User Equilibrium Problem Research Articles

Computation and application of the paired combinatorial logit stochastic user equilibrium problem

The paired combinatorial logit (PCL) model is one of the recent extended logit models adapted to resolve the overlapping problem in the route choice problem, while keeping the analytical tractability of the logit choice probability function. However, the development of efficient algorithms for solving the PCL model under congested and realistic networks is quite challenging, since it has large-dimensional solution variables as well as a complex objective function. In this paper, we examine the computation and application of the PCL stochastic user equilibrium (SUE) problem under congested and realistic networks. Specifically, we develop an improved path-based partial linearization algorithm for solving the PCL SUE problem by incorporating recent advances in line search strategies to enhance the computational efficiency required to determine a suitable stepsize that guarantees convergence. A real network in the city of Winnipeg is applied to examine the computational efficiency of the proposed algorithm and the robustness of various line search strategies. In addition, in order to acquire the practical implications of the PCL SUE model, we investigate the effectiveness of how the PCL model handles the effects of congestion, stochasticity, and similarity in comparison with the multinomial logit stochastic traffic equilibrium problem and the deterministic traffic equilibrium problem.

Traffic signal timing problems with environmental and equity considerations

SUMMARYTraffic signal timings in a road network can not only affect total user travel time and total amount of traffic emissions in the network but also create an inequity problem in terms of the change in travel costs of users traveling between different locations. This paper proposes a multi‐objective bi‐level programming model for design of sustainable and equitable traffic signal timings for a congested signal‐controlled road network. The upper level of the proposed model is a multi‐objective programming problem with an equity constraint that maximizes the reserve capacity of the network and minimizes the total amount of traffic emissions. The lower level is a deterministic network user equilibrium problem that considers the vehicle delays at signalized intersections of the network. To solve the proposed model, an approach for normalizing incommensurable objective functions is presented, and a heuristic solution algorithm that combines a penalty function approach and a simulated annealing method is developed. Two numerical examples are presented to show the effects of reserve capacity improvement and green time proportion on network flow distribution and transportation system performance and the importance of incorporating environmental and equity objectives in the traffic signal timing problems. Copyright © 2013 John Wiley & Sons, Ltd.

An N-path user equilibrium for transportation networks

The traditional trip-based approach to transportation modeling has been employed for the past decade. The last step of the trip-based modeling approach is traffic assignment, which has been typically formulated as a user equilibrium (UE) problem. In the conventional perspective, the definition of UE traffic assignment is the condition that no road user can unilaterally change routes to reduce their travel time. An equivalent definition is that the travel times of all the used paths between any given origin–destination pair are equal and less than those of the unused paths. The underlying assumption of the UE definition is that road users have full information on the available transportation paths and can potentially use any path if the currently used path is overly congested. However, a more practical scenario is that each road user has a limited path set within which she/he can choose routes from. In this new scenario, we call the resulting user equilibrium an N-path user equilibrium (NPUE), in which each road user has only N paths to select from when making route choices in the network. We introduce a new formulation of the NPUE and derive optimality conditions based on this formulation. Different from traditional modeling framework, the constraints of the proposed model are of linear form, which makes it possible to solve the problem with conventional convex programming techniques. We also show that the traditional UE is a special case of an NPUE and prove the uniqueness of the resulting flow pattern of the NPUE. To efficiently solve this problem, we devise path-based and link-based solution algorithms. The proposed solution algorithms are empirically applied to networks of various sizes to examine the impact of constrained user path sets. Numerical results demonstrate that NPUE results can differ significantly from UE results depending on the number of paths available to road users. In addition, we observed an interesting phenomenon, where increasing the number of paths available to road users can sometimes decrease the overall system performance due to their selfish routing behaviors. This paradox demonstrates that network information should be provided with caution, as such information can do more harm than good in certain transportation systems.

C-logit stochastic user equilibrium model with elastic demand

Modeling the elasticity of travel demand in network equilibrium analysis has several important transportation applications. In this paper, we provide a mathematical programming formulation for the C-logit stochastic user equilibrium problem with elastic demand (CL-SUE-ED) in the route domain. The proposed model is capable of explicitly modeling the elasticity of travel demand and the effect of route overlapping on travel choice and route choice simultaneously. Some qualitative properties of the model, including the equivalency and uniqueness of the solution, are also rigorously proved. To solve the CL-SUE-ED model, a partial linearization method is developed to handle the elastic demand and route overlapping considerations. In addition, a self-regulated averaging stepsize scheme is adopted to smartly determine the stepsize while avoiding evaluating the complex objective function. Numerical examples are also provided to demonstrate the features of the proposed model and solution algorithm.

A Bi-level Formulation for the Combined Dynamic Equilibrium based Traffic Signal Control

This paper formulates the combined dynamic user equilibrium and signal control problem (DUESC) as a bi-level optimization problem. The signal control operator in the upper level optimizes the signal setting to minimize the system travel time whereas the road users in the lower level minimize their own costs (by changing departure times, paths or both) leading to dynamic user equilibrium behavior. Three components of the bi-level formulation are discussed including network loading model, the dynamic user equilibrium model and the signal control model. Then the combined problems are formulated as a Nash-Cournot game and a Stackelberg game. A solution technique based on the iterative optimization and assignment (IOA) method is proposed to solve the DUESC problem. We use the projection algorithm to solve the lower level and the mixed integer programming solver to solve the upper level. Extensive numerical results demonstrate the benefits of using this model.

A percentile system optimization approach with and without path enumeration

In this paper we deal with the travel time reliability PUE (probabilistic user equilibrium) problem studied by Lo et al. (2006) [12] and Nie (2011) [15] and we propose an alternative model that assumes a location-scale family for the path travel times, whose means and variances are evaluated in terms of link travel times. This avoids the use of the central limit theorem and convolutions providing a flexible and simple alternative. Contrary to the most existing models that require path enumeration or an iterative method to add paths sequentially, we present a percentile system optimization in its two versions: with and without path enumeration. Two examples of applications, one of them real, are used to illustrate the power of the proposed method. The cpu times required to solve the problem seem reasonable. In addition, we answer an open question raised by Nie (2011) [15] about the permutability of percentiles and partial derivatives of route travel times with respect to route flows. A family of counterexamples is given to demonstrate that the two operations: (a) obtain percentiles and (b) partial derivation of route travel times do not commute. Finally, to reproduce the trial-and-error sequence followed by users when selecting paths, we also present an algorithm that simulates this iterative process and shows that the final long-term user behavior coincides with PUE (probabilistic user equilibrium) problem resulting from some existing models.

Asymmetric stochastic user equilibrium problem with elastic demand and link capacity constraints

This paper focuses on model development and algorithm design for the general stochastic user equilibrium (SUE) problem with elastic demand, asymmetric link travel time functions and link capacity constraints. It first defines the generalised SUE conditions using generalised link travel time. An equivalent variational inequality (VI) model for these generalised SUE conditions is then developed and it is rigorously proven to be monotone and uniform Lipschitz-continuous. These two properties of the proposed VI model ensure the global convergence of the self-adaptive prediction–correction algorithm incorporating cost averaging method as a solution algorithm. Finally, a numerical example is utilised to assess the performance of the proposed VI model and solution algorithm.

Global optimization methods for the discrete network design problem

This paper addresses the discrete network design problem (DNDP) with multiple capacity levels, or multi-capacity DNDP for short, which determines the optimal number of lanes to add to each candidate link in a road network. We formulate the problem as a bi-level programming model, where the upper level aims to minimize the total travel time via adding new lanes to candidate links and the lower level is a traditional Wardrop user equilibrium (UE) problem. We propose two global optimization methods by taking advantage of the relationship between UE and system optimal (SO) traffic assignment principles. The first method, termed as SO-relaxation, exploits the property that an optimal network design solution under SO principle can be a good approximate solution under UE principle, and successively sorts the solutions in the order of increasing total travel time under SO principle. Optimality is guaranteed when the lower bound of the total travel time of the unexplored solutions under UE principle is not less than the total travel time of a known solution under UE principle. The second method, termed as UE-reduction, adds the objective function of the Beckmann-McGuire-Winsten transformation of UE traffic assignment to the constraints of the SO-relaxation formulation of the multi-capacity DNDP. This constraint is convex and strengthens the SO-relaxation formulation. We also develop a dynamic outer-approximation scheme to make use of the state-of-the-art mixed-integer linear programming solvers to solve the SO-relaxation formulation. Numerical experiments based on a two-link network and the Sioux-Falls network are conducted.

Acceleration of Solving the Dynamic Multi-Objective Network Design Problem Using Response Surface Methods

Optimization of externalities and accessibility using dynamic traffic management measures on a strategic level is a specific example of solving a multi-objective network design problem. Solving this optimization problem is time consuming, because heuristics like evolutionary multi objective algorithms are needed and solving the lower level requires solving the dynamic user equilibrium problem. Using function approximation like response surface methods (RSM) in combination with evolutionary algorithms could accelerate the determination of the Pareto optimal set. Three algorithms in which RSM are used in different ways in combination with the Strength Pareto Evolutionary Algorithm 2+ (SPEA2+) are compared with employing the SPEA2+ without the use of these methods. The results show that the algorithms using RSM methods accelerate the search considerably at the start, but tend to converge more quickly, possibly to a local optimum, and therefore loose their head start. Therefore, usage of function approximation is mainly of interest if a limited number of exact evaluations can be done or this can be used as a pre phase in a hybrid approach.

Solving the Multiclass Percentile User Equilibrium Traffic Assignment Problem

The multiclass percentile user equilibrium (MCPUE) problem discussed in this paper assumes that travelers, subject to uncertainty in a transportation network, strive to minimize reserved travel time (also known as the travel time budget) to ensure a probability of arriving at their destinations on time. An MCPUE, defined as an extension of the Wardrop equilibrium in a probabilistic network, is achieved when no travelers, regardless of their preferred on-time arrival probability, can reduce their reserved travel time by unilaterally changing their routes. Efficient numerical procedures for computing the MCPUE are studied in this paper. Specifically, a proposed new gradient projection algorithm avoids path enumeration through a column generation procedure based on a reliable shortest path algorithm. Implementation details of inner iterations, which are critical to the overall efficiency of the algorithm, are also discussed. Numerical experiments are conducted to test the computational performance of the proposed algorithm.

Joint Implementation of Signal Control and Congestion Pricing in Transportation Network

The policy of jointly implementing signal control and congestion pricing in the transportation network is investigated. Bilevel programs are developed to model the simultaneous optimization of signal setting and congestion toll. The upper level aims to maximize the network reserve capacity or minimize the total travel time, subject to signal setting and toll constraints. The lower level is a deterministic user equilibrium problem given a plan of signal setting and congestion charge. Then the bilevel programs are transferred into the equivalent single level programs, and the solution methods are discussed. Finally, a numerical example is presented to illustrate the concepts and methods, and it is shown that the joint implementation policy can achieve promising results.

Solving the traffic assignment problem using real data for a segment of Medellin’s transportation network

This paper is based on data obtained from most recent transportation studies done in the Metropolitan Area of Valle de Aburrá, city of Medellín and other 9 municipalities. The studies were based on an Origin/Destination Survey (2005), Analysis of bus routes (2006), and Mobility Master Plan (2006). This paper explains the process of writing a software application for a given network (Network of Medellin) that solves the deterministic user equilibrium problem. The software code was implemented in Visual Basic .NET®,supported by some operations using Microsoft Excel®, and hardcoded for a segment of the Medellin network. The user equilibrium distribution of flow was found by using the Frank-Wolfe algorithm. The applied algorithm was analyzed in some aspects such as number of iterations, convergence patterns, response time, as well as changes in network demand. The traffic assignment models were analyzed by using the algorithm during the P.M. peak hour (hour of highest traffic congestion). The analysis was compared with the resultsfrom the traffic assignment procedure using TransCAD® (well-known and used transportation demand software) for the 2005 database and it was found that the software is somewhat faster than the algorithm, but the latter could be a good tool for practitioners and students for modeling small networks.

Stochastic traffic assignment, Lagrangian dual, and unconstrained convex optimization

In this paper, traffic assignment problems with stochastic travel cost perceptions are reformulated and investigated in a new unconstrained nonlinear programming formulation. The objective function of the unconstrained formulation consists of two terms, in which the first term specifies the routing principle of the target problem through a satisfaction function and the sum of the first and second terms denotes the system cost or optimization objective. This formulation proves to be the Lagrangian dual of a generic primal formulation proposed by Maher et al. (2005) for the stochastic system-optimal problem. The primal–dual modeling framework presents such a common functional form that can accommodate a wide range of different traffic assignment problems. Our particular attention is given to the dual formulation in that its unconstrained feature opens the door of applying unconstrained optimization algorithms for its embraced traffic assignment problems. Numerical examples are provided to support the insights and facts derived from applying the primal and dual formulations to model stochastic system-optimal and user-equilibrium problems and justify the conjugate relationship between the primal and dual models.

Local User Cost Equilibrium: a bush-based algorithm for traffic assignment

This article presents a new algorithm for traffic assignment, called Local User Cost Equilibrium (LUCE), which iteratively solves a sequence of user-equilibrium problems associated with flows exiting from a node. The method is based on the idea of assigning users directed towards each destination separately; these flows form a bush, i.e. an acyclic sub-graph that connects every node to that destination. For each node, the algorithm considers the arcs of its forward star as the set of travel alternatives available to users and seeks a deterministic equilibrium of flows towards the same destination. The cost function associated with each of these local route choices expresses the average impedance to reaching the destination if a user continues the trip on a particular arc. The method is ‘local’ in an analytical sense, because the cost function is linearised at the current flow pattern, as if it was independent from the other splitting rates of the same node. The method is also ‘local’ in a topological sense, as nodes are processed through a polynomial visit of the current bush, inspired by dynamic programming. The node problem is formulated as a quadratic program in terms of destination-specific flows. We prove that its solution recursively applied in topological order provides a descent direction with respect to the sum-integral objective function of traffic assignment. The local equilibrium problem at nodes is solved through a greedy algorithm resembling the ad-hoc method used to compute shortest hyperpaths in transit assignment. The latter is the main contribution of this article. The main advantage of LUCE is to achieve a fast convergence rate that compares favourably with the existing methods, and to implicitly assign the demand flow of each origin-destination pair on several paths at once.

Mathematical models and computational algorithms for probit-based asymmetric stochastic user equilibrium problem with elastic demand

This article addresses model development and computational algorithm design for the probit-based asymmetric stochastic user equilibrium (SUE) problem with elastic demand. Two variational inequality (VI) models are first proposed for the SUE problem and then existence and uniqueness of their solutions are examined. These two VI models are, in reality, built by means of a probit-based stochastic network loading (SNL) map. Since there is no computational procedure available for calculating the SNL map, we thus propose a two-stage Monte Carlo simulation-based method to estimate the SNL map. To compromise computational time with accuracy in the estimation, a lower bound of sample size required by the Monte Carlo simulation is also investigated. Based on these two VI models and Monte Carlo simulation-based method, we design two hybrid prediction–correction (PC) — cost averaging (CA) algorithms for solving the SUE problem. Finally, two numerical examples are carried out to assess performance of the proposed algorithms.

A Temporal Domain Decomposition Algorithmic Scheme for Large-Scale Dynamic Traffic Assignment

ABSTRACT With emergent interest of Simulation-Based Dynamic Traffic Assignment (SBDTA) in the field of transportation network modeling, deployment of SBDTA models for traffic operations and transportation planning have increased significantly in recent years. In parallel, research and development of innovative approaches of the SBDTA model have enhanced the quality of both the assignment component, i.e, improvement of convergence quality of the Dynamic User Equilibrium (DUE) problem, and the traffic simulation element. However, computational requirement remains to be one of the great challenges for DTA implementations on large-scale networks with a long analysis period. This paper presents a temporal decomposition scheme for large spatial- and temporal-scale dynamic traffic assignment, in which the entire analysis period is divided into Epochs. Vehicle assignment is performed sequentially in each Epoch, thus improving the model scalability and confining the peak run-time memory requirement regardless of the total analysis period. A proposed self-turning scheme adaptively searches for the run-time-optimal Epoch setting during iterations regardless of the characteristics of the modeled network. Extensive numerical experiments confirm the promising performance of the proposed algorithmic schemes.

Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach

This paper is the second of a two-part research wherein we undertake a mathematically rigorous investigation of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs). Based on the thorough study of continuous-time single-destination point-queue models in the previous part, we first extend this special case to multiple destinations respecting the First-In–First-Out property of travel flows. A DCS with constant time delay is then introduced to formulate the continuous-time model of instantaneous dynamic traffic equilibria (IDUE) with a fixed demand profile. We develop a time decomposition scheme based on link free flow travel times to convert the delay DCS to a series of DCSs without time delays that are solved by a numerical time-stepping method. We provide rigorous numerical treatment of the time-decomposed IDUE model, including solvability of the discrete-time complementarity problems and convergence of the numerical trajectories to a continuous-time solution. We present numerical results to validate the IDUE on a small network and also on the Sioux Falls network.

Continuous-time point-queue models in dynamic network loading

Extending a recent work (Pang et al., in press) pertaining to a simple single-bottleneck model, this paper is the first of a two-part research wherein we undertake a mathematically rigorous study of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs) (Pang and Stewart, 2008). The first step in this comprehensive research is to gain a thorough understanding of some continuous-time point-queue models, which will be used as the building block of a computationally tractable model for the continuous-time DUE problem that we will study in detail in the accompanying paper (Ban et al., in press). Starting with the original point-queue model introduced by Vickrey (1969), we summarize some desirable properties that a continuous-time point-queue model should possess, and show that one of these properties—the nonnegativity of the queue lengths—is violated by Vickrey’s original model. As a remedy to this drawback of Vickrey’s model and with the goal of extending it to a continuous-time setting, we introduce two continuous-time point-queue models and show that they satisfy the properties we propose. Discretizations of the continuous-time models are discussed and construction of numerical trajectories is presented; convergence of such trajectories as the time step approaches zero is established; regularity of a solution to the continuous-time problem is clarified, and numerical results are presented.

Retarded weighted quasi-variational inequalities for traffic equilibrium problems

The aim of the paper is to consider user equilibrium problems in transportation networks in the most complete and realistic situations. In fact, the presented model allows, for the dependence of data on time, the presence of elastic travel demands, the capacity restrictions and delay effects. Moreover, the weights give the possibility to study the problem in the presence of congestion. The equilibrium conditions for such a model are given and the equivalent formulation in terms of a weighted quasi-variational inequality is discussed. Then, a theorem for the existence of solutions is shown. Moreover, by using the set convergence in Kuratowski’s sense, a continuity result for equilibrium solutions is obtained. At last, a numerical example is provided.

Path-Based Algorithms to Solve C-Logit Stochastic User Equilibrium Assignment Problem

This paper develops path-based algorithms to solve the C-logit stochastic user equilibrium (SUE) problem on the basis of an adaptation of the gradient projection method. The algorithms' strategies for step size determination differ. Three strategies are investigated: (a) predetermined step size, (b) Armijo line search, and (c) self-adaptive line search. The algorithms are tested on the well-known Winnipeg (Manitoba, Canada) network. Two sets of experiments are conducted: (a) a computational comparison of different line search strategies and (b) the impact of different modeling specifications for route overlapping (a flow-independent or a flow-dependent commonality factor). The results indicate that the path-based algorithm with the self-adaptive step size strategy performs better than the other step size strategies. The paper shows that, depending on the model parameters, particularly the commonality factor parameter, the C-logit SUE flows may be quite different from the multinomial logit SUE flows.