-order condition of the lower level trafficassignment problem

119891119894

119903119908sdot (119888119894

119903119908minus 120583119894

119908) = 0 119888

119894

119903119908minus 120583119894

119908= 0 forall119903 119908 119894 (16)

The ldquoif-thenrdquo complementary constraint can be trans-formed into an equivalent set of constraints by introducinga set of binary variables shown as

119871 sdot 120590119894

119903119908+ 120576 le 119891

119894

119903119908le 119880 sdot (1 minus 120590

119894

119903119908)

119871 sdot 120590119894

119903119908le 119888119894

119903119908minus 120583119894

119908le 119880120590119894

119903119908

119888119894

119903119908minus 120583119894

119908ge 0

integer 120590119894119903119908isin 0 1 119894 = 119898 119890

(17)

Evidently in (17) if 120590119894119903119908= 0 we have 119891119894

119903119908gt 0 and 119888119894

119903119908minus

120583119894

119908= 0 otherwise120590119894

119903119908= 1 we have119891119894

119903119908= 0 and 119888119894

119903119908minus120583119894

119908ge 0

That is the user equilibrium condition holds

(2) Definitional ConstaintsWe have

sum

119903isin119877119894

119908

119891119894

119903119908= 119902119894

119908

V119894119886= sum

119908isin119882119894

sum

119903isin119877119894

119908

119891119894

119903119908120575119894

119886119903119908

119888119894

119903119908= sum

119886isin119860

119905119894

119886(V119894119886 119909119886) 120575119894

119886119903119908

V119894119886ge 0 119891

119894

119903119908ge 0

(18)

119905119894

119886ge 1199050

119886 119894 isin 119898 119890 (19)

The definitional constraints in (18) are all linear con-straints due to their additive properties

(3) Other Side ConstraintsWe have

0 le 119909119886le 119909119906

119886 0 le sum

119886isin119860

120574119886119909119886le 119861 (20)

So far we have completely transformed the lower leveluser equilibrium traffic assignment problem into an equiv-alent set of mixed integer linear constraints Since the designobjective function (1) is also linear for the NDP with fixeddemand the bilevel NDP model simultaneously consideringmorning and evening peak-hour demands can be perfectlytransformed into the equivalent MILP

33 Solution Algorithm Comparing to a nonlinear andnonconvex bilevel NDP it is simple and effective to solvethe transformed MILP problem A more attractive meritof solving the NDP problem by transforming into MILPis that a global solution can be guaranteed The globaloptimal solution is helpful and convincing for exploring

6 Mathematical Problems in Engineering

Table 1 O-D demands in two peaks

O-D pair 1rarr 3 3rarr 1 2rarr 3 3rarr 2Commute demandin the morning peak (119902119898

119908) 25 10 20 10

Commute demandin the evening peak (119902119890

119908) 10 25 10 20

Table 2 Parameters used for numerical examples

Link number 1 2 3 41199050

11988650 100 150 100

1198880

11988615 20 20 25

120574119886

50 100 150 100

2

2

31

1

3

4

Figure 3 The transportation network used in numerical examples

the design effect of the proposed NDP model The MILPmodel can easily be solved by off-the-shelf MIP solversincluding IBM ILOG CPLEX LINGO and GUROBI It hasbeen demonstrated that the bilevel optimization problem canbe fully transformed into equivalent MILP and solved by theMIP solvers efficiently and precisely (eg [27]) [29] In thispaper the proposed bilevel NDP model will be solved as theequivalent MILP by the CPLEX solver

4 Numerical Examples

41 Preliminary The numerical examples are used to illus-trate the difference between the proposed NDP model andthe traditional one considering traffic patterns in one peakIn the numerical study a small network shown in Figure 3 isused to demonstrate the property of the proposedmodelThistransportation network comprises of 3 nodes 4 links and 4O-D pairs in each commute peak All 4 links are consideredin the candidate set of the capacity improvement schemenamely 119860 = 119860 The travel demands for each O-D pairin morning and evening peaks are given in Table 1 Table 2provides the link performance parameters 1199050

119886and 1198880119886 and

the link capacity expansion cost coefficient 120574119886 The weighted

parameters in design objective function are set as 120572 = 05120573 = 05 The total budget invested on the link capacityimprovement scheme is 300 The commercial optimizationpackage CPLEX-125 is used to solve the MILP model witha gap tolerance of 01 All experiments run on Windows7 system with the following attributes Intel Core i5-252025 GHz times 2 and 4GB RAM

42 Comparison of the NDP Schemes We investigate threeNDP schemes and make a comparison of them in terms ofnetwork design effect These NDP schemes are the proposedNDP simultaneously consideringmorning and evening peak-hour commuting demands the traditional NDP only con-sidering morning peak-hour demand matrix and the tra-ditional NDP only considering evening peak-hour demandmatrix The outcomes of three NDP schemes are provided inTables 3 4 and 5 respectively

In the NDP considering morning peak-hour demandthe total travel time in the morning peak is 860551 Thesefour links will be improved by adding capacities of 974918202 2998 and 2427 respectively Once this NDP schemeis implemented the total travel time for the network userscommuting in the evening peak is 1296560 It can be foundthat travelers of O-D pairs 1-3 and 2-3 will largely benefit fromSchemeA in themorning and evening peaks But the travelersof another two O-D pairs obtain little benefit from the NDPScheme A

In the NDP considering evening peak-hour demand thetotal travel time in the evening peak is 851822 Links 1 and4 will be expanded by adding capacities of 7637 and 26181respectively and links 2 and 3maintain their initial capacitiesOnce the NDP Scheme B is performed the total travel timefor the network users commuting in the morning peak is958753 We can see that travelers of O-D pairs 1-3 3-1 and3-2 will benefit from Scheme A in two commute peaks in thattheir path travel times will be reduced by the NDP schemeBut Scheme B does not bring any benefit for the travelers ofO-D pair 2-3

In the NDP simultaneously considering two peak-hourdemands the total travel time in themorning peak is 902092and the total travel time in the evening peak is 877038Links 1 2 and 4 will be expanded by adding capacities of6152 6201 and 20725 respectively and link 3 will not beconsidered to be improved It is easy to see that travelers ofall O-D pairs will benefit from the NDP Scheme C in twocommute peaks

By comparing the NDPs considering one peak-hourdemand namely Schemes A and B it is shown that SchemeB is better than Scheme A in terms of reducing total traveltime on the network Specifically the sum of total travel timeof two peaks in Scheme B is far less than that in SchemeA in the sense that the travelers commuting in the eveningpeak largely reduce their travel costs in Scheme B Thisclearly indicates that the effects of the NDPs considering onepeak-hour demand will be affected by the network structureand link sensitivity which bring some troubles in choosingappropriate peak-hour demand matrix The fact that SchemeA is far inferior to Scheme B on design effect also gives us areminder that it should be careful to select the commute peakin performing traffic data collection

We also compare the performance of the NDP simul-taneously considering two peak-hour demands (Scheme C)with the NDP considering one peak-hour demand (SchemeB) As expected the sum of total travel time of two peaksin Scheme C is less than that in Scheme B although totaltravel time in the evening peak of Scheme C is slightly morethan that of Scheme BThe reason is that Scheme B overlooks

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

For any feasible region [ℎ 119899] the link travel time function(12) can be approximated as a linear function by Taylorexpansion

119905119894

119886(V119894119886 119909119886) = 119886119894ℎ119899

119886V119894119886+ 119887119894ℎ119899

119886119909119886+ 119888119894ℎ119899

119886

if 119870119894119886ℎle V119894119886le 119870119894

119886ℎ+1 119871119886119899le 119909119886le 119871119886119899+1

(13)

where coefficients 119886119894ℎ119899119886

119887119894ℎ119899119886

119888119894ℎ119899119886

can be obtained by deter-mining the partial derivatives of the BPR link performancefunction

119886119894ℎ119899

119886=120597119905119894

119886

120597V119894119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119887119894ℎ119899

119886=120597119905119894

119886

120597119909119886

100381610038161003816100381610038161003816100381610038161003816(119870119894119886ℎ119871119886119899)

119888119894ℎ119899

119886= 119905119894

119886(119870119894

119886ℎ 119871119886119899) minus 119870119894

119886ℎsdot 119886119894ℎ119899

119886minus 119871119886119899sdot 119887119894ℎ119899

119886

(14)

It is clear that link travel time can be precisely estimatedby (13) if very large 119867 119873 are set to guarantee sufficientnumber of binary space splits The link travel time functionthus can be replaced by an equivalent set of mixed integerlinear constraints For link 119886 119886 isin 119860 (ℎ = 1 119867 119899 =1 119873 119894 = 119898 119890) we have

119871 sdot 120585119894

119886ℎle V119894119886minus 119870119894

119886ℎle 119880 sdot (1 minus 120585

119894

119886ℎ) minus 120576

120581119894

119886ℎ= 120585119894

119886ℎ+1minus 120585119894

119886ℎ

119871 sdot 120591119886119899le 119909119886minus 119871119886119899le 119880 sdot (1 minus 120591

119886119899) minus 120576

120582119886119899= 120591119886119899+1

minus 120591119886119899

120595119894ℎ119899

119886= 120581119894

119886ℎ+ 120582119886119899

119871 sdot (2 minus 120595119894ℎ119899

119886) le 119905119886minus (119886119894ℎ119899

119886sdot V119894119886+ 119887119894ℎ119899

119886sdot 119909119886+ 119888119894ℎ119899

119886)

le 119880 sdot (2 minus 120595119894ℎ119899

119886)

integer 120585119894119886ℎ 120591119886119899isin 0 1

(15)

where 119871 119880 are respectively a very large negative constantand a very large positive constant 120576 is a very small positiveconstant It is not difficult to prove that (15) is equivalent to thelinear approximation function (13) We here do not attemptto present the proof again because the detailed proof can befound in [27]

32 Linearization of the Constraints The proposed NDPmodel includes three kinds of constraints deterministic userequilibrium constraint definitional constaints (eg demandconservation) and other side constraints (invested budgetconstaint and boundary constraints of design variables)

(1) Deterministic User Equilibrium Constraint Recall thatthe deterministic user equilibrium principle can also beexpressed by complementary constraint which is derived

from the first-order condition of the lower level trafficassignment problem

119891119894

119903119908sdot (119888119894

119903119908minus 120583119894

119908) = 0 119888

119894

119903119908minus 120583119894

119908= 0 forall119903 119908 119894 (16)

The ldquoif-thenrdquo complementary constraint can be trans-formed into an equivalent set of constraints by introducinga set of binary variables shown as

119871 sdot 120590119894

119903119908+ 120576 le 119891

119894

119903119908le 119880 sdot (1 minus 120590

119894

119903119908)

119871 sdot 120590119894

119903119908le 119888119894

119903119908minus 120583119894

119908le 119880120590119894

119903119908

119888119894

119903119908minus 120583119894

119908ge 0

integer 120590119894119903119908isin 0 1 119894 = 119898 119890

(17)

Evidently in (17) if 120590119894119903119908= 0 we have 119891119894

119903119908gt 0 and 119888119894

119903119908minus

120583119894

119908= 0 otherwise120590119894

119903119908= 1 we have119891119894

119903119908= 0 and 119888119894

119903119908minus120583119894

119908ge 0

That is the user equilibrium condition holds

(2) Definitional ConstaintsWe have

sum

119903isin119877119894

119908

119891119894

119903119908= 119902119894

119908

V119894119886= sum

119908isin119882119894

sum

119903isin119877119894

119908

119891119894

119903119908120575119894

119886119903119908

119888119894

119903119908= sum

119886isin119860

119905119894

119886(V119894119886 119909119886) 120575119894

119886119903119908

V119894119886ge 0 119891

119894

119903119908ge 0

(18)

119905119894

119886ge 1199050

119886 119894 isin 119898 119890 (19)

The definitional constraints in (18) are all linear con-straints due to their additive properties

(3) Other Side ConstraintsWe have

0 le 119909119886le 119909119906

119886 0 le sum

119886isin119860

120574119886119909119886le 119861 (20)

So far we have completely transformed the lower leveluser equilibrium traffic assignment problem into an equiv-alent set of mixed integer linear constraints Since the designobjective function (1) is also linear for the NDP with fixeddemand the bilevel NDP model simultaneously consideringmorning and evening peak-hour demands can be perfectlytransformed into the equivalent MILP

33 Solution Algorithm Comparing to a nonlinear andnonconvex bilevel NDP it is simple and effective to solvethe transformed MILP problem A more attractive meritof solving the NDP problem by transforming into MILPis that a global solution can be guaranteed The globaloptimal solution is helpful and convincing for exploring

6 Mathematical Problems in Engineering

Table 1 O-D demands in two peaks

O-D pair 1rarr 3 3rarr 1 2rarr 3 3rarr 2Commute demandin the morning peak (119902119898

119908) 25 10 20 10

Commute demandin the evening peak (119902119890

119908) 10 25 10 20

Table 2 Parameters used for numerical examples

Link number 1 2 3 41199050

11988650 100 150 100

1198880

11988615 20 20 25

120574119886

50 100 150 100

2

2

31

1

3

4

Figure 3 The transportation network used in numerical examples

the design effect of the proposed NDP model The MILPmodel can easily be solved by off-the-shelf MIP solversincluding IBM ILOG CPLEX LINGO and GUROBI It hasbeen demonstrated that the bilevel optimization problem canbe fully transformed into equivalent MILP and solved by theMIP solvers efficiently and precisely (eg [27]) [29] In thispaper the proposed bilevel NDP model will be solved as theequivalent MILP by the CPLEX solver

4 Numerical Examples

41 Preliminary The numerical examples are used to illus-trate the difference between the proposed NDP model andthe traditional one considering traffic patterns in one peakIn the numerical study a small network shown in Figure 3 isused to demonstrate the property of the proposedmodelThistransportation network comprises of 3 nodes 4 links and 4O-D pairs in each commute peak All 4 links are consideredin the candidate set of the capacity improvement schemenamely 119860 = 119860 The travel demands for each O-D pairin morning and evening peaks are given in Table 1 Table 2provides the link performance parameters 1199050

119886and 1198880119886 and

the link capacity expansion cost coefficient 120574119886 The weighted

parameters in design objective function are set as 120572 = 05120573 = 05 The total budget invested on the link capacityimprovement scheme is 300 The commercial optimizationpackage CPLEX-125 is used to solve the MILP model witha gap tolerance of 01 All experiments run on Windows7 system with the following attributes Intel Core i5-252025 GHz times 2 and 4GB RAM

42 Comparison of the NDP Schemes We investigate threeNDP schemes and make a comparison of them in terms ofnetwork design effect These NDP schemes are the proposedNDP simultaneously consideringmorning and evening peak-hour commuting demands the traditional NDP only con-sidering morning peak-hour demand matrix and the tra-ditional NDP only considering evening peak-hour demandmatrix The outcomes of three NDP schemes are provided inTables 3 4 and 5 respectively

In the NDP considering morning peak-hour demandthe total travel time in the morning peak is 860551 Thesefour links will be improved by adding capacities of 974918202 2998 and 2427 respectively Once this NDP schemeis implemented the total travel time for the network userscommuting in the evening peak is 1296560 It can be foundthat travelers of O-D pairs 1-3 and 2-3 will largely benefit fromSchemeA in themorning and evening peaks But the travelersof another two O-D pairs obtain little benefit from the NDPScheme A

In the NDP considering evening peak-hour demand thetotal travel time in the evening peak is 851822 Links 1 and4 will be expanded by adding capacities of 7637 and 26181respectively and links 2 and 3maintain their initial capacitiesOnce the NDP Scheme B is performed the total travel timefor the network users commuting in the morning peak is958753 We can see that travelers of O-D pairs 1-3 3-1 and3-2 will benefit from Scheme A in two commute peaks in thattheir path travel times will be reduced by the NDP schemeBut Scheme B does not bring any benefit for the travelers ofO-D pair 2-3

In the NDP simultaneously considering two peak-hourdemands the total travel time in themorning peak is 902092and the total travel time in the evening peak is 877038Links 1 2 and 4 will be expanded by adding capacities of6152 6201 and 20725 respectively and link 3 will not beconsidered to be improved It is easy to see that travelers ofall O-D pairs will benefit from the NDP Scheme C in twocommute peaks

By comparing the NDPs considering one peak-hourdemand namely Schemes A and B it is shown that SchemeB is better than Scheme A in terms of reducing total traveltime on the network Specifically the sum of total travel timeof two peaks in Scheme B is far less than that in SchemeA in the sense that the travelers commuting in the eveningpeak largely reduce their travel costs in Scheme B Thisclearly indicates that the effects of the NDPs considering onepeak-hour demand will be affected by the network structureand link sensitivity which bring some troubles in choosingappropriate peak-hour demand matrix The fact that SchemeA is far inferior to Scheme B on design effect also gives us areminder that it should be careful to select the commute peakin performing traffic data collection

We also compare the performance of the NDP simul-taneously considering two peak-hour demands (Scheme C)with the NDP considering one peak-hour demand (SchemeB) As expected the sum of total travel time of two peaksin Scheme C is less than that in Scheme B although totaltravel time in the evening peak of Scheme C is slightly morethan that of Scheme BThe reason is that Scheme B overlooks

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 1 O-D demands in two peaks

O-D pair 1rarr 3 3rarr 1 2rarr 3 3rarr 2Commute demandin the morning peak (119902119898

119908) 25 10 20 10

Commute demandin the evening peak (119902119890

119908) 10 25 10 20

Table 2 Parameters used for numerical examples

Link number 1 2 3 41199050

11988650 100 150 100

1198880

11988615 20 20 25

120574119886

50 100 150 100

2

2

31

1

3

4

Figure 3 The transportation network used in numerical examples

the design effect of the proposed NDP model The MILPmodel can easily be solved by off-the-shelf MIP solversincluding IBM ILOG CPLEX LINGO and GUROBI It hasbeen demonstrated that the bilevel optimization problem canbe fully transformed into equivalent MILP and solved by theMIP solvers efficiently and precisely (eg [27]) [29] In thispaper the proposed bilevel NDP model will be solved as theequivalent MILP by the CPLEX solver

4 Numerical Examples

41 Preliminary The numerical examples are used to illus-trate the difference between the proposed NDP model andthe traditional one considering traffic patterns in one peakIn the numerical study a small network shown in Figure 3 isused to demonstrate the property of the proposedmodelThistransportation network comprises of 3 nodes 4 links and 4O-D pairs in each commute peak All 4 links are consideredin the candidate set of the capacity improvement schemenamely 119860 = 119860 The travel demands for each O-D pairin morning and evening peaks are given in Table 1 Table 2provides the link performance parameters 1199050

119886and 1198880119886 and

the link capacity expansion cost coefficient 120574119886 The weighted

parameters in design objective function are set as 120572 = 05120573 = 05 The total budget invested on the link capacityimprovement scheme is 300 The commercial optimizationpackage CPLEX-125 is used to solve the MILP model witha gap tolerance of 01 All experiments run on Windows7 system with the following attributes Intel Core i5-252025 GHz times 2 and 4GB RAM

42 Comparison of the NDP Schemes We investigate threeNDP schemes and make a comparison of them in terms ofnetwork design effect These NDP schemes are the proposedNDP simultaneously consideringmorning and evening peak-hour commuting demands the traditional NDP only con-sidering morning peak-hour demand matrix and the tra-ditional NDP only considering evening peak-hour demandmatrix The outcomes of three NDP schemes are provided inTables 3 4 and 5 respectively

In the NDP considering morning peak-hour demandthe total travel time in the morning peak is 860551 Thesefour links will be improved by adding capacities of 974918202 2998 and 2427 respectively Once this NDP schemeis implemented the total travel time for the network userscommuting in the evening peak is 1296560 It can be foundthat travelers of O-D pairs 1-3 and 2-3 will largely benefit fromSchemeA in themorning and evening peaks But the travelersof another two O-D pairs obtain little benefit from the NDPScheme A

In the NDP considering evening peak-hour demand thetotal travel time in the evening peak is 851822 Links 1 and4 will be expanded by adding capacities of 7637 and 26181respectively and links 2 and 3maintain their initial capacitiesOnce the NDP Scheme B is performed the total travel timefor the network users commuting in the morning peak is958753 We can see that travelers of O-D pairs 1-3 3-1 and3-2 will benefit from Scheme A in two commute peaks in thattheir path travel times will be reduced by the NDP schemeBut Scheme B does not bring any benefit for the travelers ofO-D pair 2-3

In the NDP simultaneously considering two peak-hourdemands the total travel time in themorning peak is 902092and the total travel time in the evening peak is 877038Links 1 2 and 4 will be expanded by adding capacities of6152 6201 and 20725 respectively and link 3 will not beconsidered to be improved It is easy to see that travelers ofall O-D pairs will benefit from the NDP Scheme C in twocommute peaks

By comparing the NDPs considering one peak-hourdemand namely Schemes A and B it is shown that SchemeB is better than Scheme A in terms of reducing total traveltime on the network Specifically the sum of total travel timeof two peaks in Scheme B is far less than that in SchemeA in the sense that the travelers commuting in the eveningpeak largely reduce their travel costs in Scheme B Thisclearly indicates that the effects of the NDPs considering onepeak-hour demand will be affected by the network structureand link sensitivity which bring some troubles in choosingappropriate peak-hour demand matrix The fact that SchemeA is far inferior to Scheme B on design effect also gives us areminder that it should be careful to select the commute peakin performing traffic data collection

We also compare the performance of the NDP simul-taneously considering two peak-hour demands (Scheme C)with the NDP considering one peak-hour demand (SchemeB) As expected the sum of total travel time of two peaksin Scheme C is less than that in Scheme B although totaltravel time in the evening peak of Scheme C is slightly morethan that of Scheme BThe reason is that Scheme B overlooks

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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

Mathematical Problems in Engineering 7

Table 3 Network design considering morning peak-hour demand (Scheme A)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 860551

1 9749 18108 1 1-2 8108 156552 18202 28108 2 3 16892 156553 2998 16892 3 4 10000 104244 2427 20000 4 2 20000 10440mdash mdash mdash 5 1ndash4 10000 15639

Evening peak 1296560

1 9749 20000 1 1-2 0000 153272 18202 10000 2 3 10000 150803 2998 10000 3 4 25000 208734 2427 45000 4 2 10000 10007mdash mdash mdash 5 1ndash4 20000 26193

Sum 2157111 mdash mdash mdash mdash mdash mdash mdash

Table 4 Network design considering evening peak-hour demand (Scheme B)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 958753

1 7637 13551 1 1-2 3551 179802 0000 23551 2 3 21449 179793 0000 21449 3 4 10000 100354 26181 20000 4 2 20000 12884mdash mdash mdash 5 1ndash4 10000 15131

Evening peak 851822

1 7637 20000 1 1-2 0000 155512 0000 10000 2 3 10000 151413 0000 10000 3 4 25000 108964 26181 45000 4 2 10000 10094mdash mdash mdash 5 1ndash4 20000 16353

Sum 1810575 mdash mdash mdash mdash mdash mdash mdash

the traffic congestion in the morning peak that is it doesnot take into account the benefits of the travelers in morningpeak while designing NDP scheme In reality Schemes Aand B can be regarded as two special cases of the proposedmodel if one of the weighted parameters is zero In summarythe proposed NDP model simultaneously considering twopeak-hour demands can well characterize the practical trafficsituation and also bring about promising design effect interms of improving the transportation system performanceNote that the NDP model can be extended to consider morethan two peaks But theNDP simultaneously considering twopeak-hour demands is believed to be good enough when thedata collection costs in each peak and model flexibility aretaken into account

43 Impact Analysis of Weighted Parameters We here con-duct the impact analysis of the weighted parameter settingfor the design objective function Without loss of generalityit is assumed that 120572+120573 = 10 by normalizationThe weightedparameter 120573 is set to be increased from 00 to 10 with eachincrement of 01 The variation of total travel time for eachpeak with different weighted parameter setting is depicted in

Figure 4 As shown in Figure 5 the total travel time of twopeaks changes with the weighted parameter setting

It is shown in Figure 4 that as expected the total traveltime of evening peak decreases monotonically with theincreasing weight of 120573 since an increasing priority will beput on improving the traffic congestion in the evening peakIn turn the total travel time of morning peak continuouslyincreases with the weighted parameter 120573 that is less empha-sis will be paid on reducing the traffic congestion in themorning peak It should be stressed that the improvementeffect of the NDP scheme in great measure depends on thenetwork structure and demand distribution In this regardwe repeat that the NDP scheme A greatly overlooks thesocial welfare of the travelers in the evening peak Thereforethe network planner should avoid implementing the NDPscheme A

In Figure 5 we can clearly see how important it is toaccount for the traffic congestions in both commuting peaksAlthough the weight of considering the traffic pattern in theevening peak is small (eg 120573 = 01) the network perform-ance for whole daily commuting will be greatly improvedThat is the NDP only considering morning peak-hour

8 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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 Mathematical Problems in Engineering

Table 5 Network design simultaneously considering two peak-hour demands (Scheme C)

Commute peak TTC Link PathNumber Expanded capacity (119909

119886) Flow (V119894

119886) Number Link component Flow (119891119894

119903119908) Equilibrium cost (120583119894

119908)

Morning peak 902092

1 6152 16198 1 1-2 6198 167572 6201 26198 2 3 18804 167583 0000 18804 3 4 10000 100554 20725 20000 4 2 20000 11499mdash mdash mdash 5 1ndash4 10000 15313

Evening peak 877038

1 6152 20000 1 1-2 0000 156312 6201 10000 2 3 10000 151413 0000 10000 3 4 25000 114074 20725 45000 4 2 10000 10032mdash mdash mdash 5 1ndash4 20000 17007

Sum 1779130 mdash mdash mdash mdash mdash mdash mdash

1300

1200

1100

1000

900

800

0 01 02 03 04 05 06 07 08 09 1

NDP Scheme A

NDP Scheme C

NDP Scheme B

Tota

l tra

vel t

ime f

or ea

ch p

eak

Weighted parameter setting (120573)

TTC of morning peakTTC of evening peak

Figure 4 Total travel time for each peak with different weightedparameter settings

demand is far inferior to the NDPs taking into account thetraffic congestion in the evening peak Meanwhile it canbe found that how to determine the weighted parametersetting is important for developing a reasonableNDP schemeRecall that the network decision-maker only concerns thetraffic congestions in two commuting peaks although sometravelers might be more sensitive to the traffic congestion inthe morning To achieve the design objective of improvingthe traffic situation of two peaks as much as possible theweighted parameter setting is preferred to set as 120572 = 120573120572 120573 gt 0 no matter whether the weighted parameters arenormalized or not In other words if 120572 = 120573 120572 120573 gt 0 theNDP considering two peak-hour demands always performsno worse than other NDP schemes (including the NDPsconsidering one peak-hour demand) in terms of minimizingthe total travel time of two commuting peaks It is not difficultto verify this conclusion By revisiting the design objectivefunction in (1) it can be seen that when 120572 = 120573 120572 120573 gt 0

001600

1700

1800

1900215711

184035

180111

178823

178134

177913

178056

178718

180429

181036

181057

2000

2100

2200

01 02 03 04 05 06 07 08 09 10

Weighted parameter setting (120573)

Tota

l tra

vel t

ime o

f two

pea

ks

Optimal weighted parameter setting withbest network performance (120572 = 120573 = 05)

Figure 5 Total travel time of two peaks with different weightedparameter settings

the design objective function is to directly optimize thenetwork performance in terms of total travel time of twopeaks with no error due to introducing priority of any peakThe outcome in the numerical examples also indicates thatNDP scheme will have the best effect when 120572 = 120573 = 05

5 Conclusions and Further Studies

In the previous NDP studies it was always assumed thatthe NDP considering one peak-hour demand distributionis capable and effective to characterize real traffic situationon the network However we found that the NDP modelconsidering only one peak-hour demand matrix might beunable to describe the real traffic patterns due to the asym-metric traffic characteristics in the morning and eveningpeaks and the impacts of network structure and link sen-sitivity The traditional NDP model considering one peak-hour demand matrix thus may lead to misleading outcomesfor transportation management and planning To address

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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

Mathematical Problems in Engineering 9

these problems we proposed an NDP model simultaneouslyconsidering morning and evening peak-hour demands Theproposed model can be used to some extent to avoid theimpacts of asymmetric demand distributions in the morningand evening commute peaks the network structure andlink sensitivity The NDP problem is formulated as a bilevelprogrammingmodel in which the upper level is to minimizethe weighted sum of total travel time for the network userstravelling in both morning and evening commute peaks andthe lower level is to characterize the user equilibrium choicebehaviors of the travelers in two peaks Some extensions onelastic and uncertain peak-hour demands for two peaks arealso discussedThe proposed NDPmodel is transformed intoan equivalent MILP which can be solved by optimizationsolvers (eg CPLEX) Through this transformation a globalsolution can be guaranteed Numerical examples are finallyperformed to demonstrate the effectiveness of the developedmodel It was shown that the proposed NDP model cangenerate more promising design effect than the traditionalNDP model considering one peak-hour demand and avoidthe misleading decision Meanwhile it was found that how tochoose a surveyed peak for data collection is very importantfor the traditional NDP model considering one peak-hourdemand That is we should carefully determine which peak-hour demand is less likely to result in misleading outcomesFurthermore for weighted parameter setting we recommendto set 120572 = 120573 120572 120573 gt 0 which leads to the best network designeffect

Further studies could be carried out to extend theproposed model in the following aspects First although weintroduce a conceptual framework of the NDP consideringpeak-hour uncertain demands in two peaks a tractable andspecific NDP model considering two peak-hour uncertaindemands could be an interesting work Second the distri-bution of multiclass users with different value-of-times andmixed vehicles would also show a large diversity in themorn-ing and evening commute peaks To address such diversity inthe NDP problem reveals important investigations

Notations

Design Variables

xk = ( 119909119886 ) 119886 isin 119860 Design variables namely link

capacity expansions

Variables to Be Determined in Each Equilibrium

V119898119886 V119890119886 Flow on link 119886 isin 119860 in the morning peak

and evening peak respectively119905119898

119886(V119898119886) 119905119890119886(V119890119886) Travel time on link 119886 isin 119860 in the morningpeak and evening peak respectively

119891119898

119903119908 Traffic flow on route 119903 isin 119877119898

119908 119908 isin 119882119898in

the morning peak119891119890

119903119908 Traffic flow on route 119903 isin 119877119890

119908 119908 isin 119882119890 in

the evening peak

119888119894

119903119908 Travel time on route 119903 for O-D pair 119908 in the

morning if 119894 = 119898 or in the evening peak if 119894 = 119890120583119898

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119898 in the morning peak

120583119890

119908 Equilibrium minimal travel cost between O-D pair

119908 isin 119882119890 in the evening peak

119887119886(119909119886) Construction cost for each improved link 119886 isin 119860

Parameter Given

119886 Link 119886 isin 119860119860 Set of candidate links to be improved 119860 sub 119860119882119898119882119890 Sets of O-D pairs in morning and even peaks

respectively119908 O-D pair 119908 isin 119882119898⋃119882119890119877119898

119908 Set of routes connecting O-D pair 119908 isin 119882119898 in

morning commute peak119877119890

119908 Set of routes connecting O-D pair 119908 isin 119882119890 in

evening commute peak119903 Route 119903 isin 119877119898

119908⋃119877119890

119908

120575119898

119886119903119908 Link-route indicator 120575119898

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119898 uses link 119886 isin 119860 atmorning peak and 0 otherwise

120575119890

119886119903119908 Link-route indicator 120575119890

119886119903119908equals 1 if route 119903

between O-D pair 119908 isin 119882119890 uses link 119886 isin 119860 atevening peak and 0 otherwise

119902119898

119908 119902119890119908 Travel demand for O-D pair in the morning

peak and evening peak respectively120572 120573 Weighted parameter in objective function1199050

119886 Free flow travel time for link 119886 isin 1198601198880

119886 Existing capacity for link 119886 isin 119860

120574119886 Parameter in link capacity improvement

construction cost

Conflict of Interests

All the authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgments

This study is supported by a program for new centuryexcellent talents in university (Grant no NCET-10-0637) aresearch Grant for 2013 Shanghai Postdoctoral SustentationFund (no 13R21416400) and a research Grant for ChinaPostdoctoral Science Foundation (no 2013M531222)

References

[1] Q Meng H Yang and M G H Bell ldquoAn equivalent continu-ously differentiable model and a locally convergent algorithmfor the continuous network design problemrdquo TransportationResearch Part B vol 35 no 1 pp 83ndash105 2001

[2] M Abdulaal and L J LeBlanc ldquoContinuous equilibrium net-work design modelsrdquo Transportation Research Part B vol 13no 1 pp 19ndash32 1979

[3] T L Magnanti and R T Wong ldquoNetwork design and trans-portation-planningmdashmodels and algorithmsrdquo TransportationScience vol 18 no 1 pp 1ndash55 1984

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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

10 Mathematical Problems in Engineering

[4] H Yang and M G H Bell ldquoModels and algorithms forroad network design a review and some new developmentsrdquoTransport Reviews vol 18 no 3 pp 257ndash278 1998

[5] A Karoonsoontawong and S Travis Waller ldquoComparison ofsystem- and user-optimal stochastic dynamic network designmodels using Monte Carlo bounding techniquesrdquo Transporta-tion Research Record vol 1923 pp 91ndash102 2005

[6] S V Ukkusuri and S T Waller ldquoLinear programming modelsfor the user and system optimal dynamic network designproblem formulations comparisons and extensionsrdquoNetworksand Spatial Economics vol 8 no 4 pp 383ndash406 2008

[7] A Chen and X Xu ldquoGoal programming approach to solvingnetwork design problem with multiple objectives and demanduncertaintyrdquo Expert Systems with Applications vol 39 no 4 pp4160ndash4170 2012

[8] A Chen and C Yang ldquoStochastic transportation networkdesign problem with spatial equity constraintrdquo TransportationResearch Record vol 1882 pp 97ndash104 2004

[9] H K Lo X W Luo and B W Y Siu ldquoDegradable transportnetwork travel time budget of travelers with heterogeneous riskaversionrdquoTransportation Research Part B vol 40 no 9 pp 792ndash806 2006

[10] H Shao W H K Lam Q Meng and M L Tam ldquoDemand-driven traffic assignment problem based on travel time relia-bilityrdquo Transportation Research Record vol 1985 pp 220ndash2302006

[11] H Wang W H K Lam X Zhang and H Shao ldquoSustainabletransportation network design with stochastic demands andchance constraintsrdquo International Journal of Sustainable Trans-portation 2013

[12] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign a study of budget sensitivityrdquo Journal of the Eastern AsiaSociety For Transportation Studies vol 5 pp 1124ndash1139 2003

[13] H K Lo and W Y Szeto ldquoTime-dependent transport networkdesign under cost-recoveryrdquo Transportation Research Part Bvol 43 no 1 pp 142ndash158 2009

[14] W Y Szeto and H K Lo ldquoTime-dependent transport networkimprovement and tolling strategiesrdquo Transportation ResearchPart A vol 42 no 2 pp 376ndash391 2008

[15] S V Ukkusuri and G Patil ldquoMulti-period transportationnetwork design under demand uncertaintyrdquo TransportationResearch Part B vol 43 no 6 pp 625ndash642 2009

[16] S T Waller K C Mouskos D Kamaryiannis and A K Zil-iaskopoulos ldquoA linearmodel for the continuous network designproblemrdquo Computer-Aided Civil and Infrastructure Engineeringvol 21 no 5 pp 334ndash345 2006

[17] S T Waller and A K Ziliaskopoulos ldquoStochastic dynamicnetwork design problemrdquo Transportation Research Record vol1771 pp 106ndash113 2001

[18] Q Meng H Yang and H Wang ldquoThe transportation networkdesign problemsrdquo Transportation Research Part B 2013

[19] A Chen Z Zhou P Chootinan S Ryu C Yang and S CWong ldquoTransport network design problem under uncertaintya review and new developmentsrdquo Transport Reviews vol 31 no6 pp 743ndash768 2011

[20] Highway Capacity Manual ldquoTransportation Research BoardrdquoWashington DC USA 2000

[21] X Zhang H Yang H-J Huang and H M Zhang ldquoInte-grated scheduling of daily work activities andmorning-eveningcommuteswith bottleneck congestionrdquoTransportationResearchPart A vol 39 no 1 pp 41ndash60 2005

[22] X Zhang H-J Huang and H M Zhang ldquoIntegrated dailycommuting patterns and optimal road tolls and parking fees ina linear cityrdquo Transportation Research Part B vol 42 no 1 pp38ndash56 2008

[23] X Zhang and B van Wee ldquoEfficiency comparison of variousparking charge schemes considering daily travel cost in a linearcityrdquo European Journal of Transport and Infrastructure Researchvol 11 no 2 pp 234ndash255 2011

[24] R Venkatanarayana B L Smith and M J DemetskyldquoQuantum-frequency algorithm for automated identification oftraffic patternsrdquo Transportation Research Record vol 2024 pp8ndash17 2007

[25] X Zhang ldquoEffects of queue spillover in networks consideringsimultaneous departure time and route choicesrdquoTransportationPlanning and Technology vol 36 no 3 pp 267ndash286 2013

[26] Y Sheffi Urban Transportation Networks Equilibrium AnalysiswithMathematical ProgrammingMethods PrenticeHall Engle-wood Cliff NJ USA 1985

[27] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part B vol 44 no 4 pp 482ndash492 2010

[28] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B vol 45 no 5 pp808ndash827 2011

[29] X Zhang and B van Wee ldquoEnhancing transportation networkcapacity by congestion pricing with simultaneous toll locationand toll level optimizationrdquo Engineering Optimization vol 44no 4 pp 477ndash488 2012

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