Linear Complementarity Formulation for Single Bottleneck Model with Heterogeneous Commuters
TL;DR: The authors formulates the dynamic equilibrium conditions for a single bottleneck model with heterogeneous commuters as a linear complementarity problem and presents theoretical proofs for solution existence and uniqueness, and numerical results and insights for different heterogeneity assumptions.
Abstract: This paper formulates the dynamic equilibrium conditions for a single bottleneck model with heterogeneous commuters as a linear complementarity problem. This novel formulation offers a formal framework for the rigorous study and solution of a single bottleneck model with general heterogeneity parameter assumptions, enabling the adoption of well established complementarity theory and methods to analyze the model, and providing a significant contribution to the existing literature that either lacks a rigorous formulation or solves the problem under a limited set of heterogeneity parameter assumptions. The paper presents theoretical proofs for solution existence and uniqueness, and numerical results and insights for different heterogeneity assumptions.
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TL;DR: In this paper, the authors demonstrate the efficiency and effectiveness of a tradable credit system in managing the morning commute congestion with identical and non-identical commuters, and prove that an optimal credit charging scheme always exists despite how commuters vary in their value-of-time (VOT).
Abstract: We demonstrate the efficiency and effectiveness of a tradable credit system in managing the morning commute congestion with identical and nonidentical commuters. The credit system consists of a time-varying credit charged at the bottleneck and an initial credit distribution to the commuters, where the credits are universal in terms of time. Credits are tradable between the commuters and the credit price is determined by a competitive market. Under the assumption that late-arrival is not allowed, we prove that an optimal credit charging scheme, which completely eliminates the bottleneck queue, always exists despite how commuters vary in their value-of-time (VOT). The optimal charge rate is strictly increasing and convex with time, which therefore drives the commuters to depart in the increasing order of their VOT. The optimal credit charging scheme is pareto-improving, but may cause undesirable welfare distribution among the commuters. Our study shows that a combination of an initial credit distribution and an optimal credit charging scheme can simultaneously achieve system optimum and certain forms of equality (e.g., “numerical” or “proportional” equality), and that the commuters in the middle VOT bracket will receive the most credits under the proportionally equitable credit distribution.
130 citations
TL;DR: A bibliometric analysis approach is adopted for identifying the distribution of all journal publications, influential papers, top contributing authors, and leading topics in the past half century of bottleneck model research since its inception.
Abstract: The bottleneck model introduced by Vickrey in 1969 has been recognized as a benchmark representation of the peak-period traffic congestion due to its ability to capture the essence of congestion dynamics in a simple and tractable way. This paper aims to provide a 50th anniversary review of the bottleneck model research since its inception. A bibliometric analysis approach is adopted for identifying the distribution of all journal publications, influential papers, top contributing authors, and leading topics in the past half century. The literature is classified according to recurring themes into travel behavior analysis, demand-side strategies, supply-side strategies, and joint strategies of demand and supply sides. For each theme, typical extended models developed to date are surveyed. Some potential directions for further studies are discussed.
96 citations
Cites methods from "Linear Complementarity Formulation ..."
...uniqueness, but not provide solution method Mun and Yonekawa (2006) Uniform Same α, β and γ Analytical Leurent and Wagner (2009) Continuous Same α, β and γ Numerical (heuristic algorithm) Ramadurai et al., (2010) Discrete Discrete groups with different values of α, β and γ Numerical (Lemke’s algorithm) van den Berg and Verhoef (2011a) Identical Continuously distributed α, and same β and γ Analytical...
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TL;DR: In this paper, the authors developed a formulation for the network level dynamic traffic equilibrium model with departure time choice and route choice, where the embedded network loading procedure follows the cell transmission model without the holding-back issues by using detailed representations of flows at merges and diverges.
Abstract: This paper develops a formulation for the network level dynamic traffic equilibrium model with departure time choice and route choice. The embedded network loading procedure follows the cell transmission model without the holding-back issues by using detailed representations of flows at merges and diverges. The problem is modeled using a complementarity approach. The existence of the equilibrium solution is discussed using techniques from generalized variational inequalities. Computational results are performed using state of the art solvers. Since these solvers fail to solve any reasonable size networks, a specialized projection algorithm is developed to solve the problem. Numerical results are presented to demonstrate the performance of the algorithm in various starting with simple networks and extending to reasonable size networks with different traffic parameters. It is shown that the solution procedure produces good dynamic equilibrium solutions for general transportation networks.
93 citations
TL;DR: A bottleneck model in which the capacity of the bottleneck is assumed stochastic and follows a uniform distribution is investigated, and the analytical and numerical results show that the capacity variability would indeed change the commuters' travel behavior by increasing the mean trip cost and lengthening the peak period.
Abstract: In this paper we investigate a bottleneck model in which the capacity of the bottleneck is assumed stochastic and follows a uniform distribution. The commuters' departure time choice is assumed to follow the user equilibrium principle according to mean trip cost. The analytical solution of the proposed model is derived. Both the analytical and numerical results show that the capacity variability would indeed change the commuters' travel behavior by increasing the mean trip cost and lengthening the peak period. We then design congestion pricing schemes within the framework of the new stochastic bottleneck model, for both a time-varying toll and a single-step coarse toll, and prove that the proposed piecewise time-varying toll can effectively cut down, and even eliminate, the queues behind the bottleneck. We also find that the single-step coarse toll could either advance or postpone the earliest departure time. Furthermore, the numerical results show that the proposed pricing schemes can indeed improve the efficiency of the stochastic bottleneck through decreasing the system's total travel cost.
92 citations
Cites background from "Linear Complementarity Formulation ..."
...Arnott et al. (1993) and Braid (1989) extended the basic bottleneck model to consider elastic demand, whilst Lindsey (2004) and Ramadurai et al (2010) developed a single bottleneck model with heterogeneous commuters....
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TL;DR: This paper formulate the dynamic user equilibrium problem with an embedded cell transmission model on a network with a single OD pair, multiple parallel paths, multiple user classes with elastic demand, based on ideas from complementarity theory.
Abstract: In this paper we formulate the dynamic user equilibrium problem with an embedded cell transmission model on a network with a single OD pair, multiple parallel paths, multiple user classes with elastic demand. The formulation is based on ideas from complementarity theory. The travel time is estimated based on two methods which have different transportation applications: (1) maximum travel time and (2) average travel time. These travel time functions result in linear and non-linear complementarity formulations respectively. Solution existence and the properties of the formulations are rigorously analyzed. Extensive computational experiments are conducted to demonstrate the benefits of the proposed formulations on various test networks.
78 citations
References
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Book•
18 Feb 1992
TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.
Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.
2,897 citations
Posted Content•
TL;DR: In this paper, the authors discuss the investment designed to relieve congestion and distinguish six types of congested situations, which are: simple interaction, multiple interaction, bottleneck, triggerneck, network and control, and general density.
Abstract: Investment in transit facilities necessarily begins by being largely investment in the provision of new routes or new services under conditions of substantial indivisibilities and increasing returns to scale. Under these conditions the usual profitability tests for determining the desirability of specific investments lead generally to under- rather than to over-investment in transit facilities. As investment proceeds, however, larger and larger proportions of transportation investment are made primarily to relieve congestion on existing routes and to expand overall capacity. This paper discusses the investment designed to relieve congestion. For purposes of economic analysis it is useful to distinguish at least six types of congested situations, which are: simple interaction, multiple interaction, bottleneck, triggerneck, network and control, and general density.
2,006 citations
"Linear Complementarity Formulation ..." refers background in this paper
...The simplicity of the single bottleneck model allowed the development of generalizable, qualitative, theoretical insights about policy measures such as congestion pricing (Vickrey, 1969; Arnott et al., 1990; Cohen, 1987; Yang and Huang, 1997; Mun, 1999, 2006; Daganzo and Garcia, 2000), metering (Vickrey, 1973; Arnott et al....
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...Vickrey (1973) demonstrated the differences in equilibrium pricing when commuters are heterogeneous compared to his earlier work on homogeneous commuters (Vickrey, 1969)....
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Posted Content•
TL;DR: In this article, the authors demonstrate that the scheduling of activities by consumers can be explicitly modeled in theoretically satisfactory and empirically productive way, and demonstrate that this approach can be applied productively to other goods subject to peak demands.
Abstract: The purpose of this paper is to demonstrate that the scheduling of activities by consumers can be explicitly modeled in theoretically satisfactory and empirically productive way. Even in the case of urban work trips, probably one of the most tightly constrained of everyday activities, schedule shifting is found to be of quantitative importance for the understanding of urban transportation systems. Considerable effort will be required to assess fully the implications of this type of behavior for such important areas of transportation analysis as demand studies, value-of-time measurement, policy simulation, and cost-benefit analysis. Meanwhile, it seems likely that this approach can be applied productively to other goods subject to peak demands.
1,148 citations
"Linear Complementarity Formulation ..." refers background in this paper
...Empirical studies have shown that travelers differ in terms of their sensitivities to travel and schedule delay based on different socioeconomic characteristics (Small, 1982)....
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...Both the scenarios below assume that rβg < 1 or βg < αg ∀g ∈ G and rγg > 1 or γg > αg ∀g ∈ G Small (1982). 3....
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TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this paper, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.
Abstract: An algebraic proof is given of the existence of equilibrium points for bimatrix (or two-person, non-zero-sum) games. The proof is constructive, leading to an efficient scheme for computing an equilibrium point. In a nondegenerate case, the number of equilibrium points is finite and odd. The proof is valid for any ordered field.
1,087 citations
TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.
Abstract: Some simple constructive proofs are given of solutions to the matric system Mz − ω = q; z ≧ 0; ω ≧ 0; and zT ω = 0, for various kinds of data M, q, which embrace the quadratic programming problem and the problem of finding equilibrium points of bimatrix games. The general scheme is, assuming non-degeneracy, to generate an adjacent extreme point path leading to a solution. The scheme does not require that some functional be reduced.
966 citations