This paper deals with the modelling of the Dynamic Traffic Assignment Problem (DTAP) for analyzin... more This paper deals with the modelling of the Dynamic Traffic Assignment Problem (DTAP) for analyzing the day time varying flows of urban transportation networks. During the past two decades, many models have been proposed in the literature, but some of them are based on heuristic concepts, and most of them incorporate important limitations. In this paper, we propose a Dynamic Traffic Assignment Model which is mainly based on the following assumption: the time spent by a vehicle on a link may be decomposed into a fixed travel time and a waiting time. The fixed travel time is the free flow travel time, after which vehicles are put in the link exit queue, until there is room for them to proceed their trip. We show that this model leads to a network structure (a temporal expansion of the base network, including the queues), and then formulate the DTAP as a network equilibrium problem over the expanded network. The mathematical formulation is achieved through a variational inequality where the variables are the path (or link) flows over the space-time expanded network. Numerical results show that the model may handle large networks, so it may be used in practice to analyze the traffic congestion moves in real cities, as well in space (physical links) as in day time.
The problem of dynamic traffic assignment is to predict the evolution of the flow pattern in a ne... more The problem of dynamic traffic assignment is to predict the evolution of the flow pattern in a network where travel demands and costs vary over time and space. Using optimal control theory two continuous time formulations of the dynamic traffic assignment problem are considered, one corresponding to system optimization and the other to user optimization. Optimality conditions are derived by the Pontryagin minimum principle and given economic interpretations which correspond to intuitive notions regarding dynamic system optimized and user optimized traffic flow patterns. We further establish that Pontryagin\u27s necessary conditions are also sufficient under commonly encountered regularity conditions. The existence of singular control is also considered. Notably, we offer in the form of an optimal control problem the first dynamic generalization of Beckmann\u27s equivalent optimization problem for a static user equilibrium traffic assignment. This dynamic model is extended to include time-varying elastic demands and penalties for late arrivals. Finally, another type of dynamic generalization of Wardrop\u27s first principle is considered and the existence of the dynamic user equilibrium flow pattern is conjectured in an infinite-dimensional variational inequality formulation
Optimal control theory is applied to the problem of dynamic traffic assignment, corresponding to ... more Optimal control theory is applied to the problem of dynamic traffic assignment, corresponding to user optimization, in a congested network with one origin-destination pair connected by N parallel arcs. Two continuous time formulations are considered, one with fixed demand and the other with elastic demand. Optimality conditions are derived by Pontryagin's maximum principle and interpreted as a dynamic generalization of Wardrop's first principle. The existence of singular controls is examined, and the optimality of singular controls is assured by the generalized convexity conditions. Under the steady-state assumptions, a dynamic model with elastic demand is shown to be a proper extension of Beckmann's equivalent optimization problem with elastic demand. Finally, the derivation of the dynamic user optimization objective functional is demonstrated, which is analogous to the derivation of the objective function of Beckmann's mathematical programming formulation for user ...
This paper considers the problem of dynamic congestion pricing which determines an optimal toll s... more This paper considers the problem of dynamic congestion pricing which determines an optimal toll schedule to be charged at the entrance of a bottleneck on a single-destination traffic network. A two-person nonzero-sum Stacke!berg differential game model is formulated when the underlying information structure is open loop. Characteristics of the Stackelberg equilibrium solution are analyzed. The Pontryagin minimum principle is applied to derive the necessary conditions from which the open-loop Stackelberg equilibrium strategy can be obtained. The heuristic algorithm which obviates an evaluation of the gradient vector of the Hamiltonian is also proposed. Copyright
We consider the competition among a finite number of firms who must transport the fixed volume of... more We consider the competition among a finite number of firms who must transport the fixed volume of traffic over a prescribed planning horizon on a congested transportation network with one origin-destination pair connected by parallel routes. It is assumed that each firm attempts to minimize inruvidual transportation cost by making a sequence of simultaneous decisions of departure time, route, and Bow rate based on the trade-off between arc traversal time and schedule delay penalty. The model is formulated as an N-person nonzero-sum ruscrete-time dynamic game. A Cournot-Nash network equilibrium is defined under the open-loop information structure. Optimality conrutions are derived using the Kuhn-Tucker theorem and given economic interpretation. Future extensions of the proposed model are also ruscussed.
Transportation Research Part B: Methodological, 2002
In this paper, a dynamic user equilibrium traffic assignment model with simultaneous departure ti... more In this paper, a dynamic user equilibrium traffic assignment model with simultaneous departure time/ route choices and elastic demands is formulated as an arc-based nonlinear complementarity problem on congested traffic networks. The four objectives of this paper are (1) to develop an arc-based formulation which obviates the use of path-specific variables, (2) to establish existence of a dynamic user equilibrium solution to the model using Brouwer's fixed-point theorem, (3) to show that the vectors of total arc inflows and associated minimum unit travel costs are unique by imposing strict monotonicity conditions on the arc travel cost and demand functions along with a smoothness condition on the equilibria, and (4) to develop a heuristic algorithm that requires neither a path enumeration nor a storage of path-specific flow and cost information. Computational results are presented for a simple test network with 4 arcs, 3 nodes, and 2 origin-destination pairs over the time interval of 120 periods.
Thesis (Ph. D.)--University of Pennsylvania, 1988. Includes bibliographical references (leaves 73... more Thesis (Ph. D.)--University of Pennsylvania, 1988. Includes bibliographical references (leaves 73-76). Microfiche.
Planning report (M.U.P.)--San Jose State University, 1985. Includes bibliographical references (l... more Planning report (M.U.P.)--San Jose State University, 1985. Includes bibliographical references (leaves 105-107).
Transportation Research Part C: Emerging Technologies, 2007
ABSTRACT This paper considers the problem of dynamic congestion pricing that determines optimal t... more ABSTRACT This paper considers the problem of dynamic congestion pricing that determines optimal time-varying tolls for a pre-specified subset of arcs with bottleneck on a congested general traffic network. A two-person nonzero-sum dynamic Stackelberg game model is formulated with the assumption that the underlying information structure is open loop. Characteristics of the Stackelberg equilibrium solution are analyzed. The Hooke–Jeeves algorithm that obviates an evaluation of the gradient vector of the objective function is presented with a numerical example. The paper concludes with its future extensions.
This paper considers the problem of the competition among a finite number of players who must tra... more This paper considers the problem of the competition among a finite number of players who must transport the fixed volume of traffic on a simple network over a prescribed planning horizon. Each player attempts to minimize his total transportation cost by making simultaneous decisions of departure time, route, and flow rate over time. The problem is modeled as a N-person nonzero-sum differential game. Two solution concepts are applied: [ l ] the open-loop Nash equilibrium solution and [2] the feedback Nash equilibrium solution. Optimality conditions are derived and given an economic interpretation as a dynamic game theoretic generalization of Wardrop's second principle. Future extensions of the model are also discussed. The model promises potential applications to Intelligent Vehicle Highway Systems (IVHS) and air traffic control systems. 0 7993 by
ABSTRACT The dynamic mixed behavior traffic network equilibrium model is formulated as a noncoope... more ABSTRACT The dynamic mixed behavior traffic network equilibrium model is formulated as a noncooperative N-person nonzero-sum differential game under the open-loop information structure. A simple network is considered where one origin-destination pair is connected by parallel arcs and two types of players - User Equilibrium (UE) and Cournot-Nash (C-N) - interact through the congestion phenomenon. Each of UE and C-N players attempts to achieve its own prescribed objective by making a continuum of simultaneous decisions of departure time, route, and departure flow rate over a fixed time interval. The necessary and sufficient conditions are derived and given economic interpretation as a dynamic game theoretic generalization of the mixed behavior traffic network equilibrium principle which requires equilibration of average costs for UE players and equilibration of marginal costs for C-N players. An approximate iterative algorithm is proposed for solving the model in discrete time, which makes use of the augmented Lagrangian method and the gradient method. A numerical example is presented and future extensions of the model and the algorithm are also discussed.
In this paper, a dynamic system-optimal traffic assignment model is formulated as a convex contro... more In this paper, a dynamic system-optimal traffic assignment model is formulated as a convex control problem for a congested general network with many origins and many destinations. Analytical and computational difficulties caused by the nonconvexity of the previous models are eliminated. The modeling of arc traffic dynamics is improved to prohibit instantaneous flow propagation on each arc even though the concave exit functions are still employed to represent the physical process of traffic congestion. An economic interpretation of the optimality conditions is given as a dynamic assignment principle which requires equilibration of actual marginal costs on all the paths that are used. A numerical example is also presented.
Transportation Research Part B: Methodological, 1990
An equivalent continuous time optimal control problem is formulated to predict the temporal evolu... more An equivalent continuous time optimal control problem is formulated to predict the temporal evolution of traffic flow pattern on a congested multiple origin-destination network, corresponding to a dynamic generalization of Wardropian user equilibrium. Optimality conditions are derived using the Pontryagin minimum principle and given economic interpretations, which are generalizations of similar results previously reported for single-destination networks. Analyses of sufficient conditions for optimality and of singular controls are also given. Under the steady-state assumptions, the model is shown to be a proper dynamic extension of Beckmann's mathematical programming problem for a static user equilibrium traffic assignment.
ABSTRACT The objective of this paper is to develop a quantitative methodology for the study of dy... more ABSTRACT The objective of this paper is to develop a quantitative methodology for the study of dynamic oligopolistic competition among spatially separated firms serving spatially separated markets. The problem is modeled as a N-person nonzero-sum differential game in the open-loop and closed-loop strategy spaces. Optimality conditions are derived and provided with economic interpretations. Iterative algorithms and numerical results are presented for the open-loop and closed-loop models. Citation Copyright 1997 by Kluwer Academic Publishers.
In this paper we formulate the dynamic network user equilibrium problem as a variational inequali... more In this paper we formulate the dynamic network user equilibrium problem as a variational inequality problem in discrete time in terms of unit path cost functions. We then show how arc exit flow functions and nested cost operators can be used to calculate unit path costs given the departure ...
Transportation Science is the foremost journal in the field of transportation analysis, featuring... more Transportation Science is the foremost journal in the field of transportation analysis, featuring comprehensive, timely articles and surveys that cover all levels of planning and all modes of transportation. It is a quarterly journal published by the Institute for Operations Research and ...
Transportation Research Part C: Emerging Technologies, 1995
One way to estimate the potential benefits of new traffic control and management systems is to co... more One way to estimate the potential benefits of new traffic control and management systems is to compare the total cost incurred in equilibrium with the system optimized total cost. To do this, we formulate the dynamic traffic assignment models with schedule delays under the system optimum and user equilibrium principles and solve them using numerical methods. System optimum and user equilibrium dynamic assignments on an 18-arc test network are then compared in terms of total travel times and schedule delays at different levels of traffic congestion. This comparison provides important implications for the success of the intelligent transportation systems (ITS) in reducing traffic congestion.
Transportation Research Part B: Methodological, 1998
AbstractÐIn this paper we develop two types of dynamic congestion pricing model, based on the the... more AbstractÐIn this paper we develop two types of dynamic congestion pricing model, based on the theory of marginal cost pricing. The ®rst model is appropriate for situations where commuters have the ability to learn the best route choices through day-to-day explorations on a network with arc capacities and travel demands that are stable from day to day. The second model is appropriate for situations where commuters optimize their routing decisions each day on a network with arc capacities and travel demands that¯uctuate sig-ni®cantly from day to day. We show that two types of time-varying congestion tolls can be determined by solving a convex control formulation of the dynamic system optimal trac assignment problem on a network with many origins and many destinations. We also show that the dynamic system optimal trac assignment is an equilibrium for commuters under the tolls in both cases. #
This paper deals with the modelling of the Dynamic Traffic Assignment Problem (DTAP) for analyzin... more This paper deals with the modelling of the Dynamic Traffic Assignment Problem (DTAP) for analyzing the day time varying flows of urban transportation networks. During the past two decades, many models have been proposed in the literature, but some of them are based on heuristic concepts, and most of them incorporate important limitations. In this paper, we propose a Dynamic Traffic Assignment Model which is mainly based on the following assumption: the time spent by a vehicle on a link may be decomposed into a fixed travel time and a waiting time. The fixed travel time is the free flow travel time, after which vehicles are put in the link exit queue, until there is room for them to proceed their trip. We show that this model leads to a network structure (a temporal expansion of the base network, including the queues), and then formulate the DTAP as a network equilibrium problem over the expanded network. The mathematical formulation is achieved through a variational inequality where the variables are the path (or link) flows over the space-time expanded network. Numerical results show that the model may handle large networks, so it may be used in practice to analyze the traffic congestion moves in real cities, as well in space (physical links) as in day time.
The problem of dynamic traffic assignment is to predict the evolution of the flow pattern in a ne... more The problem of dynamic traffic assignment is to predict the evolution of the flow pattern in a network where travel demands and costs vary over time and space. Using optimal control theory two continuous time formulations of the dynamic traffic assignment problem are considered, one corresponding to system optimization and the other to user optimization. Optimality conditions are derived by the Pontryagin minimum principle and given economic interpretations which correspond to intuitive notions regarding dynamic system optimized and user optimized traffic flow patterns. We further establish that Pontryagin\u27s necessary conditions are also sufficient under commonly encountered regularity conditions. The existence of singular control is also considered. Notably, we offer in the form of an optimal control problem the first dynamic generalization of Beckmann\u27s equivalent optimization problem for a static user equilibrium traffic assignment. This dynamic model is extended to include time-varying elastic demands and penalties for late arrivals. Finally, another type of dynamic generalization of Wardrop\u27s first principle is considered and the existence of the dynamic user equilibrium flow pattern is conjectured in an infinite-dimensional variational inequality formulation
Optimal control theory is applied to the problem of dynamic traffic assignment, corresponding to ... more Optimal control theory is applied to the problem of dynamic traffic assignment, corresponding to user optimization, in a congested network with one origin-destination pair connected by N parallel arcs. Two continuous time formulations are considered, one with fixed demand and the other with elastic demand. Optimality conditions are derived by Pontryagin's maximum principle and interpreted as a dynamic generalization of Wardrop's first principle. The existence of singular controls is examined, and the optimality of singular controls is assured by the generalized convexity conditions. Under the steady-state assumptions, a dynamic model with elastic demand is shown to be a proper extension of Beckmann's equivalent optimization problem with elastic demand. Finally, the derivation of the dynamic user optimization objective functional is demonstrated, which is analogous to the derivation of the objective function of Beckmann's mathematical programming formulation for user ...
This paper considers the problem of dynamic congestion pricing which determines an optimal toll s... more This paper considers the problem of dynamic congestion pricing which determines an optimal toll schedule to be charged at the entrance of a bottleneck on a single-destination traffic network. A two-person nonzero-sum Stacke!berg differential game model is formulated when the underlying information structure is open loop. Characteristics of the Stackelberg equilibrium solution are analyzed. The Pontryagin minimum principle is applied to derive the necessary conditions from which the open-loop Stackelberg equilibrium strategy can be obtained. The heuristic algorithm which obviates an evaluation of the gradient vector of the Hamiltonian is also proposed. Copyright
We consider the competition among a finite number of firms who must transport the fixed volume of... more We consider the competition among a finite number of firms who must transport the fixed volume of traffic over a prescribed planning horizon on a congested transportation network with one origin-destination pair connected by parallel routes. It is assumed that each firm attempts to minimize inruvidual transportation cost by making a sequence of simultaneous decisions of departure time, route, and Bow rate based on the trade-off between arc traversal time and schedule delay penalty. The model is formulated as an N-person nonzero-sum ruscrete-time dynamic game. A Cournot-Nash network equilibrium is defined under the open-loop information structure. Optimality conrutions are derived using the Kuhn-Tucker theorem and given economic interpretation. Future extensions of the proposed model are also ruscussed.
Transportation Research Part B: Methodological, 2002
In this paper, a dynamic user equilibrium traffic assignment model with simultaneous departure ti... more In this paper, a dynamic user equilibrium traffic assignment model with simultaneous departure time/ route choices and elastic demands is formulated as an arc-based nonlinear complementarity problem on congested traffic networks. The four objectives of this paper are (1) to develop an arc-based formulation which obviates the use of path-specific variables, (2) to establish existence of a dynamic user equilibrium solution to the model using Brouwer's fixed-point theorem, (3) to show that the vectors of total arc inflows and associated minimum unit travel costs are unique by imposing strict monotonicity conditions on the arc travel cost and demand functions along with a smoothness condition on the equilibria, and (4) to develop a heuristic algorithm that requires neither a path enumeration nor a storage of path-specific flow and cost information. Computational results are presented for a simple test network with 4 arcs, 3 nodes, and 2 origin-destination pairs over the time interval of 120 periods.
Thesis (Ph. D.)--University of Pennsylvania, 1988. Includes bibliographical references (leaves 73... more Thesis (Ph. D.)--University of Pennsylvania, 1988. Includes bibliographical references (leaves 73-76). Microfiche.
Planning report (M.U.P.)--San Jose State University, 1985. Includes bibliographical references (l... more Planning report (M.U.P.)--San Jose State University, 1985. Includes bibliographical references (leaves 105-107).
Transportation Research Part C: Emerging Technologies, 2007
ABSTRACT This paper considers the problem of dynamic congestion pricing that determines optimal t... more ABSTRACT This paper considers the problem of dynamic congestion pricing that determines optimal time-varying tolls for a pre-specified subset of arcs with bottleneck on a congested general traffic network. A two-person nonzero-sum dynamic Stackelberg game model is formulated with the assumption that the underlying information structure is open loop. Characteristics of the Stackelberg equilibrium solution are analyzed. The Hooke–Jeeves algorithm that obviates an evaluation of the gradient vector of the objective function is presented with a numerical example. The paper concludes with its future extensions.
This paper considers the problem of the competition among a finite number of players who must tra... more This paper considers the problem of the competition among a finite number of players who must transport the fixed volume of traffic on a simple network over a prescribed planning horizon. Each player attempts to minimize his total transportation cost by making simultaneous decisions of departure time, route, and flow rate over time. The problem is modeled as a N-person nonzero-sum differential game. Two solution concepts are applied: [ l ] the open-loop Nash equilibrium solution and [2] the feedback Nash equilibrium solution. Optimality conditions are derived and given an economic interpretation as a dynamic game theoretic generalization of Wardrop's second principle. Future extensions of the model are also discussed. The model promises potential applications to Intelligent Vehicle Highway Systems (IVHS) and air traffic control systems. 0 7993 by
ABSTRACT The dynamic mixed behavior traffic network equilibrium model is formulated as a noncoope... more ABSTRACT The dynamic mixed behavior traffic network equilibrium model is formulated as a noncooperative N-person nonzero-sum differential game under the open-loop information structure. A simple network is considered where one origin-destination pair is connected by parallel arcs and two types of players - User Equilibrium (UE) and Cournot-Nash (C-N) - interact through the congestion phenomenon. Each of UE and C-N players attempts to achieve its own prescribed objective by making a continuum of simultaneous decisions of departure time, route, and departure flow rate over a fixed time interval. The necessary and sufficient conditions are derived and given economic interpretation as a dynamic game theoretic generalization of the mixed behavior traffic network equilibrium principle which requires equilibration of average costs for UE players and equilibration of marginal costs for C-N players. An approximate iterative algorithm is proposed for solving the model in discrete time, which makes use of the augmented Lagrangian method and the gradient method. A numerical example is presented and future extensions of the model and the algorithm are also discussed.
In this paper, a dynamic system-optimal traffic assignment model is formulated as a convex contro... more In this paper, a dynamic system-optimal traffic assignment model is formulated as a convex control problem for a congested general network with many origins and many destinations. Analytical and computational difficulties caused by the nonconvexity of the previous models are eliminated. The modeling of arc traffic dynamics is improved to prohibit instantaneous flow propagation on each arc even though the concave exit functions are still employed to represent the physical process of traffic congestion. An economic interpretation of the optimality conditions is given as a dynamic assignment principle which requires equilibration of actual marginal costs on all the paths that are used. A numerical example is also presented.
Transportation Research Part B: Methodological, 1990
An equivalent continuous time optimal control problem is formulated to predict the temporal evolu... more An equivalent continuous time optimal control problem is formulated to predict the temporal evolution of traffic flow pattern on a congested multiple origin-destination network, corresponding to a dynamic generalization of Wardropian user equilibrium. Optimality conditions are derived using the Pontryagin minimum principle and given economic interpretations, which are generalizations of similar results previously reported for single-destination networks. Analyses of sufficient conditions for optimality and of singular controls are also given. Under the steady-state assumptions, the model is shown to be a proper dynamic extension of Beckmann's mathematical programming problem for a static user equilibrium traffic assignment.
ABSTRACT The objective of this paper is to develop a quantitative methodology for the study of dy... more ABSTRACT The objective of this paper is to develop a quantitative methodology for the study of dynamic oligopolistic competition among spatially separated firms serving spatially separated markets. The problem is modeled as a N-person nonzero-sum differential game in the open-loop and closed-loop strategy spaces. Optimality conditions are derived and provided with economic interpretations. Iterative algorithms and numerical results are presented for the open-loop and closed-loop models. Citation Copyright 1997 by Kluwer Academic Publishers.
In this paper we formulate the dynamic network user equilibrium problem as a variational inequali... more In this paper we formulate the dynamic network user equilibrium problem as a variational inequality problem in discrete time in terms of unit path cost functions. We then show how arc exit flow functions and nested cost operators can be used to calculate unit path costs given the departure ...
Transportation Science is the foremost journal in the field of transportation analysis, featuring... more Transportation Science is the foremost journal in the field of transportation analysis, featuring comprehensive, timely articles and surveys that cover all levels of planning and all modes of transportation. It is a quarterly journal published by the Institute for Operations Research and ...
Transportation Research Part C: Emerging Technologies, 1995
One way to estimate the potential benefits of new traffic control and management systems is to co... more One way to estimate the potential benefits of new traffic control and management systems is to compare the total cost incurred in equilibrium with the system optimized total cost. To do this, we formulate the dynamic traffic assignment models with schedule delays under the system optimum and user equilibrium principles and solve them using numerical methods. System optimum and user equilibrium dynamic assignments on an 18-arc test network are then compared in terms of total travel times and schedule delays at different levels of traffic congestion. This comparison provides important implications for the success of the intelligent transportation systems (ITS) in reducing traffic congestion.
Transportation Research Part B: Methodological, 1998
AbstractÐIn this paper we develop two types of dynamic congestion pricing model, based on the the... more AbstractÐIn this paper we develop two types of dynamic congestion pricing model, based on the theory of marginal cost pricing. The ®rst model is appropriate for situations where commuters have the ability to learn the best route choices through day-to-day explorations on a network with arc capacities and travel demands that are stable from day to day. The second model is appropriate for situations where commuters optimize their routing decisions each day on a network with arc capacities and travel demands that¯uctuate sig-ni®cantly from day to day. We show that two types of time-varying congestion tolls can be determined by solving a convex control formulation of the dynamic system optimal trac assignment problem on a network with many origins and many destinations. We also show that the dynamic system optimal trac assignment is an equilibrium for commuters under the tolls in both cases. #
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