The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation

Wie, Byung-Wook; Tobin, Roger L.; Carey, Malachy

doi:10.1016/s0191-2615(01)00041-8

Cited by 68 publications

(46 citation statements)

References 9 publications

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“…As pointed out by Han (2013), there are two essential components within the notion of DUE: (i) the mathematical expression of Nashlike equilibrium conditions, and (ii) a network performance model, which is, in effect, an embedded dynamic network loading (DNL) problem. There are multiple means of expressing the Nash-like notion of a dynamic equilibrium mathematically, including a variational inequality (Friesz et al, 1993;Wisten, 1994, 1995), an evolutionary dynamic (Mounce, 2006;Smith and Wisten, 1995), a nonlinear complementarity problem (Wie et al, 2002;Han et al, 2011), a differential variational inequality (Friesz et al, 2011(Friesz et al, , 2013Friesz and Mookherjee, 2006), and a differential complementarity system (Pang et al, 2011). Clearly, another key component of the DUE is the path delay operator, typically obtained from dynamic network loading (DNL), which is a sub-problem of a complete DUE model.…”

Section: The Dynamic User Equilibrium As the Lower-level Problemmentioning

confidence: 99%

A bi-level model of dynamic traffic signal control with continuum approximation

Han

Sun

Liu

et al. 2015

Transportation Research Part C: Emerging Technologies

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This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955;Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in , which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure.

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Section: The Dynamic User Equilibrium As the Lower-level Problemmentioning

confidence: 99%

A bi-level model of dynamic traffic signal control with continuum approximation

Han

Sun

Liu

et al. 2015

Transportation Research Part C: Emerging Technologies

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“…These include variational inequality (Friesz et al 1993(Friesz et al , 2013Han et al 2013c;Szeto and Lo 2004); differential variational inequality (Freisz et al 2001(Freisz et al , 2010Friesz and Mookherjee 2006); nonlinear complementarity problem (Ukkusuri et al 2012;Wie et al 2002); and differential complementarity problem (Pang et al 2011). In addition, several extensions of the DUE problem pertaining to bounded user rationality (Han et al 2015b), demand elasticity (Friesz and Meimand 2014;Han et al 2015a) and en-route update (Kachroo andÖzbay 2005), have been formulated using (differential) variational inequalities (which are closely related to open-loop optimal control problems) and feedback (closed-loop) control formulations.…”

Section: Introductionmentioning

confidence: 99%

“…The (effective) delay operators play a pivotal role in the mathematical formulations of DUE problem, including variational inequality (Friesz et al 1993), differential variational inequality (Friesz et al 2001;Friesz and Mookherjee 2006), and complementarity problems (Pang et al 2011;Wie et al 2002). Continuity of the effective delay operator, as we focus in this paper, is critical to the DUE models as it is necessary for the existence of dynamic user equilibria (Browder 1968;Han et al 2013c;Zhu and Marcotte 2000).…”

Section: Introductionmentioning

confidence: 99%

Continuity of the Effective Delay Operator for Networks Based on the Link Delay Model

Han

Friesz

2017

Netw Spat Econ

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This paper is concerned with a dynamic traffic network performance model, known as dynamic network loading (DNL), that is frequently employed in the modeling and computation of analytical dynamic user equilibrium (DUE). As a key component of continuous-time DUE models, DNL aims at describing and predicting the spatial-temporal evolution of traffic flows on a network that is consistent with established route and departure time choices of travelers, by introducing appropriate dynamics to flow propagation, flow conservation, and travel delays. The DNL procedure gives rise to the path delay operator, which associates a vector of path flows (path departure rates) with the corresponding path travel costs. In this paper, we establish strong continuity of the path delay operator for networks whose , our continuity proof is constructed without assuming a priori uniform boundedness of the path flows. Such a more general continuity result has a few important implications to the existence of simultaneous route-and-departure-time DUE without a priori boundedness of path flows, and to any numerical algorithm that allows convergence to be rigorously analyzed.

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“…Since Merchant and Nemhauser (see, [30] and [31]) first proposed their model in 1978, there have been a number of papers (see, e.g., [13], [6], [44], [14], [24], [23], [48], [17], [27], [25], and [42]) discussing the variational inequalities or mathematical programming formulations for the dynamic traffic assignment problem with the assumption that the planning horizon is a set of discrete points instead of a continuous interval. Many of these papers use a dynamic or time-expanded network (see, e.g., [1]) to simultaneously capture the topology of the transportation network and the evolution of traffic over time.…”

Section: Introductionmentioning

confidence: 99%

“…First, some (e.g., [30], [13], [23], [6], [48], [25], [21]) seek a system optimal solution and others (e.g., [24], [44], [14], and [17]) compute a user equilibrium instead. The other factor is the travel cost function used by these models.…”

Section: Introductionmentioning

confidence: 99%

Discrete-time dynamic traffic assignment models with periodic planning horizon: system optimum

Nahapetyan

Lawphongpanich

2006

J Glob Optim

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This paper proposes a system optimal dynamic traffic assignment model that does not require the network to be empty at the beginning or at the end of the planning horizon.The model assumes that link travel times depend on traffic densities and uses a discretized planning horizon. The resulting formulation is a nonlinear program with binary variables and a time-expanded network structure. Under a relatively mild condition, the nonlinear program has a feasible solution. When necessary, constraints can be added to ensure that the solution satisfies the First-In-First-Out condition. Also included are approximation schemes based on linear integer programs that can provide solutions arbitrarily close to that of the original nonlinear problem.

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The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation

Cited by 68 publications

References 9 publications

A bi-level model of dynamic traffic signal control with continuum approximation

A bi-level model of dynamic traffic signal control with continuum approximation

Continuity of the Effective Delay Operator for Networks Based on the Link Delay Model

Discrete-time dynamic traffic assignment models with periodic planning horizon: system optimum

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