Computing Dynamic User Equilibria on LargeScale Networks with Software Implementation
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Abstract
Dynamic user equilibrium (DUE) is the most widely studied form of dynamic traffic assignment (DTA), in which road travelers engage in a noncooperative Nashlike game with departure time and route choices. DUE models describe and predict the timevarying traffic flows on a network consistent with traffic flow theory and travel behavior. This paper documents theoretical and numerical advances in synthesizing traffic flow theory and DUE modeling, by presenting a holistic computational theory of DUE, which is numerically implemented in a MATLAB package. In particular, the dynamic network loading (DNL) subproblem is formulated as a system of differential algebraic equations based on the LighthillWhithamRichards fluid dynamic model, which captures the formation, propagation and dissipation of physical queues as well as vehicle spillback on networks. Then, the fixedpoint algorithm is employed to solve the DUE problems with simultaneous route and departure time choices on several largescale networks. We make openly available the MATLAB package, which can be used to solve DUE problems on userdefined networks, aiming to not only facilitate benchmarking a wide range of DUE algorithms and solutions, but also offer researchers a platform to further develop their own models and applications. The MATLAB package and computational examples are available at https://github.com/DrKeHan/DTA.
Keywords
Dynamic traffic assignment Dynamic user equilibrium Dynamic network loading Traffic flow model Fixedpoint algorithm Software1 Introduction
This paper is concerned with a class of models known as dynamic user equilibrium (DUE). DUE problems have been studied within the broader context of dynamic traffic assignment (DTA), which is viewed as the modeling of timevarying flows on traffic networks consistent with established travel demand and traffic flow theory.

Wardrop’s first principle, also known as the user optimal principle, views travelers as Nash agents competing on a network for road capacity. Specifically, the travelers selfishly seek to minimize their own travel costs by adjusting route choices. A user equilibrium is envisaged where the travel costs of all travelers in the same origindestination (OD) pair are equal, and no traveler can lower his/her cost by unilaterally switching to a different route.

Wardrop’s second principle, known as the system optimal principle, assumes that travelers behave cooperatively in making their travel decisions such that the total travel cost on the entire network is minimized. In this case, the travel costs experienced by travelers in the same OD pair are not necessarily identical.
Since the seminal work of Merchant and Nemhauser (1978a, b) the DTA literature has been focusing on the dynamic extension of Wardrop’s principles, which gives rise to the notions of dynamic user equilibrium (DUE) and dynamic system optimal (DSO) models. The DUE model stipulates that experienced travel cost, including travel time and early/late arrival penalties, is identical for those route and departure time choices selected by travelers between a given OD pair. The DSO model seeks systemwide minimization of travel costs incurred by all the travelers subject to the constraints of fixed travel demand and network flow dynamics.
For a comprehensive review of DTA models, the reader is referred to Boyce et al. (2001); Peeta and Ziliaskopoulos (2001); Szeto and Lo (2005, 2006); Jeihani (2007) and Bliemer et al. (2017). A discussion of DTA from the perspective of intelligent transportation system can be found in Ran and Boyce (1996a). Chiu et al. (2011) present a primer on simulationbased DTA modeling. Garavello et al. (2016) focus on the traffic flow modeling aspect of DTA, namely the hydrodynamic models for vehicular traffic and their network extensions. Wang et al. (2018) review relevant DTA literature concerning environmental sustainability.
Dynamic user equilibrium (DUE), which is one type of DTA, remains a modern perspective on traffic network modeling that enjoys wide scholarly support. It is conventionally studied as a openloop, nonatomic Nashlike game (Friesz et al. 1993). The notion of open loop refers to the assumption that the travelers’ route choices do not change in response to dynamic network conditions after they leave the origin. The nonatomic nature refers to the prevailing technique of flowbased modeling, instead of treating the traffic as individual vehicles; this is in contrast to agentbased modeling (Balmer et al. 2004; Cetin et al. 2003; Shang et al. 2017). DUE captures two aspects of travel behavior quite well: departure time choice and route choice. Within the DUE model, travel cost for the same trip purpose is identical for all utilized path and departure time choices. The relevant notion of travel cost is a weighted sum of travel time and arrival penalty.
 (1)
routechoice (RC) DUE (Bliemer and Bovy 2003; Chen and Hsueh 1998; Lo and Szeto 2002; Long et al. 2013; Ran and Boyce 1996b; Tong and Wong 2000; Varia and Dhingra 2004; Zhu and Marcotte 2000); and
 (2)
simultaneous routeanddeparturetime (SRDT) choice DUE (Friesz et al. 1993, 2001, 2013, 2011; Han et al. 2013b, 2015a, b; Huang and Lam 2002; Nie and Zhang 2010; Szeto and Lo 2004; Ukkusuri et al. 2012; Wie 2002).
 1.
variational inequalities (Friesz et al. 1993, 2013; Han et al. 2013b, 2015a, b; Smith and Wisten 1994, 1995);
 2.
nonlinear complementarity problems (Han et al. 2011; Pang et al. 2011; Ukkusuri et al. 2012; Wie et al. 2002);
 3.
differential variational inequalities (Friesz and Meimand 2014; Friesz and Mookherjee 2006; Han et al. 2015a);
 4.
differential complementarity systems (Ban et al. 2012);
 5.
 6.
 1.
some form of link and/or path dynamics;
 2.
an analytical relationship between flow/speed/density and link traversal time
 3.
flow propagation constraints;
 4.
a model of junction dynamics and delays;
 5.
a model of path traversal time; and
 6.
appropriate initial conditions.
Computational algorithms for DUE. The algorithms are arranged in an increasing order in terms of the generality of the relevant notions of monotonicity
DUE type  Computational  Convergence  

Friesz et al. (2011)  SRDT DUE  Fixedpoint  Lipschitz cont. 
algorithm  strongly monotone  
Lo and Szeto (2002)  RC DUE  Alternating direction  Cocoercive 
algorithm  
Szeto and Lo (2004)  SRDT DUE  descent algorithm  Cocoercive 
elastic demand  (projection)  
Szeto and Lo (2006)  RC DUE  Routeswapping  Continuous 
bounded rationality  algorithm  monotone  
Huang and Lam (2002)  SRDT DUE  Routeswapping  Continuous 
algorithm  monotone  
Tian et al. (2012)  SRDT DUE  Routeswapping  Continuous 
algorithm  monotone  
Long et al. (2013)  RC DUE  Extragradient/  Lipschitz cont. 
double projection  pseudo monotone  
Han et al. (2015b)  SRDT DUE  Selfadaptive  Continuous 
bounded rationality  projection  Dproperty  
Han et al. (2015a)  SRDT DUE  Proximal point  Dual solvable 
elastic demand  method 
This paper documents theoretical and numerical advances in synthesizing traffic flow theory and traffic assignment models, by presenting a computational theory of DUE, which includes algorithms and software implementation. While there have been numerous studies on the modeling and computation of DUEs, including those reviewed in this paper, little agreement exists regarding an appropriate mathematical formulation of DUE or DNL models, as well as the extent to which certain models should/can be applied. This is partially due to the lack of opensource solvers and a set of benchmarking test problems for DUE models. In addition, largescale computational examples of DUE were rarely reported and, when they were, little detail was provided that allows the results to be validated, reproduced and compared.
This paper aims to bridge the aforementioned gaps by presenting a computable theory for the simultaneous routeanddeparturetime (SRDT) choice DUE model along with opensource software packages. In particular, the DNL model is based on the LighthillWhithamRichards fluid dynamic model (Lighthill and Whitham 1955; Richards 1956), and is formulated as a system of differential algebraic equations (DAEs) by invoking the variational theory for kinematic wave models. This technique allows the DAE system to be solved with straightforward timestepping without the need to solve any partial differential equations. Moreover, this paper presents the fixedpoint algorithm for solving the DUE problem, which was derived from the differential variational inequality formalism (Friesz and Han 2018). Both the DNL procedure and the fixedpoint algorithm are implemented in MATLAB, and we present the computational results on several test networks, including the Chicago Sketch network with 250,000 paths. To our knowledge, this is the largest instance of SRDT DUE computation reported in the literature to date.
In addition, we make openly available the MATLAB package, which can be used to solve DUEs on userdefined networks. The package is documented in this paper (Appendix), and the codes and examples are available at https://github.com/DrKeHan/DTA. It is our intention that the opensource package will not only help DTA modelers with benchmarking a wide range of algorithms and solutions, but also offer researchers a platform to further develop their own models and applications.
The rest of the paper is organized as follows. Section 2 introduces some key notions and mathematical formulations of DUE. Section 3 details the dynamic network loading procedure and the DAE system formulation. The fixedpoint algorithm for computing DUE is presented in Section 4. The computational results obtained from the proposed DUE solver are presented in Section 5. Section 6 offers some concluding remarks. Finally, the MATLAB software package is documented in the Appendix.
2 Formulations of Dynamic User Equilibrium
We introduce a few notations and terminologies for the ease of presentation below.
 \(\mathcal {P}\)

set of paths in the network
 \(\mathcal {W}\)

set of OD pairs in the network
 Q _{ i j }

fixed OD demand between \((i, j)\in \mathcal {W}\)
 \(\mathcal {P}_{ij}\)

subset of paths that connect OD pair (i, j)
 t

continuous time parameter in a fixed time horizon [t_{0}, t_{f}]
 h_{p}(t)

departure rate along path p at time t
 h(t)

complete vector of departure rates \(h(t)=(h_{p}(t): p\in \mathcal {P})\)
 Ψ_{p}(t, h)

travel cost along path p with departure time t, under departure profile h
 v_{ij}(h)

minimum travel cost between OD pair (i, j) for all paths and departure times
Definition 2.1
where ‘a.e.’, standing for ‘for almost every’, is a technical term employed by measuretheoretic arguments to indicate that Eq. (2.4) only needs to hold in [t_{0}, t_{f}] with the exception of any subset that has zero Lebesgue measure.
2.1 Variational Inequality Formulation of DUE
2.2 Nonlinear Complementarity Formulation of Due
2.3 Differential Variational Inequality Formulation of DUE
2.4 Fixedpoint Formulation of DUE
3 Dynamic Network Loading
An integral component of the DUE formulation is the effective delay operator Ψ, which is constructed using the dynamic network loading (DNL) procedure. This section details one type of DNL models based on the fluid dynamic approximation of traffic flow on networks, known as the LighthillWhithamRichards (LWR) model (Lighthill and Whitham 1955; Richards 1956). The model, including its various forms (Newell 1993a; Daganzo 1994, 1995; Yperman et al. 2005; Osorio et al. 2011), is widely studied in the DTA literature. This section presents a complete DNL procedure based on the LWR model and its variational representation, which lead to a differential algebraic equation (DAE) system.
3.1 The LighthillWhithamRichards Link Model
While (3.1) captures the withinlink dynamics, the interlink propagation of congestion requires a careful treatment of junction dynamics, which is underpinned by the notions of link demand and supply.
3.2 Link Demand and Supply
3.3 The Variational Representation of Link Dynamics
3.4 Junction Dynamics that Incorporate Route Information
Essential to the network extension of the LWR model are the junction dynamics. Unlike many existing junction models such as those reviewed in Section 3.2, in a pathbased DNL formulation, one must incorporate established routing information into the junction model. Such information is manifested in an endogenous flow distribution matrix, which specifies the proportion of exit flow from a certain incoming link that advances into a given outgoing link. This can be done by explicitly tracking the route composition in every unit of flow along the link.
3.5 Dynamics at the Origin Nodes
A model at the origin (source) nodes is needed since the path flows h_{p}(⋅), defined by Eq. (2.3), are not bounded from above. In this case, a queuing model is needed at the origin node to accommodate departure flow exceeding the supply of relevant downstream link.
3.6 Calculating Path Travel Times
3.7 The Differential Algebraic Equation System Formulation of DNL
As a summary of the individual sections presented so far, we formulate the complete system of differential algebraic equations (DAEs). We begin with the following list of key notations.
 \(\mathcal {P}\)

set of all paths
 \(\mathcal {S}\)

set of origins
 \(\mathcal {P}^{o}\)

set of paths originating from \(o\in \mathcal {S}\)
 \(\mathcal {I}^{J}\)

set of incoming links of a junction J
 \(\mathcal {O}^{J}\)

set of outgoing links of a junction J
 A ^{ J }

flow distribution matrix of junction J
 h_{p}(t)

departure rate along path \(p\in \mathcal {P}\)
 \({f}_{i}^{\text {in}}(t)\)

inflow of link i
 \({f}_{i}^{\text {out}}(t)\)

outflow of link i
 \({N}_{i}^{\text {up}}(t)\)

cumulative link entering count
 \({N}_{i}^{\text {dn}}(t)\)

cumulative link exiting count
 D_{i}(t)

demand of link i
 S_{i}(t)

supply of link i
 \({{\mu }_{i}^{p}}(t, x)\)

percentage of flow on link i that belongs to path p
 q_{o}(t)

point queue at the origin node \(o\in \mathcal {S}\)
 τ_{i}(t)

entry time of link i corresponding to exit time t
 λ_{i}(t)

exit time of link i corresponding to entry time t
Equations (3.15)–(3.23) form the DAE system for the DNL procedure. Compared to the partial differential algebraic equation (PDAE) system presented in Han et al. (2016a), the DAE system does not involve any spatial derivative as one would expect from the LWRtype equations, by virtue of the variational formulation.
4 The FixedPoint Algorithm for Computing DUE
Remark 4.1
5 Computational Examples of DNL and DUE
Key attributes of the test networks
Nguyen network  Sioux Falls  Anaheim sketch  Chicago  

No. of links  19  76  914  2,950 
No. of nodes  13  24  416  933 
No. of zones  4  24  38  387 
No. of OD pairs  4  528  1,406  86,179 
No. of paths  24  6,180  30,719  250,000 
We apply the fixedpoint algorithm (Section 4) with the embedded DNL procedure (Section 3). The fixedpoint algorithm is chosen among many other alternatives in the literature, as our extensive experience with DUE computation suggests that this method tends to exhibit satisfactory empirical convergence within limited number of iterations, despite that its theoretical convergence requires strong monotonicity of the delay operator. The DNL submodel is solved as a DAE system (3.15)–(3.23), following the time stepping logic in Fig. 1. All the computations reported in this section were performed using the MATLAB (R2017b) package on a standard desktop with Intel i5 processor and 8 GB of RAM.
5.1 Performance of the FixedPoint Algorithm
Performance of the fixedpoint algorithm on different networks
Nguyen network  Sioux Falls  Anaheim  Chicago Sketch  

No. of iterations  54  73  45  69 
Computational time  5.9 s  6.1 min  23.3 min  4.8 hr 
Avg. time per DNL  0.1 s  4.3 s  25.7 s  163.9 s 
Avg. time per FP update  0.007 s  0.6 s  3.1 s  81.2 s 
Figure 3 shows the relative gaps, i.e. left hand side of Eq. (5.1), for a total of 100 fixedpoint iterations on the four networks. It can be seen that for relatively small networks (Nguyen and Sioux Falls), the convergence can be achieved relatively quickly and to a satisfactory degree; and the corresponding curves are monotonically decreasing and smooth. For Anaheim and Chicago Sketch networks, the decreasing trend of the gap can sometimes stall and experience fluctuations locally.
5.2 DUE Solutions
6 Conclusion
This paper presents a computational theory for dynamic user equilibrium (DUE) on largescale networks. We begin by presenting a complete and generic dynamic network loading (DNL) procedure based on the network extension of the LWR model and the variational theory, which allows us to formulate the DNL problem as a system of differential algebraic equations (DAEs). The DNL model is capable of capturing the formation, propagation and dissipation of physical queues as well as vehicle spillback. The DAE system can be discretized and solved in a timeforward fashion. In addition, the fixedpoint algorithm for solving DUE problems is presented.
Both the DAE system and the fixedpoint algorithm are implemented in MATLAB, and the programs are developed in such a way that they can be applied to solve DUE and DNL problems on any userdefined networks. The MATLAB package is documented in the Appendix of this paper, which provides detailed instructions for using the solvers.
The MATLAB program is applied to solve DUE problems on several test networks of varying sizes. The largest one is the Chicago Sketch network with 86,179 OD pairs and 250,000 paths. To the authors’ knowledge this is by far the largest instance of SRDT DUE solution reported in the literature. Hopefully, our efforts in making these codes and data openly accessible could facilitate the testing and benchmarking of dynamic traffic assignment algorithms, and promote synergies between model development and applications.
Footnotes
 1.
The DNL code employs junction models that incorporate the continuous signal model proposed by Han et al. (2014) and Han and Gayah (2015). For any given junction (node), each of its incoming links, including the source if the node happens to be an origin, will be assigned a priority parameter between 0 and 1, such that the sum of relevant priority parameters equals 1. Once the priority of the source is specified, the priorities of the remaining incoming links are set proportional to the links’ capacities. These parameters may be changed dynamically to accommodate different signal control scenarios or strategies.
 2.
For large networks, the path set can be generated by using the kshortest path or FrankWolfe algorithms.
Notes
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