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This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill-Whitham-Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations (PDAEs). We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model., Comment: 29 pages, 9 figures, Transportation Research Part B: Methodological (2015)

This paper analyzes simultaneous route-and-departure-time (SRDT) dynamic user equilibrium (DUE) that incorporates the notion of boundedly rational (BR) user behavior in the selection of departure time and route choices. Intrinsically, the boundedly rational dynamic user equilibrium (BR-DUE) model we present assumes that travelers do not always seek the least costly route-and-departure-time choice. Rather, their perception of travel cost is affected by an indifference band describing travelers' tolerance of the difference between their experienced travel costs and the minimum travel cost. An extension of the BR-DUE problem is the so-called variable tolerance dynamic user equilibrium (VT-BR-DUE) wherein endogenously determined tolerances may depend not only on paths, but also on the established path departure rates. This paper presents a unified approach for modeling both BR-DUE and VT-BR-DUE, which makes significant contributions to model formulation, analysis of existence, solution characterization, and heuristic numerical computation of such problems. The VT-BR-DUE problem, together with the BR-DUE problem as a special case, is formulated as a variational inequality. We provide a very general existence result for VT-BR-DUE and BR-DUE that relies on assumptions weaker than those required for mere DUE models. Moreover, a characterization of the solution set is provided based on rigorous topological analysis. Finally, three computational algorithms are proposed based on the VI and DVI formulations. Numerical studies are conducted to assess the proposed algorithms in terms of solution quality, convergence, and computational efficiency., Comment: 49 pages, 16 figures

In the modeling of traffic networks, a signalized junction is typically treated using a binary variable to model the on-and-off nature of signal operation. While accurate, the use of binary variables can cause problems when studying large networks with many intersections. Instead, the signal control can be approximated through a continuum approach where the on-and-off control variable is replaced by a priority parameter. Advantages of such approximation include elimination of the need for binary variables, lower time resolution requirements, and more flexibility and robustness in a decision environment. It also resolves the issue of discontinuous travel time functions arising from the context of dynamic traffic assignment. Despite these advantages in application, it is not clear from a theoretical point of view how accurate is such continuum approach; i.e., to what extent is this a valid approximation for the on-and-off case. The goal of this paper is to answer these basic research questions and provide further guidance for the application of such continuum signal model. In particular, by employing the Lighthill-Whitham-Richards model (Lighthill and Whitham, 1955; Richards, 1956) on a traffic network, we investigate the convergence of the on-and-off signal model to the continuum model in regimes of diminishing signal cycles. We also provide numerical analyses on the continuum approximation error when the signal cycles are not infinitesimal. As we explain, such convergence results and error estimates depend on the type of fundamental diagram assumed and whether or not vehicle spillback occurs in a network. Finally, a traffic signal optimization problem is presented and solved which illustrates the unique advantages of applying the continuum signal model instead of the on-and-off one., Comment: 35 pages, 16 figures, 5 tables

This paper is concerned with dynamic user equilibrium (DUE) with elastic travel demand (E-DUE). We present and prove a variational inequality (VI) formulation of E-DUE using measure-theoretic argument. Moreover, existence of the E-DUE is formally established with a version of Brouwer's fixed point theorem in a properly defined Hilbert space. The existence proof requires the effective delay operator to be continuous, a regularity condition also needed to ensure the existence of DUE with fixed demand (Han et al., 2013c). Our proof does not invoke the a priori upper bound of the departure rates (path flows)., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1211.4614, arXiv:1304.5286

This paper is concerned with dynamic user equilibrium with elastic travel demand (E-DUE) when the trip demand matrix is determined endogenously. We present an infinite-dimensional variational inequality (VI) formulation that is equivalent to the conditions defining a continuous-time E-DUE problem. An existence result for this VI is established by applying a fixed-point existence theorem (Browder, 1968) in an extended Hilbert space. We present three algorithms based on the aforementioned VI and its re-expression as a differential variational inequality (DVI): a projection method, a self-adaptive projection method, and a proximal point method. Rigorous convergence results are provided for these methods, which rely on increasingly relaxed notions of generalized monotonicity, namely mixed strongly-weakly monotonicity for the projection method; pseudomonotonicity for the self-adaptive projection method, and quasimonotonicity for the proximal point method. These three algorithms are tested and their solution quality, convergence, and computational efficiency compared. Our convergence results, which transcend the transportation applications studied here, apply to a broad family of infinite-dimensional VIs and DVIs, and are the weakest reported to date., Comment: 32 pages, 6 figures, 2 tables

This paper is concerned with adaptive signal control problems on a road network, using a link-based kinematic wave model (Han et al., 2012). Such a model employs the Lighthill-Whitham-Richards model with a triangular fundamental diagram. A variational type argument (Lax, 1957; Newell, 1993) is applied so that the system dynamics can be determined without knowledge of the traffic state in the interior of each link. A Riemann problem for the signalized junction is explicitly solved; and an optimization problem is formulated in continuous-time with the aid of binary variables. A time-discretization turns the optimization problem into a mixed integer linear program (MILP). Unlike the cell-based approaches (Daganzo, 1995; Lin and Wang, 2004; Lo, 1999b), the proposed framework does not require modeling or computation within a link, thus reducing the number of (binary) variables and computational effort. The proposed model is free of vehicle-holding problems, and captures important features of signalized networks such as physical queue, spill back, vehicle turning, time-varying flow patterns and dynamic signal timing plans. The MILP can be efficiently solved with standard optimization software., Comment: 15 pages, 7 figures, current version is accepted for presentation at the 92nd Annual Meeting of Transportation Research Board

This paper is concerned with highway traffic estimation using traffic sensing data, in a Lagrangian-based modeling framework. We consider the Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956) in Lagrangian-coordinates, and provide rigorous mathematical results regarding the equivalence of viscosity solutions to the Hamilton-Jacobi equations in Eulerian and Lagrangian coordinates. We derive closed-form solutions to the Lagrangian-based Hamilton-Jacobi equation using the Lax-Hopf formula (Daganzo, 2005; Aubin et al., 2008), and discuss issues of fusing traffic data of various types into the Lagrangian-based H-J equation. A numerical study of the Mobile Century field experiment (Herrera et al., 2009) demonstrates the unique modeling features and insights provided by the Lagrangian-based approach., Comment: 17 pages, 7 figures, current version submitted to Transportation Research Part B

In this paper, we establish and prove analytical properties of the state operator embedded in an optimal control problem, in the context of dynamic user equilibrium (DUE) models (Friesz et al. 1993)., Comment: 9 pages

In this paper we present a differential variational inequality formulation of dynamic network user equilibrium with elastic travel demand. We discuss its qualitative properties and provide algorithms for and examples of its solution., Comment: This paper has been withdrawn by the author due to incompleteness and some overlap with contents already published in journal papers

We consider an analytical signal control problem on a signalized network whose traffic flow dynamic is described by the Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956). This problem explicitly addresses traffic-derived emissions as side constraints. We seek to tackle this problem using a mixed integer mathematical programming approach. Such a class of problems, which we call LWR-Emission (LWR-E), has been analyzed before to certain extent. Since mixed integer programs are practically efficient to solve in many cases (Bertsimas et al., 2011b), the mere fact of having integer variables is not the most significant challenge to solving LWR-E problems; rather, it is the presence of the potentially nonlinear and nonconvex emission-related constraints/objectives that render the program computationally expensive. To address this computational challenge, we proposed a novel reformulation of the LWR-E problem as a mixed integer linear program (MILP). This approach relies on the existence of a statistically valid macroscopic relationship between the aggregate emission rate and the vehicle occupancy of the same link. This relationship is approximated with certain functional forms and the associated uncertainties are handled explicitly using robust optimization (RO) techniques. The RO allows emissions-related constraints and/or objectives to be reformulated as linear forms under mild conditions. To further reduce the computational cost, we employ the link transmission model to describe traffic dynamics with the benefit of fewer (integer) variables and less potential traffic holding. The proposed MILP explicitly captures vehicle spillback, avoids traffic holding, and simultaneously minimizes travel delay and addresses emission-related concerns., Comment: 35 pages, 7 figures, 4 tables

We consider a continuous supply chain network consisting of buffering queues and processors first proposed by [D. Armbruster, P. Degond, and C. Ringhofer, SIAM J. Appl. Math., 66 (2006), pp. 896--920] and subsequently analyzed by [D. Armbruster, P. Degond, and C. Ringhofer, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), pp. 433--460] and [D. Armbruster, C. De Beer, M. Freitag, T. Jagalski, and C. Ringhofer, Phys. A, 363 (2006), pp. 104--114]. A model was proposed for such a network by [S. G\"ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545--559] using a system of coupling ordinary differential equations and partial differential equations. In this article, we propose an alternative approach based on a variational method to formulate the network dynamics. We also derive, based on the variational method, a computational algorithm that guarantees numerical stability, allows for rigorous error estimates, and facilitates efficient computations. A class of network flow optimization problems are formulated as mixed integer programs (MIPs). The proposed numerical algorithm and the corresponding MIP are compared theoretically and numerically with existing ones [A. F\"ugenschuh, S. G\"ottlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490--1507; S. G\"ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545--559], which demonstrates the modeling and computational advantages of the variational approach., Comment: 25 pages, 13 figures

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 arc flows are described by the link delay model (Friesz et al., 1993). Unlike result established in Zhu and Marcotte (2000), 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 choice DUE without a priori boundedness of path flows, and to any numerical algorithm that allows convergence to be rigorously analyzed., Comment: 12 pages, 1 figure

In this paper we propose a dynamic Stackelberg game-theoretic model for urban freight transportation planning which is able to characterize the interaction between freight and personal transportation in an urban area. The problem is formulated as a bi-level dynamic mathematical program with equilibrium constraints (MPEC) which belongs to a class of computationally challenging problems. The lower level is dynamic user equilibrium (DUE) with inhomogeneous traffic that characterizes traffic system optimum (SO) freight transportation planning problem which aims at minimizing the total cost to a truck company. A mathematical program with complementarity constraints (MPCC) reformulation is derived and a projected gradient algorithm is designed to solve this computationally challenging problem. Numerical experiments are conducted to show that when planning freight transportation the background traffic is nonnegligible, even though the amount of trucks compared to other vehicles traveling on the same network is relatively small. What's more, in our proposed bi-level model for urban freight transportation planning, we find a dynamic case of a Braess-like Paradox which can provide managerial insights to a metropolitan planning organization (MPO) in increasing social welfare by restricting freight movement.

This paper is concerned with the existence of the simultaneous route-and-departure choice dynamic user equilibrium (SRDC-DUE) in continuous time, first formulated as an infinite-dimensional variational inequality in Friesz et al. (1993). In deriving our existence result, we employ the generalized Vickrey model (GVM) introduced in and to formulate the underlying network loading problem. As we explain, the GVM corresponds to a path delay operator that is provably strongly continuous on the Hilbert space of interest. Finally, we provide the desired SRDC-DUE existence result for general constraints relating path flows to a table of fixed trip volumes without invocation of a priori bounds on the path flows., Comment: 21 pages

Mobile sensing enabled by GPS or smart phones has become an increasingly important source of traffic data. For sufficient coverage of the traffic stream, it is important to maintain a reasonable penetration rate of probe vehicles. From the standpoint of capturing higher-order traffic quantities such as acceleration/deceleration, emission and fuel consumption rates, it is desirable to examine the impact on the estimation accuracy of sampling frequency on vehicle position. Of the two issues raised above, the latter is rarely studied in the literature. This paper addresses the impact of both sampling frequency and penetration rate on mobile sensing of highway traffic. To capture inhomogeneous driving conditions and deviation of traffic from the equilibrium state, we employ the second-order phase transition model (PTM). Several data fusion schemes that incorporate vehicle trajectory data into the PTM are proposed. And, a case study of the NGSIM dataset is presented which shows the estimation results of various Eulerian and Lagrangian traffic quantities. The findings show that while first-order traffic quantities can be accurately estimated even with a low sampling frequency, higher-order traffic quantities, such as acceleration, deviation, and emission rate, tend to be misinterpreted due to insufficiently sampled vehicle locations. We also show that a correction factor approach has the potential to reduce the sensing error arising from low sampling frequency and penetration rate, making the estimation of higher-order quantities more robust against insufficient data coverage of the highway traffic., Comment: 36 pages, 14 figures, 8 tables

The problem of robust dynamic pricing of an abstract commodity, whose inventory is specified at an initial time but never subsequently replenished, originally studied by Perakis and Sood (2006) in discrete time, is considered from the perspective of continuous time. We use a multiplicative demand function to model the uncertain demand, and develop a robust counterpart to replace the uncertain demand constraint. The sellers' robust best response problem yields a generalized Nash equilibrium problem, which can be formulated as an equivalent, continuous-time quasi-variational inequality. We demonstrate that, for appropriate regularity conditions, a generalized robust Nash equilibrium exists. We show that the quasi-variational inequality may be replaced by an equivalent variational inequality, and use a fixed-point algorithm to solve the variational inequality. We also demonstrate how explicit time lags associated with price updating in real-world decision environments, as well as specific pricing decision rules, may be introduced to create a dual time scale formulation and the associated solutions computed. We illustrate, via numerical examples, how robust pricing based on our DPFI formulation offers generally superior and never inferior worst case performance compared to nominal pricing.

This paper proposes a dynamic congestion pricing model that takes into account mobile source emissions. We consider a tollable vehicular network where the users selfishly minimize their own travel costs, including travel time, early/late arrival penalties and tolls. On top of that, we assume that part of the network can be tolled by a central authority, whose objective is to minimize both total travel costs of road users and total emission on a network-wide level. The model is formulated as a mathematical program with equilibrium constraints (MPEC) problem and then reformulated as a mathematical program with complementarity constraints (MPCC). The MPCC is solved using a quadratic penalty-based gradient projection algorithm. A numerical study on a toy network illustrates the effectiveness of the tolling strategy and reveals a Braess-type paradox in the context of traffic-derived emission., Comment: 23 pages, 9 figures, 5 tables. Current version to appear in the Proceedings of the 20th International Symposium on Transportation and Traffic Theory, 2013, the Netherlands