A variational approach for continuous supply chain networks

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. Fre- itag, 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. Fu ̈genschuh, S. Go ̈ttlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490–1507; S. Go ̈ttlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559], which demonstrates the modeling and computational advantages of the variational approach

Data-driven linear decision rule approach for distributionally robust optimization of on-line signal control

We propose a two-stage, on-line signal control strategy for dynamic networks using a linear decision rule (LDR) approach and a distributionally robust optimization (DRO) technique. The first (off-line) stage formulates a LDR that maps real-time traffic data to optimal signal control policies. A DRO problem is solved to optimize the on-line performance of the LDR in the presence of uncertainties associated with the observed traffic states and ambiguity in their underlying distribution functions. We employ a data-driven calibration of the uncertainty set, which takes into account historical traffic data. The second (on-line) stage implements a very efficient linear decision rule whose performance is guaranteed by the off-line computation. We test the proposed signal control procedure in a simulation environment that is informed by actual traffic data obtained in Glasgow, and demonstrate its full potential in on-line operation and deployability on realistic networks, as well as its effectiveness in improving traffic

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

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 Han et al. (2014), 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

On the continuum approximation of the on-and-off signal control on dynamic traffic networks

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 continuous 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 to the signalized intersection in question. 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 model

The one-round Voronoi game replayed

We consider the one-round Voronoi game, where player one (``White'', called ``Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (``Black'', called ``Barney''), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al., who showed that for large enough $n$ and r=1, Barney has a strategy that guarantees a fraction of 1/2+a, for some small fixed a. We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2 and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.Comment: 14 pages, 6 figures, Latex; revised for journal version, to appear in Computational Geometry: Theory and Applications. Extended abstract version appeared in Workshop on Algorithms and Data Structures, Springer Lecture Notes in Computer Science, vol.2748, 2003, pp. 150-16

Lagrangian-based Hydrodynamic Model for Traffic Data Fusion on Freeways

This paper conducts a comprehensive study of the Lagrangian-based hydrodynamic model with application to highway state estimation. Our analysis is motivated by the practical problems of freeway traffic monitoring and estimation using multi-source data measured from mobile devices and fixed sensors. We conduct rigorous mathematical analysis on the Hamilton-Jacobi representation of the Lighthill-Whitham-Richards model in the transformed coordinates, and derive explicit and closed-form solutions with piecewise affine initial, boundary, and internal conditions, based on the variational principle. A numerical study of the Mobile Century field experiment demonstrates some unique features and the effectiveness in traffic estimation of the Lagrangian-based model

Doubly dynamic traffic assignment: simulation modelling framework and experimental results

Properties of a doubly dynamic simulation assignment model were investigated. The model involves specifying a day-to-day route choice model as a discrete time stochastic process, combining a between-day driver learning and adjusting model with a continuous-time, within-day dynamic network loading. Such a simulation model may be regarded as the realization of a stochastic process, which under certain mild conditions admits a unique stationary probability distribution (i.e., an invariant probability distribution over time of network flows and travel times). Such a stationary state of the stochastic process is of interest to transport modelers, as one can then describe the stochastic process by its moments such as the means, variances, and covariances of the flow and travel time profiles. The results of a simulation experiment are reported in which the process of individual drivers' day-to-day route choices are based on the aggregate learning of the experienced within-day route costs by all drivers departing in the same period. Experimental results of the stationarity of the stochastic process are discussed, along with an analysis of the sensitivity of autocorrelations of the route flows to the route choice model parameters. The results illustrate the consistency of the link flow model with properties such as first in-first out, and a simple network is used to illustrate the properties

A model for integrating home-work tour scheduling with time-varying network congestion and marginal utility profiles for home and work activities.

The existing literature on activity-based modeling emphasizes that individuals schedule their activities by keeping the whole-day activity pattern in mind. Several attempts have been made to integrate this with network congestion; however, for explicit explanation of travel behavior of individuals, further improvements are required. The proposed model is a combined model that addresses the scheduling of the home-work tour with time-varied network congestion in a fixed-point problem frame-work. Marginal utility profiles that represent individual time-of-day preferences and satiation effect of the activities are incorporated for the measurement of the utility of activity engagement along with the disutility of travel. Consideration of only time-of-day dependent marginal utility profiles of activities in the utility function does not appropriately integrate activities and travel within the tour. The proof is shown analytically and numerically. This finding contradicts earlier research into integration of morning and evening commutes with network congestion. Additionally, the results of two numerical experiments are presented. In the first experiment, an arbitrary dynamic tolling strategy is assumed, and then a detailed analysis is performed to show variation in the balance of trade-offs involved in the process. The second experiment assesses the sensitivity of the combined model through incorporation of different dynamic traffic loading models. Some meaningful observations are drawn from these experiments and are discussed with the identification of avenues for future research

Investigating undesired spatial and temporal boundary effects of congestion charging.

Two types of reported problems are related to the existing congestion charging projects that levy traffic only in a certain area within one or a few time periods during the day. One is that travellers depart earlier or later than a charging period to avoid paying full or part of the congestion charging tolls, which creates two undesired demand peaks that are often greater than available capacity. One peak comes just before the start of congestion charging and the other follows the end of it. We term this phenomenon ‘temporal boundary effect’ of congestion charging. The other reported problem is that travellers would rather stay away from a charging zone than pay congestion charging tolls, which causes undesired congestion on those roads or paths on the edge of the charging zone. We call this phenomenon ‘spatial boundary effect’ generated by congestion charging. This research investigates these boundary effects in the context of simultaneous route and departure time choice dynamic user equilibrium (SRD-DUE) network flows with an aim to gain new insights into congestion charging design. Numerical experiments investigating constant and time-varying congestion charging toll profiles are presented in this paper. This investigation shows that congestion charging may not be able to eliminate hypercongestion efficiently if schemes are not well designed, and can unfortunately give rise to undesired boundary effects and that a simply designed congestion charging scheme with small level toll or time-varying toll profiles can reduce the magnitude of boundary effects but may not be able to fully eliminate such undesired effect