A Dual Approach for Solving the Combined Distribution and Assignment Problem with Link Capacity...

A Dual Approach for Solving the Combined Distributionand Assignment Problem with Link Capacity Constraints

Seungkyu Ryu & Anthony Chen & Xiangdong Xu &

Keechoo Choi

Published online: 23 January 2014# Springer Science+Business Media New York 2014

Abstract In this paper, we consider the combined distribution and assignment (CDA)problem with link capacity constraints modeled as a hierarchical logit choice problembased on random utility theory. The destination and route choices are calculated basedon the multi-nominal logit probability function, which forms the basis for constructingthe side constrained CDA (SC-CDA) problem as an equivalent mathematical program-ming (MP) formulation. A dual MP formulation of the SC-CDA problem is developedas a solution algorithm, which consists of an iterative balancing scheme and a columngeneration scheme, for solving the SC-CDA problem. Due to the entropy-type objec-tive function, the dual formulation has a simple nonlinear constrained optimizationstructure, where the feasible set only consists of nonnegative orthants. The iterativebalancing scheme explicitly makes use of the optimality conditions of the dual formu-lation to analytically adjust the dual variables and update the primal variables, while acolumn generation scheme is used to iteratively generate routes to the working route setas needed to satisfy the side constraints. Two numerical experiments are conducted todemonstrate the features of the SC-CDA model and the computational performance ofthe solution algorithm. The results reveal that imposing link capacity constraints canhave a significant impact on the network equilibrium flow allocations, and the dualapproach is a practical solution algorithm for solving the complex SC-CDA problem.

Keywords Combined distribution and assignment problem . Side constraints . Capacityconstraints . Dual approach . Iterative balancing

Netw Spat Econ (2014) 14:245–270DOI 10.1007/s11067-013-9218-2

S. Ryu : A. Chen (*)Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110,USAe-mail: [email protected]

X. XuSchool of Transportation, Southeast University, Nanjing 210096, China

K. ChoiDepartment of Transportation Engineering, Ajou University, Suwon 442-749, Korea

1 Introduction

As an integral component of urban transportation system, various types of side con-straints are typically required in order to improve realism in the travel demand forecast-ing procedure. For example, to consider the queuing effect at signalized intersections orbehind bottlenecks (e.g., bridges, tunnels, or when two or more roadways merge into asingle roadway section), imposing physical link capacity constraints may be a betterchoice compared to modifying the link travel time functions. On the other hand, toreflect the environmental requirements imposed by the government, certain environ-mental constraints can also be applied to restrict vehicles entering certain areas (e.g.,central business district). Hearn and Ribera (1980) suggested that modifying the trafficassignment procedure by adding side constraints (e.g., link capacity constraints) is agood approach to obtain reasonable traffic flow pattern in a road network.

Side constraints, in general, describe the limited supply of certain resources (e.g., linkcapacities) in a road network which are shared by several activities (e.g., link flows), orcertain natural or technological restrictions (e.g., signal control, emission restraints, etc.).By imposing side constraints, the effect of different traffic control policies can bedescribed, the existing traffic equilibrium models can be improved, and the flowrestrictions that the central authority wishes to impose upon the network users can bedescribed (Larsson and Patriksson 1999). According to Larsson and Patriksson (1999),side constraints can be distinguished into two principally different types in the trafficassignment models: prescriptive (hard) and descriptive (soft). The prescriptive sideconstraints are typically imposed upon the users of traffic system. It strictly constrainsthe traffic flows and it is used for traffic management and control policies (e.g., trafficsignals with fixed cycle times). On the other hand, descriptive side constraints areintroduced to refine additional traffic flow restrictions into the model (e.g., jointcapacities in roundabouts, merging capacities, etc.), to better model congestion effect,and to incorporate some a priori knowledge of traffic flow (e.g., replicating observedflows on some links and/or intersections within certain error bounds). Modeling sideconstraints in the traffic assignment problem has been studied in various areas assummarized below:

& Queuing and congestion effects (Bell 1995; Yang and Lam 1996; Larsson et al.2004; Chen et al. 2011)

& Link toll pricing (Ferrari 1995; Yang and Lam 1996; Yang and Bell 1997)& Restraining traffic flows (Ferrari 1995; Yang and Bell 1997; Chen et al. 2011)& Replicating observed flows (Bell et al. 1997; Chen et al. 2005, 2009, 2010, 2012a;

Chootinan et al. 2005)& Environmental side constraints (Ferrari 1995; Chen et al. 2011)

Although side constraints have been incorporated into various applications, moststudies focused on the modeling of side constraints in the traffic assignment problemfollowing Wardrop’s first principle (Wardrop 1952). In contrast, only a few studiesconsidered side constraints in the combined distribution and assignment (CDA) prob-lem in the literature (e.g., Tam and Lam 2000, 2004; Li et al. 2007). Note that CDAmodels have several applications in transportation systems analysis. Among others, thefollowing applications need to use CDA models to capture travelers’ route choice anddestination choice behaviors simultaneously.

246 S. Ryu et al.

(1) Network design problem (NDP). For example, in the bi-level program (BLP)developed by Lin and Feng (2003) for the land use-NDP, a CDA model was usedto simulate travel demands according to the land use distribution and networklayout. Also, in the BLP developed by Yim et al. (2011) for the reliability-basedland use (including residential and job allocations) and NDP, a CDA model wasused to map the land-use pattern to the link-loading pattern in the network.

(2) Analysis of network capacity, capacity reliability, and capacity flexibility. Forexample, Yang et al. (2000) used a CDAmodel as the lower-level subprogram in aBLP model to determine the network capacity and level of service. Chen et al.(2013) and Chen and Kasikitwiwat (2011) further used the CDA model inmodeling capacity reliability and capacity flexibility, respectively.

(3) Taxi operation modeling. For example, Wong et al. (2001, 2008) developed a BLPmodel for taxi movements in congested road networks. A CDA model was usedto describe the simultaneous movements of vacant and occupied taxis as well asnormal traffic for a given total customer generation from each origin and a giventotal attraction to each destination.

For additional information of the combined network equilibrium models, readers aredirected to Boyce and Bar-Gera (2004) and Hasan and Dashti (2007) on the multiclasscombined models, Briceño et al. (2008) on an integrated behavioral model of land-use andtransport system, de Grange et al. (2010) on the calibration and spatial aggregation ofcombined models, Chen et al. (2007) on the network-based accessibility measures using thecombined travel demandmodel, Siefel et al. (2006) on the comparison of urban travel forecastsbetween the sequential four-step model and the combined travel demand model, and Boyce(2007) on a general review of network equilibrium models for travel demand forecasting.

In a few existing studies considered side constraints in the CDA problem (e.g., Tamand Lam 2000, 2004; Li et al. 2007), travelers’ route choice behavior (trip assignment)is characterized by the deterministic user equilibrium (DUE) model, whereas thedestination choice behavior (trip distribution) is governed by a multinomial logit model(or an entropy model). There exist some behavioral inconsistencies between these twotravel choices. In terms of solution algorithm, the penalty method was adopted tohandle the side constraints (see Li et al. 2007). As is well known, the penalty methodmay have difficulty converging if inappropriate penalty value is used.

Motivated by the above observations, we consider the CDA problem with link capacityconstraints by extending Bell (1995)’s logit assignment model to a hierarchical logit-baseddistribution and assignment model (i.e., both destination choice and route choice aregoverned by the logit model in a hierarchical structure) with link capacity constraints, andfurther developingBell’s approach based on the dual formulation as a solution algorithm thatexplicitly makes use of the optimality conditions to analytically adjust the dual variables andupdate the primal variables (i.e., route flows andO-D flows) alongwith a column generationscheme. Specifically, the logit-based CDA model based on random utility theory resolvesthe behavioral choice inconsistencies of the existing CDAmodels and enhances the realismof the CDA model by explicitly incorporating physical capacity restrictions as side con-straints. Different from directly solving the SC-CDA problem (e.g., penalty method), a dualMP formulation is developed as a solution algorithm for solving the SC-CDA problem. Dueto the entropy-type objective function, the dual formulation of the SC-CDA problem has asimple nonlinear constrained optimization structure, where the feasible set only consists of

Solving the Combined Distribution and Assignment Problem 247

nonnegative orthants. The proposed solution algorithm consists of an iterative balancingscheme and a column generation scheme. The iterative balancing scheme explicitly makesuse of the optimality conditions of the dual formulation to analytically adjust the dualvariables and update the primal variables, while a column generation scheme is used toiteratively generate routes to the working route set as needed to satisfy the side constraints.Two numerical experiments are also conducted to demonstrate the features of the SC-CDAmodel and the computational performance of the solution algorithm.

The remainder of the paper is organized as follows. After the introduction, the next sectionprovides the MP formulation of the SC-CDA problem. Next, the dual MP formulation of theSC-CDA problem is provided to develop a solution algorithm for solving the SC-CDAproblem. Finally, numerical results are presented along with some concluding remarks.

2 Combined Distribution and Assignment Model with Link Capacity Constraints

This section describes the combined distribution and assignment problem with linkcapacity constraints. Notation is provided first for convenience, followed by thehierarchical destination choice and route choice structure, mathematical programmingformulation, and network equilibrium conditions.

2.1 Notation

Consider a transportation network G = (N, A), where N is the set of nodes and A is theset of links on a given network. We define the following list of notations.

Sets

R set of origins, R⊆NS set of destinations, S⊆NRS set of origin-destination (O-D) pairsKrs set of routes connecting O-D pair rsA set of capacity constrained links, A⊆A

Parameters and inputs

θr dispersion parameter for route choiceθd impedance parameter for destination choiceθt rescaled parameter, where 1

θt¼ 1

θd− 1

θr

� �Pr trip production of origin rCa capacity on link a

Variables

xa flow on link ata(⋅) travel time on link aδkars route-link indicator, 1 if link a is on route k between O-D pair rs and 0 otherwisef krs flow on route k connecting O-D pair rs

248 S. Ryu et al.

ckrs cost on route k connecting O-D pair rsqrs O-D flow between origin r and destination s

2.2 Hierarchical Distribution and Assignment Model

Figure 1 shows the hierarchical structure of the combined distribution andassignment problem without link capacity constraints. It seeks to determineconsistent level-of-service and flow values of the trip distribution and tripassignment steps by resolving the inconsistency issue of the sequential traveldemand forecasting procedure between the destination choice and routechoice steps. Following Oppenheim (1995), each traveler’s destination-routedecision process is assumed to follow a top-down hierarchical structure asfollows:

1. Given a traveler at origin r, a given time period (e.g., hour, day, etc.), and anactivity (e.g., shopping, work, recreation, etc.), he/she first decides a destination sto conduct the activity ( ps|r).

2. Given the destination choice made at the first level, he/she then chooses a route kfor traveling from origin r to destination s to conduct the activity (pk|rs).

To account for the perception error of destination choice and route choice,the hierarchical logit choice problem based on random utility theory isassumed. Hence, the probabilities at each choice step are calculated basedon the multinomial logit choice function. The number of trips from origin rto destination s, qrs, is calculated by multiplying the destination choiceprobability (ps|r) with the number of productions from origin r (Pr) at thefirst choice step (i.e., destination choice). The destination choice probability(ps|r) is calculated as follows:

psjr ¼qrsPr

¼ e−θd eWrs

� �Xs∈S

e−θd eWrs

� � ;∀r∈R; s∈S; ð1Þ

Fig. 1 Hierarchical structure of CDA model


where eWrs is the expected received disutility of all routes from r to sexpressed as:

eWrs ¼ −1

θrlnXk∈Krs

e−θr crskð Þ;∀rs∈RS; ð2Þ

where ckrs is the cost on route k connecting O-D pair rs. Once each user has

determined a destination s, the proportion of users choosing a particularroute k (pk|rs) depends on the costs of all routes between origin r anddestination s as follows:

pkjrs ¼f rskqrs

¼ e−θr crskð ÞXk∈Krs

e−θr crskð Þ ;∀k∈Krs; rs∈RS: ð3Þ

Therefore, flow on route k connecting O-D pair rs can be obtained as:

f rsk ¼ Pr⋅psjr⋅pkjrs

¼ Pr⋅e−θd Wers� �Xs∈S

e−θd Wers� � ⋅ e−θr crskð ÞXk∈Krs

e−θr crskð Þ ; ∀k∈Krs; rs∈RS: ð4Þ

2.3 Mathematical Program

Based on the hierarchical distribution and assignment structure given in Fig. 1, theCDA problem with link capacity constraints can be formulated as a convex programwith side constraints as follows.

Minimize Z f ; qð Þ ¼Xa∈A

Z xa

0ta ωð Þdωþ 1

θr

Xrs∈RS

Xk∈Krs

f rsk ln f rsk −1� �þ 1

θt

Xrs∈RS

qrs lnqrs−1ð Þ ð5Þ

subject to :Xs∈S

qrs ¼ Pr; ∀ r∈R ð6Þ

Xk∈Krs

f rsk ¼ qrs; ∀rs∈RS ð7Þ

xa≤Ca; ∀ a∈�A ð8Þ

xa ¼Xrs∈RS

Xk∈Krs

f rsk δrska; ∀ a∈A ð9Þ

250 S. Ryu et al.

f rsk ≥0; ∀k∈Krs; rs∈RS ð10Þ

qrs≥0; ∀rs∈RS ð11ÞThe side constrained CDA (SC-CDA) formulation assumes the dispersion

parameter for route choice is larger than the impedance parameter for destina-tion choice, i.e., θr≥θd . For convenience, θt is defined as the rescaled param-eter for θr and θd. Since θt must be positive for the random utility function tohave meaning, the relationship between the rescaled parameter (θt) and the dispersion

and impedance parameters (θr and θd) implies that 1θt¼ 1

θd− 1

θr

� �≥0. For more details,

see Oppenheim (1995).The objective function (5) consists of three terms: a user equilibrium term and two

entropy terms associated with route choice and destination choice, respectively. Theentropy terms are slightly different from that used by Oppenheim (1995) due to the“−1” in the logarithmic terms. It is introduced to obtain a simpler, closed-formexpression of route flows and O-D flows in Eqs. (15) and (16). For simplicity, the O-D attractiveness hrs is not considered in the current formulation. It can be easily

included by adding an attractiveness term − ∑rs∈RS

hrs⋅qrs

� �to the objective function.

Equations (6) and (7) are the conservation constraints at the zonal level (i.e., the sum ofall destination flows from origin r equals its zonal production flows) and O-D level(i.e., the sum of all route flows between origin r and destination s equals its O-Ddemand). Compared to Oppenheim’s CDA problem under the logit-based choiceprobability expressions, a major difference is the physical link capacity constraintsintroduced in Eq. (8) to restrict the assigned link flows to be less than the link capacity.Equation (9) is a definitional constraint (i.e., the sum of all route flows passing throughlink a equals its link flow), and Eqs. (10) and (11) are the non-negativity constraints forroute flows and O-D flows. For this convex program with linear constraints, both routeflows and O-D flows are unique due to the two entropy terms in the objective functionand the strictly increasing link travel time function.

2.4 Lagrangian Function for the Mathematical Program

The Lagrangian function of the above SC-CDA mathematical programming formula-tion and its first partial derivatives with respect to the route flows and O-D flows can beexpressed as follows.

L f ; q; o; u; dð Þ ¼Xa∈A

Z xa

0ta ωð Þdωþ 1

θr

Xrs∈RS

Xk∈Krs


θt

Xrs∈RS

qrs lnqrs−1ð Þ

−Xr∈R

or Pr−Xs∈S

qrs

!−Xrs∈RS

urs qrs−Xk∈Krs

f rsk

!−Xa∈�A

da Ca−Xrs∈RS

Xk∈Krs

f rsk δrska

!ð12Þ


where or, urs, and da are the dual variables of constraints (6)–(8), respectively; or is theperceived attractiveness of origin r; urs is the perceived minimum O-D cost betweenorigin r and destination s; and da is related to the link queuing delay when the estimatedlink flow reaches its capacity (Bell and Iida 1997).

The logarithmic terms in Eq. (12) ensure that both the route flow variable fkrs and O-

D demand variable qrs are strictly positive. After taking the first-order derivatives, wehave

∂L∂ f rsk

¼ 0 ⇒Xa∈A

ta xað Þ δrska þ1

θrln f rsk þ urs þ

Xa∈�A

daδrska ¼ 0; ∀ k∈Krs; rs∈RS; ð13Þ

∂L∂qrs

¼ 0 ⇒1

θtlnqrs þ or−urs ¼ 0; ∀ rs∈RS: ð14Þ

From Eqs. (13) and (14), the optimality conditions lead to the following analyticalexpressions for both route flows and O-D flows:

f rsk ¼ exp −θrXa∈A

ta xað Þ δrska þ urs þXa∈�A

daδrska

! !; ∀ k∈Krs; rs∈RS; ð15Þ

qrs ¼ exp −θt or−ursð Þð Þ; ∀ rs∈RS: ð16ÞFrom the conservation Eq. (7), the logit-based route choice probability expression

can be obtained

pk=rs ¼f rskX

k∈Krs

f rsk¼

exp −θr crsk þXa∈�A

daδrska

! !Xk∈Krs


daδrska

! !; ∀ k∈Krs; rs∈RS; ð17Þ

where crsk ¼ ∑a∈A

ta xað Þ δrska; ∀ k∈Krs; rs∈RS . Similarly, the perceived minimum

O-D cost (urs) can be obtained by summing Eq. (15) and rearranging the terms asfollows.

urs ¼ −1

θrlnqrs−ln

Xk∈Krs

exp −θr crsk þXa∈A

daδrska

! ! !ð18Þ

From Eqs. (16) and (18) and the rescaled parameter (θt), we can rewrite the O-Dflow variables as

qrs ¼ exp −θd or−1

θrlnXk∈Krs


daδrska

! ! ! !; ∀ rs∈RS ð19Þ

252 S. Ryu et al.

Then, the logit-based destination choice probability expression can be written as

psjr ¼qrsX

s∈Sqrs

¼exp −θd −

1

θrlnXk∈Krs


daδrska

! ! ! !Xs∈S

exp −θd −1

θrlnXk∈Krs


daδrska

! ! ! !; ∀rs∈RS ð20Þ

Therefore, flow on route k connecting O-D pair rs in the SC-CDA problem can beobtained as:

f rsk ¼ Pr⋅psjr⋅pkjrs

¼ Pr⋅

exp −θd −1

θrlnXk∈Krs


daδrska

! ! ! !Xs∈S

exp −θd −1

θrlnXk∈Krs


daδrska

! ! ! !⋅


daδrska

! !Xk∈Krs


daδrska

! !; ð21Þ

∀k∈Krs; rs∈RS

3 Solution Algorithm

Note that the MP formulation of the SC-CDA problem in Eqs. (5)–(11) is quitecomplex, since the objective function has an UE term and two entropy terms.In addition, the inclusion of link capacity constraints destroys the tractableCartesian product structure of the feasible set as in the conventional trafficassignment problem with side constraints (Patriksson 1994, 2004; Nie et al.2004). Hence, solving the SC-CDA model is much more complex than theconventional CDA model without side constraints. In the literature, most solu-tion techniques for solving convex programs with side constraints adopt thepenalty method. Penalty function methods are classical methods used to solveconstrained nonlinear programming problems. Applications in the traffic assign-ment problem with link capacity constraints often used the outer and innerpenalty function methods (Hearn and Ribera 1980; Inouye 1987) embedding atraffic assignment algorithm (e.g., link-based algorithm or path-based algorithm)for solving the traffic assignment problem without link capacity constraints ineach iteration. These methods have been shown to be numerically unstable fordifferent penalty parameter updating schemes. A better method is to combinethe Lagrangian dual methods with the penalty function methods. Larsson andPatriksson (1995) developed an augmented Lagrangian multiplier method em-bedded with their disaggregate simplicial decomposition algorithm (Larsson andPatriksson 1992) for solving the traffic assignment problem with link capacityconstraints. Nie et al. (2004) numerically demonstrated with the augmentedLagrangian method and the inner penalty function method. Note that mostalgorithms were developed for solving the deterministic user equilibrium prob-lem with link capacity constraints by embedding a traffic assignment algorithmand solved a sequence of penalized (or augmented multiplier) problems. Theperformances of these algorithms depend on the efficiency of the traffic assign-ment procedure since it is iteratively invoked in each iteration.


Instead of adopting the augmented Lagrangian multiplier method, we further devel-oped the dual approach for the logit-based SUE problem with link capacity constraints(Bell 1995) to solve the hierarchical logit-based CDA problem with link capacityconstraints in this paper. The advantage is that the dual problem of the hierarchicallogit-based CDA problem with link capacity constraints is easier to solve, and thanks tothe entropy-type objective function, the primal variables (i.e., route flows and ODflows) can be analytically updated (Fang et al. 1997). Hence, it is not necessary toiteratively invoke a traffic assignment procedure (and another procedure for solving thedistribution problem) in each iteration as in the augmented Lagrangian (or penalty)methods for solving the traffic assignment problem with link capacity constraints. Notethat the convergence proof of the iterative balancing algorithm has been provided byBell (1995). Here we focus on showing how to obtain the adjustment factors forupdating the dual variables.

3.1 Dual Formulation

In order to construct the dual formulation of the SC-CDA problem, the followingsimple inequality is considered:

ln ρ≤ρ−1 ð22ÞThis inequality becomes equality if and only if ρ=1. The ratios for both route choice

and destination choice can be specified as follows.

ρr ¼hrskf rsk

¼exp −θr crsk þ urs þ

Xa∈�A

daδrska

! !f rsk

; ∀ k∈Krs; rs∈RSð23Þ

ρd ¼grsqrs

¼ exp −θt or−ursð Þð Þqrs

; ∀rs∈RS ð24Þ

Note that the above two ratios equal one only at the equilibrium solution. Bysubstituting ρr and ρd in Eq. (22), we have

−θr crsk þ urs þXa∈�A

daδrska

!−ln f rsk ≤

hrskf rsk

−1; ∀ k∈Krs; rs∈RS ð25Þ

−θt or−ursð Þ−lnqrs ≤grsqrs

−1; ∀rs∈RS ð26Þ

By summing over all routes in Eq. (25) and all O-D pairs in Eq. (26) in the network,we have

−Xrs∈RS

ursXk∈Krs

f rsk −qrs

! !−Xa∈�A

daXrs∈RS

Xk∈Krs

f rsk δrska

!−Xr∈R

orXs∈S

qrs

!−1

θr

Xrs∈RS

Xk∈Krs

hrsk

!−

1

θt

Xrs∈RS

grs

!≤Xrs∈RS

Xk∈Krs

crsk frsk þ 1

θr

Xrs∈RS

Xk∈Krs


θt

Xrs∈RS

qrs lnqrs−1ð Þ ð27Þ

254 S. Ryu et al.

Clearly, the right-hand side of Eq. (27) is equivalent to the objective function of the SC-CDA problem with constant route costs. The left-hand side of Eq. (27) is at most equal tothe objective value of the primal problem (the right-hand side of Eq. (27)). Therefore,maximizing the left-hand side of Eq. (27) is essentially the dual formulation ofthe SC-CDA problem. The dual program, which is important in developingmany solution techniques for the entropy-type formulations, can be stated as:

Maxda;or∈Ω

W ¼ −Xa∈�A

daCa−Xr∈R

orPr−1

θr

Xrs∈RS

Xk∈Krs

exp −θr crsk þ urs þXa∈�A

daδrska

! ! !

−1

θt

Xrs∈RS

exp −θt or−ursð Þð Þ !

Ω ¼ da≥0;∀a∈�A; or is unrestricted in sign;∀ r∈R

n oð28Þ

Note that the first term in Eq. (27), − ∑rs∈RS

urs ∑k∈Krs

f rsk −qrs

! !, is dropped,

because the term in the inner parenthesis equals zero according to the O-D demandconservation in Eq. (7). Also, the dual variable (urs) associated with the O-D demandconservation constraint (7) can be determined by Eq. (14) as follows.

urs ¼ 1

θtlnqrs þ or; ∀ rs∈RS: ð29Þ

Remark 1 The dual formulation in Eq. (28) is associated with a given path flowand demand pattern (and accordingly a given ck

rs and urs). We use the iterativedual formulation to design an iterative balancing scheme for solving the SC-CDA model. Specifically, the optimality conditions of the iterative dual formu-lation are used to analytically adjust the dual variables and update the primalvariables per iteration as shown in the next section.

Remark 2 The second term on the left-hand side of Eq. (27) is equal to − ∑a∈A

daxa .

Note that the dual variable is active/positive only when a link flow reaches its

own capacity; otherwise, it is equal to zero. Hence, we can decompose the term

of − ∑a∈A

daxa into two parts (active and inactive links). For the inactive links,

the multiplier da is equal to zero, and accordingly daxa=0. In other words, theinactive terms do not affect the optimal objective value (i.e., ∑

a∈Adaxa ¼ ∑

a∈AdaCa ).

On the other hand, the dual formulation (28) has a duality gap between the UE (userequilibrium) and SO (system optimal) objectives. The first term ∑

rs∈RS∑

k∈Krs

crsk frsk on

the right-hand side of Eq. (27) is not equal to the congestion term in the primal

objective function ∑a∈A

∫xa0 ta ωð Þdω . However, the iterative dual formulation is


associated with a given path flow and demand pattern (and accordingly a given ckrs

and urs) for each iteration.

3.2 Dual Variable Adjustment

The solution algorithm for solving the SC-CDA problem is based on the iterativebalancing scheme derived from the dual formulation (28). The basic idea is to maxi-mize the dual problem by adjusting the dual variables. The iterative balancing schemestarts with an arbitrarily set of dual variables (da and or) within the feasible set Ω. Inother words, the scheme seeks to satisfy one constraint at a time in the primal problem(i.e., Eqs. (6) and (8)) by adjusting one dual variable in each iteration.

Once the scheme converges, the optimal primal variables can be determined byEqs. (15) and (16). The first-order derivatives of the dual objective function withrespect to the dual variables (da, or) can be expressed as

∇daW ¼ Ca−Xrs∈RS

Xk∈Krs

exp −θrXa∈A

ta xað Þδrska þ urs þXa∈�A

daδrska

! !⋅δrska; ∀a∈�A

ð30Þ∇orW ¼ Pr−

Xs∈S

exp −θt or−ursð Þð Þ; ∀r∈R ð31Þ

By substituting urs in Eqs. (30) and (31) with Eqs. (29) and (19), respectively, thederivatives become

∇daW ¼ Ca−Xrs∈RS

Xk∈Krs

exp −θrXa∈A

ta xað Þδrska þ1

θtlnqrs þ or þ

Xa∈�A

daδrska

! !⋅δrska; ∀ a∈�A ð32Þ

∇orW ¼ Pr−Xs∈S

exp −θd or−1

θrlnXk∈Krs

exp −θrXa∈A

ta xað Þδrska þXa∈�A

daδrska

! ! ! !; ∀ r∈R ð33Þ

Based on Eqs. (32) and (33), we need to find an adjustment factor for each dualvariable such that the derivatives vanish.

(1) βa for adjusting da

Xrs∈RS

Xk∈Krs

exp −θrXa∈A


θtlnqrs þ or þ

Xa∈�A

daδrska þ βa

! !⋅δrska ¼ Ca ⇒

Xrs∈RS

Xk∈Krs

exp −θrXa∈A


θtlnqrs þ or þ

Xa∈�A

daδrska

! !⋅δrska⋅exp −θr⋅βað Þ ¼ Ca⇒X

rs∈RS

Xk∈Krs

f rsk ⋅ δrska ⋅ exp −θr⋅βað Þ ¼ Ca⇒

exp −θr⋅βað Þ ¼ Ca

xa⇒βa ¼ −

1

θrln

Ca

xa

� �; ∀ a∈�A

ð34Þ

256 S. Ryu et al.

(2) λr for adjusting orXs∈S

exp −θd or−1

θrlnXk∈Krs

exp −θrXa∈A


daδrska

! !þ λr

! !¼ Pr⇒

Xs∈S

exp −θd or−1

θrlnXk∈Krs

exp −θrXa∈A


daδrska

! ! ! !⋅exp −θdλrð Þ ¼ Pr⇒

Xs∈S

qrs⋅exp −θdλrð Þ ¼ Pr ⇒λr ¼ −1

θdln

PrXs∈S

qrs

0BB@1CCA; ∀ r∈R

ð35Þ

3.3 Solution Procedure

Using the adjustment factors derived in Eqs. (34) and (35), the iterative balancingsolution procedure can be summarized as follows.

Step 0 Initialization. Set m=0, (da)m=0 for all links in A and (or)

m=0 for all origins.Compute initial primal variables (O-D flows, route flows, and link flows).

qrsð Þm ¼ exp −θd orð Þm− 1

θrlnXk∈Krs

exp −θrXa∈A


dað Þmδrska ! ! ! !

f rsk� �m ¼ exp −θr

Xa∈A


θdln qrsð Þm þ orð Þm þ

Xa∈�A

dað Þmδrska ! !

xað Þm ¼Xrs∈RS

Xk∈Krs

f rsk� �m

δrska

Step 1 Determine adjustment factors.

λrð Þm ¼ −1

θdln

PrXs∈S

qrsð Þm

0BB@1CCA and βað Þm ¼ −

1

θrln

Ca

xað Þm� �

Step 2 Update dual variables.

orð Þmþ1 ¼ orð Þm þ λrð Þm and dað Þmþ1 ¼ Max 0; dað Þm þ βað Þmf gStep 3 Update primal variables (O-D flows, route flows, and link flows).

qrsð Þmþ1 ¼ exp −θd orð Þmþ1−1

θrlnXk∈Krs

exp −θrXa∈A

ta xma� �

δrska þXa∈�A

dað Þmþ1δrska

! ! ! !

f rsk� �mþ1 ¼ exp −θr

Xa∈A


θdln qrsð Þmþ1 þ orð Þmþ1 þ

Xa∈�A

dað Þmþ1δrska

! !xað Þmþ1 ¼

Xrs∈RS

Xk∈Krs

f rsk� �mþ1

δrska

Step 4 Convergence test. Determine the average excess cost for the SC-CDA problemas follows.

ε ¼ 1

D

Xrs∈RS

Xk∈Krs

f rsk� �mþ1 eckrs� �mþ1

−mink∈Krs

eckrs� �mþ1� � �


where ε is the average excess cost, D is the total demand, and ecrsk is the

generalized route cost ecrsk ¼ crsk þ 1θrln f rsk þ ∑

a∈Adaδ

rska

!.

If max orð Þmþ1− orð Þm ; dað Þmþ1− dað Þm n o≥η (e.g., 104, an upper limit

for detecting divergence), terminate the iterative balancing scheme and go tocolumn generation (route generation) step. Otherwise, set m=m+1 and go tostep 1 until ε≤η (e.g., 10−8, a lower limit for stopping criteria).

It is worth noting that the dual variable adjustment factors in Step 1 explicitly make useof the optimality conditions of the dual formulation. The above iterative balancing schemeassumes that a working route set is given. For large networks, a column (or route)generation procedure also can be augmented to the iterative balancing scheme as shownin Fig. 2. Basically, the solution algorithm introduces an outer loop (or iteration) toiteratively generate routes to the working route set as needed to satisfy the link capacityconstraints and congestion effects, while the iterative balancing scheme iteratively adjuststhe primal (O-D flows, route flows, and link flows) and dual variables (perceived originattractivenesses, perceived O-D minimum costs, and link queuing delays) in the innerloop for a given working route set from the outer loop. Note that there are many ways todevelop the column generation scheme to create the working route set in Fig. 2. In thisstudy, we adopted the actual link cost scheme, which is based on not only the link costsbut also the dual variables (i.e., link queuing delays) from the active side constraints in

Iterative Balancing

Inner Loop

Update Link CostsUsing Dual variables

Initialization

Column Generation

Update Working Route Set Converge?

YES

NO

Output

Outer Loop

a a at t d

,a rd o

`, ,rsrs k aq f x

Compute Dual Variables :

Compute Primal Variables :

Initial Primal Variables :

Initial Dual Variables :

, ,rsrs k aq f x

,a rd o

n=0 (outer iteration )

Update Link Costs

( )a a at t xn=n+1

Fig. 2 Flowchart of the solution algorithm

258 S. Ryu et al.

Eq. (8). The dual variables force the column generation scheme to generate routes thatsatisfy the link capacity constraints. Damberg et al. (1996) described this scheme as “avalid approach especially when the random components are less significant; this appliesto the situation where the value of the (dispersion) parameter is large and travelers aresensitive to the travel costs (e.g., when the congestion level is high)”. Note that Bell (1995)also adopted the actual route cost to generate routes for solving the SUE problemwith linkcapacity constraints. Another way suggested by Chen et al. (2009) is to implement ashortest “simple path” with the presence of negative cycle by including a scanningoperation to the shortest path algorithm to eliminate negative cost cycles (e.g., see thelabeling and scanning method by Tarjan (1983) for details). It is also possible to use aninitial route set with routes generated by a choice set generation scheme based onbehavioral survey (Bekhor et al. 2008a) in the Initialization Step to minimize the occur-rence of generating paths with negative cost loop. As for the convergence test in Step 4, weadopted the average excess cost used by Bar-Gera and Boyce (2003) in the origin-basedalgorithm. Note that the average excess cost makes use of the generalized route cost todetermine the equilibrium solution. To handle the numerical problems such as underflow/overflow, we restrict the error to be within a computable range (e.g., 104 as an upper limitfor detecting divergence and 10−8 (or 10−10) as a lower bound for stopping the algorithm).

4 Numerical Results

To demonstrate the SC-CDA model and the dual approach as a solution algorithm, twonetworks are adopted in the numerical experiments. First, a small network is used toillustrate the features of the SC-CDA model and its differences compared to the originalCDAmodel without link capacity constraints. Then, a medium-sized network is employedto demonstrate the applicability of the model and solution algorithm to larger networks.

4.1 A Small Grid Network

In the first experiment, a hypothetical network shown in Fig. 3 is adopted to demon-strate the effect of including physical link capacity constraints in the CDA problem.The network contains 9 nodes, 14 links, and 9 O-D pairs. For this network, there are atotal of 33 routes, all of which will be included in solving the CDA problem withoutand with physical link capacity constraints. Travel times on each link are assumed tofollow the Bureau of Public Roads (BPR) function.

ta ¼ t0a 1þ 0:15vaCa

� �4 !

; ∀a∈A ð36Þ

where ta0 is free-flow travel time of link a. The zonal productions and link characteristics

are also shown in Fig. 3. The dispersion parameter and the impedance parameter (θr andθd) for route choice and destination choice are assumed to be 0.2 and 0.1, respectively.


4.1.1 Convergence Characteristics

To demonstrate the effect of incorporating side constraints in the CDA problem, twolinks (link 9 and link 10) are considered as physical link capacity constraints in the SC-CDA problem. First, we show the convergence characteristics of the solution algorithmin Figs. 4 and 5. For convenience, “CDA” and “SC-CDA” are used to denote the CDAproblem without and with side constraints, respectively. Figure 4 shows the conver-gence characteristics of the solution algorithm derived based on the dual approach forsolving both CDA and SC-CDA problems. In both problems, the solution algorithm

1E-101E-091E-081E-071E-061E-051E-041E-031E-021E-011E+00

1 6 11 16 21 26 31 36

Ave

rag

e E

xces

s C

ost

Iteration #

CDA SC-CDA

Fig. 4 Convergence character-istics of CDA and SC-CDAmodels

1 2 3

4 5 6

7 8 9

2

1 4

6510 12

7 9

1413

3

118

(a) A small grid network (b) Zonal productions

(c) Network characteristicsFig. 3 A small grid network with 9 O-D pairs

260 S. Ryu et al.

can converge to a highly accurate solution (i.e., 1.0E-10 in terms of the logarithmicaverage excess cost). Both cases exhibit a linear convergence rate. However, the SC-CDAproblem requires a fewmore iteration since it has to adjust two extra dual variables for thephysical capacity constraints on link 9 and link 10. Figure 5 depicts the trajectories of twoLagrangianmultipliers (i.e., dual variables for the two active capacity constraints on link 9and link 10) during the iterative process. As can be seen in Fig. 5, both dual variablesstabilize quickly after the first ten iterations. In terms of the actual dual values, link 9 has alarger value (or higher queuing delay) compared to that of link 10.

4.1.2 Effect of Link Capacity Constraints

We examine the effect of imposing physical link capacity constraints on the CDAproblemin Fig. 6 and Table 1. Figure 6 presents the volume to capacity (V/C) ratios for both CDAand SC-CDA models. From Fig. 6, flows on link 9 and link 10 exceed the allowablecapacity in the CDA model without side constraints. When physical capacity constraintsare imposed on these two links, the excess flows are diverted to other under-utilized links(e.g., links 4, 6, 8, and 13) via routes that bypass these two critical links. Table 1 provides acomplete summary of the assigned route flows and O-D flows for both CDA and SC-CDAmodels. It can be observed that route flows (i.e., routes 1, 2, and 4) that pass throughlink 9 are significantly reduced to fulfill the link capacity constraints. Also, the O-D flows(i.e., destination choice) are used as an additional means to fully utilize the spare capacityin the network (i.e., flow on O-D (1, 6) is also lower in the SC-CDA model).

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1 6 11 16 21 26 31 36

Qu

euin

g D

elay

(m

in)

Iteration #

Link 9 link 10

Fig. 5 Lagrange multiplier tra-jectories of link 9 and link 10

Flows diversion from link 9 and link 10

V/C=1.0

0.000.200.400.600.801.001.201.401.601.80

1 2 3 4 5 6 7 8 9 10 11 12 13 14

V/C

Rat

io

Link #

CDA SC-CDA

Fig. 6 V/C ratios of both CDA and SC-CDA models


4.2 Winnipeg Network

In experiment 2, the SC-CDA model and solution algorithm are further demonstratedusing the well-known Winnipeg network as depicted in Fig. 7. The network consists of

Table 1 Assigned route flows and O-D flows of both CDA and SC-CDA models

O-D Route # Link member CDA SC-CDA

Route flow O-D flow Route flow O-D flow

1–6 1 3–9 27.13 110.88 16.88 101.68

2 1 – 5 – 9 26.97 16.78

3 1 – 4 – 6 27.50 49.63

4 2 – 7 – 9 29.29 18.38

1–8 1 3 – 10 25.53 109.52 21.96 112.16

2 2 – 8 – 13 31.09 44.85

3 1 – 4 – 10 25.33 21.43

4 2 – 7 – 10 27.56 23.91

1 – 9 1 3 – 11 16.83 149.60 23.86 156.16

2 3 – 9 – 12 12.20 5.86

3 1 – 5 – 9 – 12 12.13 5.83

4 1 – 4 – 6 – 12 12.37 17.23

5 2 – 7 – 9 – 12 13.17 6.38

6 2 – 8 – 13 – 14 13.63 18.92

7 1 – 4 – 10 – 14 11.10 9.04

8 3 – 10 – 14 11.19 9.26

9 2 – 7 – 10 – 14 12.08 10.09

10 1 – 5 – 11 16.73 23.72

11 2 – 7 – 11 18.17 25.98

2–6 1 4–6 71.30 141.23 104.82 140.27

2 5–9 69.93 35.45

2–8 1 5–10 108.60 108.60 103.35 103.35

2–9 1 5–11 54.38 170.18 74.74 176.38

2 5 – 10 – 14 36.17 29.02

3 5 – 9 – 12 39.43 18.35

4 4 – 6 – 12 40.20 54.27

4–6 1 7–9 98.01 98.01 79.62 79.62

4–8 1 7–10 57.58 122.53 45.89 131.96

2 8–13 64.95 86.07

4–9 1 7–11 47.59 149.46 67.06 158.42

2 8 – 13 – 14 35.71 48.85

3 7 – 10 – 14 31.65 26.04

4 7 – 9 – 12 34.51 16.47

Total demand 1160.00 1160.00

262 S. Ryu et al.

154 zones, 2,535 links and 4,345 O-D pairs. The BPR formula with link-specificparameters is used for the link travel time function. The network and demand datawere constructed according to the Emme/2 software (INRO Consultants 1999). Theworking route set is obtained from Bekhor et al. (2008a), where the routes are generatedby using a combination of the link elimination method and the penalty method. Thereare totally 174,491 routes with an average of 40.1 routes for each O-D pair, and themaximum number of routes generated for any O-D pair is 50. Note that this workingroute set has been adopted to examine the length-based and congestion-based C-logitSUE model in Zhou et al. (2012) and Xu et al. (2012), the path-size logit (PSL) andpaired combinatorial logit (PCL) SUE models in Chen et al. (2012b), the cross-nestedlogit (CNL) SUE model in Bekhor et al. (2008b), and the nonadditive traffic equilib-rium model in Chen et al. (2012c). The O-D trip data provided by the Emme/2 softwareare aggregated to obtain the zonal production data.

To demonstrate the applicability of the SC-CDA model and solution algorithm tolarger networks with a given working route set, we first investigate the computationalperformance of the solution algorithm by varying the number of side constraints andthe values of the dispersion parameter for route choice (θr) and impedance parameterfor destination choice (θd). We set θr and θd for the Winnipeg network as 0.02 and 0.01,respectively. For the second analysis, we focus on the SC-CDA model by restricting theassigned flows on the 14 bridges (or 28 directional links) to be less than the bridgecapacities. This analysis could be useful in assessing andmanaging bridge safety in practice.

To gain an idea on the number of links exceeding its capacity in theWinnipeg network, we first run the CDA model without side constraints. Table 2

Red RiverPerimeter highway

Assiniboine River

B1

B2

B3

B4

B 7

B8

B9

B12

B13

B14B5

B6

B11

B10

Fig. 7 Winnipeg network and bridge locations


provides a summary of the V/C ratio distribution on all links (1,923 links)excluding the centroid connectors. We observe that the assigned link flows arehighly congested. More than 700 links (or 36.8 % of 1,923 links) have flowsexceeding the link capacities (i.e., V/C>1). In addition, more than 80 links (or4.2 % of 1,923 links) have a V/C ratio greater than 2.0. The highest V/C ratiois 6.25 (or 6 times above its link capacity). These high V/C ratio links are notreasonable in a realistic network.

Table 2 V/C ratio distribution on all links excluding centroid connectors by the CDA model

V/C ratio # of links Percentage (%)

0.00≤V/C≤0.25 173

293

411

338

9>>>=>>>;1215

9:00

15:24

21:37

17:58

9>>>=>>>;63:190.25<V/C≤0.500.50<V/C≤0.750.75<V/C≤1.001.00<V/C≤1.25 236

182

123

85

42

23

17

9>>>>>>>>>>>=>>>>>>>>>>>;708

12:27

9:46

6:40

4:42

2:18

1:20

0:88

9>>>>>>>>>>>=>>>>>>>>>>>;36:81

1.25<V/C≤1.501.50<V/C≤1.751.75<V/C≤2.002.00<V/C≤2.502.50<V/C≤3.003.00<V/C≤6.25Total 1,923 100.0

0

100

200

300

400

500

600

700

0 5 10 20 50 100 200 400 708

CP

U T

ime

(sec

)

# of Constrained Links

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 5 10 15

Ave

rag

e E

xces

s C

ost

Iteration #

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 10 20 30 40 50 60

Ave

rag

e E

xces

s C

ost

Iteration #

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 50 100 150 200 250 300

Ave

rag

e E

xces

s C

ost

Iteration #

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 15 30 45 60 75 90

Ave

rag

e E

xces

s C

ost

Iteration #

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 10 20 30 40 50 60

Ave

rag

e E

xces

s C

ost

Iteration #

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 50 100 150 200 250

Ave

rag

e E

xces

s C

ost

Iteration #

(a) # of constrained links =0

(b) # of constrained links =10

(c) # of constrained links =50

(f) # of constrained links =708

(e) # of constrained links =400

(d) # of constrained links =200

Fig. 8 CPU time as a function of the number of side constraints

264 S. Ryu et al.

4.2.1 Effect of Number of Link Capacity Constraints

Using the above results in Table 2, we first examine the effect of side constraints onthe computational performance of the solution algorithm for solving the SC-CDAproblem. We vary the number of side constraints from 0 to 708 using the mostcongested links first. Figure 8 depicts the computational time (CPU) in seconds as afunction of the number of side constraints. For selected numbers of side constraints(i.e., 0, 10, 50, 200, 400, 708), the convergence curves in terms of average excess costand iteration number are also presented in Fig. 8 to illustrate that computational timegenerally grows in a nonlinear fashion as the number of side constraints increases.Figure 9a depicts the link flow pattern in terms of V/C ratios of the CDA model

(a) SC-CDA without link capacity constraint (b) SC-CDA with 10 link capacity constraints

(c) SC-CDA with 50 link capacity constraints (d) SC-CDA with 200 link capacity constraints

(e) SC-CDA with 400 link capacity constraints (f) SC-CDA with 708 link capacity constraintsFig. 9 Evolution of link flow difference under various numbers of link capacity constraints


without link capacity constraints, while Fig. 9b to f plot the evolution of link flowdifference with respect to Fig. 9a (i.e., the CDA model without link capacityconstraints) for various numbers of side constraints. When the number of sideconstraints is small (less than 50), the link flow patterns are visually similar asindicated by the link flow differences in Fig. 9b to c. In contrast, the link flowpatterns show more differences when the number of link capacity constraints isincreased as indicated by the link flow differences in Fig. 9d to f. One can visualizethe congestion level in the central area is dispersing outward as the number of sideconstraints increases.

4.2.2 Effect of Dispersion and Impedance Parameters

The above analysis indicates that the proposed solution algorithm is capable ofobtaining an accurate solution for different numbers of link capacity constraints. Inthe following analyses, we examine the sensitivity of the dispersion parameter for routechoice and the impedance parameter for destination choice on computational perfor-mance of the algorithm. We first analyze the impact of different impedance parameterswith a fixed dispersion parameter of 0.02. From Fig. 10a, we can observe that the CPUtime is about the same for different impedance parameter values (i.e., not affected bythe impedance parameter). Next we examine the impact of different dispersion param-eters with a fixed impedance parameter of 0.01. From Fig. 10b, it shows CPU time isslightly increased when the dispersion parameter value is increased. Based on thissensitivity analysis, it appears the dispersion parameter has a slightly larger impact interms of CPU time compared to the impedance parameter for the Winnipeg network.

4.2.3 Effect of 14 Bridges as Capacity Constraints

For the second analysis, we focus on the SC-CDAmodel by restricting the assigned flowson the 14 bridges (or 28 directional links) to be less than the bridge capacities. Thisanalysis could be useful in assessing and managing bridge safety in practice. Table 3shows the assigned V/C ratios of the 14 bridges in both directions before and after

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 5 10 15 20

Ave

rag

e E

xces

s C

ost

CPU Time(sec)

0.015

0.010

0.005

impedanceparameter ( ) d

Dispersion parameter ( r ) =0.02

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

0 5 10 15 20

Ave

rag

e E

xces

s C

ost

CPU Time(sec)

0.025

0.020

0.015

dispersionparameter ( ) r

(b) (a) Impedance parameter ( d ) =0.01Fig. 10 Computational efforts under different impedance and dispersion parameter values

266 S. Ryu et al.

imposing the capacity constraints (i.e., CDA and SC-CDA). Eight bridges (or thirteendirectional links) in the CDA model are overloaded (as indicated by the shaded areas inTable 3) before imposing the link capacity constraints, while the flows on these 13directional links in the SC-CDA model are restricted at their capacities after imposingthe link capacity constraints. In particular, flows on B2-EB and B11-SB in the CDAmodel exceed more than two times of the capacity. These significantly overloaded bridgesmay create a safety hazard. The assigned flows of the SC-CDAmodel, on the other hand,divert the overloaded flows to other routes that may or may not use the bridges. As shownin Table 3, the flow pattern on the bridges is generally more dispersed, and moreimportantly, all assigned flows are within the capacity limit of the bridges.

Table 3 Assigned V/C ratios on 14 bridges by the CDA and SC-CDA models

Bridge ID Bridge name Direction V/C ratio by CDA V/C ratio by SC-CDA

B1 North Perimeter EB 0.68 0.92

WB 0.40 0.37

B2 Kildonan Settlers EB 2.22 1.00

WB 1.65 1.00

B3 Redwood EB 1.08 1.00

WB 1.21 1.00

B4 Disraeli EB 0.99 1.00

WB 0.50 0.48

B5 Provencher EB 1.60 1.00

WB 0.96 0.96

B6 Norwood EB 1.19 1.00

WB 1.17 0.99

B7 St. Vital EB 1.44 1.00

WB 1.01 0.96

B8 Fort Gary EB 0.73 0.88

WB 0.59 0.53

B9 South Perimeter EB 0.32 0.40

WB 0.30 0.27

B10 Mid Town NB 0.81 0.66

SB 1.38 1.00

B11 Osborne NB 1.46 1.00

SB 2.57 1.00

B12 Maryland NB 0.80 1.00

SB 0.44 0.42

B13 St. James NB 1.07 1.00

SB 0.85 0.84

B14 West Perimeter NB 0.78 0.70

SB 0.53 0.63

NB Northbound, SB Southbound, EB Eastbound and WB Westbound


5 Conclusions

In this paper, we developed a side constrained combined distribution andassignment (SC-CDA) model for modeling physical link capacity constraintsin a road network. A hierarchical logit-based choice model based on randomutility theory was adopted to describe the destination choice and route choice,which forms the basis for constructing the SC-CDA problem as a mathematicalprogramming. A dual version of the SC-CDA problem was formulated todevelop a solution algorithm (i.e., an iterative balancing scheme combined witha column generation scheme) for solving the SC-CDA problem. Numericalexperiments using a hypothetical network and a real network (Winnipeg net-work) were conducted to demonstrate the features of the proposed model andthe computational performance of the solution algorithm for solving the SC-CDA problem. From the numerical results, the SC-CDA model was shown tobe useful in modeling the congestion and queuing effects (i.e., improve captur-ing the spatial congestion pattern and avoid overloading the physical linkcapacity). Imposing link capacity constraints can have a significant impact onthe network equilibrium flow allocations. In addition, the solution algorithmwas shown to be capable of obtaining accurate solutions within reasonablecomputational effort for realistic network sizes.

Several directions for future research are worth noting. One possibility is to extendthe SC-CDAmodel to consider other travel choice dimensions (e.g., mode-route choice,destination-mode-route choice, travel-destination-mode-route choice). Another possi-bility is to consider the overlapping problem inherited in the logit-based models byexploring recent advances in the extended logit models applied to the SUE problem (seePrashker and Bekhor (2004) and Chen et al. (2012b) for a review). In addition, morerealistic cost functions (e.g., asymmetric link cost function and non-additive route costfunction) can be considered using the variational inequality formulations for combinedtravel demandmodels (see Hasan and Dashti 2007; Zhou et al. 2009). Different physicaland environmental side constraints (e.g., Chen et al. 2011) can also be considered tomodel different physical configurations (e.g., lane drop, merging, nodal capacity, etc.),traffic control policies, and environmental restraints (e.g., emission and noise) toimprove the realism of travel demand forecasting models.

Acknowledgments The authors are grateful to Professor Terry Friesz (Editor-in-Chief) and threereferees for providing useful comments and suggestions for improving the quality and clarity of thepaper. This research was partly supported by the Oriental Scholar Professorship Program sponsoredby the Shanghai Ministry of Education in China, and the National Research Foundation of Koreagrant funded by the Korea government (MEST) (NRF-2010-0029443). These supports are gratefullyacknowledged.

References

Bar-Gera H, Boyce D (2003) Origin-based algorithms for combined travel forecasting models. Transp Res B37(5):405–422

Bekhor S, Toledo T, Prashker JN (2008a) Effects of choice set size and route choice models on path-basedtraffic assignment. Transportmetrica 4(2):117–133

268 S. Ryu et al.

Bekhor S, Toledo T, Reznikova L (2008b) A path-based algorithm for the cross-nested logit stochastic userequilibrium. Comput Aided Civ Infrastruct Eng 24(1):15–25

Bell MGH (1995) Stochastic user equilibrium assignment in network with queues. Transp Res B 29:125–137Bell MGH, Iida Y (1997) Transportation network analysis. Wiley, NYBell MGH, Shield CM, Busch F, Kruse G (1997) A stochastic user equilibrium path flow estimator. Transp

Res C 5(3–4):197–210Boyce D (2007) Forecasting travel on congested urban transportation networks: review and prospects for

network equilibrium models. Networks Spat Econ 7(2):99–128Boyce D, Bar-Gera H (2004) Multiclass combined models for urban travel forecasting. Networks Spat Econ

4(1):115–124Briceño L, Cominetti R, Cortés C, Martínez FJ (2008) An integrated behavioral model of land-use and

transport system: an extended network equilibrium approach. Networks Spat Econ 8(2–3):201–224Chen A, Kasikitwiwat P (2011) Modeling network capacity flexibility of transportation networks. Transp Res

A 45(2):105–117Chen A, Chootinan P, Recker W (2005) Examining the quality of synthetic origin-destination trip table

estimated by path flow estimator. J Transp Eng 131(7):506–513Chen A, Yang C, Kongsomsaksakul S, Lee M (2007) Network-based accessibility measures for vulnerability

analysis of degradable transportation networks. Network Spat Econ 7(3):241–256Chen A, Chootinan P, Recker W (2009) Norm approximation method for handling traffic count inconsis-

tencies in path flow estimator. Transp Res B 43(8):852–872Chen A, Ryu S, Chootinan P (2010) L∞-norm path flow estimator for handling traffic count inconsistencies:

formulation and solution algorithm. J Transp Eng 136(6):565–575Chen A, Zhou Z, Ryu S (2011) Modeling physical and environmental side constraints in traffic equilibrium

problem. Int J Sust Transp 5(3):172–197Chen A, Chootinan P, Ryu S, Lee M, Recker W (2012a) An intersection turning movement estimation

procedure based on path flow estimator. J Adv Transp 46(2):161–176Chen A, Pravinvongvuth S, Xu X, Ryu S, Chootinan P (2012b) Examining the scaling effect and overlapping

problem in logit-based stochastic user equilibrium models. Transp Res A 46(8):1343–1358Chen A, Zhou Z, Xu X (2012c) A self-adaptive gradient projection algorithm for the nonadditive traffic

equilibrium problem. Comput Oper Res 39(2):127–138Chen A, Kasikitwiwat P, Yang C (2013) Alternate capacity reliability measures for transportation networks. J

Adv Transp 47(1):79–104Chootinan P, Chen A, Recker W (2005) Improved path flow estimator for estimating origin-destination trip

tables. Transp Res Rec 1923:9–17Damberg O, Lundgren JT, Patriksson M (1996) An algorithm for the stochastic user equilibrium problem.

Transp Res B 30(2):115–131de Grange L, Fernandez E, de Cea J, Irrazabal J (2010) Combined model calibration and spatial aggregation.

Networks Spat Econ 10(4):551–578Fang S-C, Rajasekera JR, Tsao HSJ (1997) Entropy optimization and mathematical programming. Kluwer

Academic Publishers, BostonFerrari P (1995) Road pricing and network equilibrium. Transp Res B 29(5):357–372Hasan MK, Dashti HM (2007) A multiclass simultaneous transportation equilibrium model. Networks Spat

Econ 7(3):197–211Hearn DW, Ribera J (1980) Bounded flow equilibrium problems by penalty methods. In: Proceedings of IEEE

International Conference on Circuits and Computers, 162–166Inouye H (1987) Traffic equilibria and its solution in congested road network. Proceedings of Proc. IFAC

Conf. on Control in Transportation Systems, Vienna, pp. 262–272INRO Consultants (1999) Emme/2 user’s manual: release 9.2. MontréalLarsson T, Patriksson M (1992) Simplicial decomposition with disaggregated representation for the traffic

assignment problem. Transp Sci 26(1):4–17Larsson T, Patriksson M (1995) An augmented Lagrangean dual algorithm for link capacity side constrained

traffic assignment problems. Transportation Research Part B 29(6):433–455Larsson T, Patriksson M (1999) Side constrained traffic equilibrium models—analysis, computation and

applications. Transp Res B 33(4):233–264Larsson T, Patriksson M, Rydegren C (2004) A column generation procedure for the side constrained traffic

equilibrium problem. Transp Res B 38(1):17–38Li Z, Huang H, Lam WHK, Wong SC (2007) A model for evaluation of transport policies in multimodal

networks with road and parking capacity constraints. J Math Model Algoritm 6(2):239–257


Lin JJ, Feng CM (2003) A bi-level programming for the land use-network design problem. Ann Reg Sci 37:93–105

Nie Y, Zhang HM, Lee D-H (2004) Models and algorithms for the traffic assignment problem with linkcapacity constraints. Transp Res B 38:285–312

Oppenheim N (1995) Urban travel demand modeling. John Wiley & Sons, Inc., NYPatriksson M (1994) The traffic assignment problem: models and methods. VSP, UtrechtPatriksson M (2004) Algorithms for computing traffic equilibria. Networks Spat Econ 4(1):23–38Prashker JN, Bekhor S (2004) Route choice models used in the stochastic user equilibrium problem: a review.

Transp Rev 24(4):437–463Siefel JD, de Cea J, Fernandez JE, Rodriguez RE, Boyce D (2006) Comparisons of urban travel forecasts

prepared with the sequential procedure and a combined model. Networks Spat Econ 6(2):135–148Tam ML, Lam WHK (2000) Maximum car ownership under constraint of road capacity and parking space.

Transp Res A 34(3):145–170Tam ML, Lam WHK (2004) Balance for car ownership under user demand and road network supply

conditions-case study in Hong Kong. J Urban Plan Dev 130(1):24–36Tarjan RE (1983) Data structures and network algorithms. CBMS 44, Society for Industrial and Applied

Mathematics, PhiladelphiaWardrop J (1952) Some theoretical aspects of road traffic research. Proc Inst Civ Eng, Part II (1): 325–378Wong KI, Wong SC, Yang H (2001) Modeling urban taxi services in congested road networks with elastic

demand. Transp Res B 35:819–842Wong KI, Wong SC, Yang H, Wu JH (2008) Modeling urban taxi services with multiple user classes and

vehicle modes. Transp Res B 42:985–1007Xu X, Chen A, Zhou Z, Behkor S (2012) Path-based algorithms for solving C-logit stochastic user equilibrium

assignment problem. Transp Res Rec 2279:21–30Yang H, Bell MGH (1997) Traffic restraint, road pricing and network equilibrium. Transp Res B 31(4):303–

314Yang H, Lam WHK (1996) Optimal road tolls under conditions of queuing and congestion. Transp Res A

30(5):319–332Yang H, Bell MGH, Meng Q (2000) Modeling the capacity and level of service of urban transportation

networks. Transp Res B 34(4):255–275Yim K, Wong SC, Chen A, Wong CK, Lam WHK (2011) A reliability-based land use and transportation

optimization model. Transp Res C 19(2):351–362Zhou Z, Chen A, Wong SC (2009) Alternative formulations of a combined trip generation, trip distribution,

modal split, and trip assignment model. Eur J Oper Res 198(1):129–138Zhou Z, Chen A, Bekhor S (2012) C-logit stochastic user equilibrium model: formulations and solution

algorithm. Transportmetrica 8(1):17–41

270 S. Ryu et al.

A Dual Approach for Solving the Combined Distribution and Assignment Problem with Link Capacity...

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