User Equilibrium Assignment Problem Research Articles

Continuous Network Design Using Partial Linearized Subgradient Methods

Although demand management has shown to be a vital tool in managing congestion, many metropolitan planning organizations (MPOs) and departments of transportation (DOTs) still pursue network modification and capacity increase as a congestion relief and a mitigation measure. The network design problem (NDP) still has, therefore, an essential and significant role in shaping and sizing urban transportation networks. The literature has traditionally treated the NDP as a bi-level mathematical programming problem or a mathematical program with equilibrium constraints (MPEC). In the bi-level optimization setting, the problem is approached as a leader-follower problem in which the lower level is a user equilibrium (UE) assignment problem as the follower, and the upper level is the network sizing problem as the leader problem. NDP has long been known as a challenging problem, and many solution algorithms have been proposed to solve it. This study proposed an efficient solution algorithm for the continuous network design problem (CNDP). The solution algorithm has been shown empirically to solve the CNDP in a shorter time using partial linearized subgradient methods. The proposed method was applied to a small network that was traditionally used to evaluate the performance of solution algorithms versus Braess’s paradox. It was then applied to the Sioux Falls network as a well-known benchmark network to compare the results with previous studies. The proposed method has been shown to run much faster than all the previous studies reviewed in this paper with minimal degradation of accuracy (0.52% lower than the best solution).

A Simultaneous Solution for Reserve Capacity Maximization and Delay Minimization Problems in Signalized Road Networks

In this study, we present a bilevel programming model in which upper level is defined as a biobjective problem and the lower level is considered as a stochastic user equilibrium assignment problem. It is clear that the biobjective problem has two objectives: the first maximizes the reserve capacity whereas the second minimizes performance index of a road network. We use a weighted-sum method to determine the Pareto optimal solutions of the biobjective problem by applying normalization approach for making the objective functions dimensionless. Following, a differential evolution based heuristic solution algorithm is introduced to overcome the problem presented by use of biobjective bilevel programming model. The first numerical test is conducted on two-junction network in order to represent the effect of the weighting on the solution of combined reserve capacity maximization and delay minimization problem. Allsop & Charlesworth’s network, which is a widely preferred road network in the literature, is selected for the second numerical application in order to present the applicability of the proposed model on a medium-sized signalized road network. Results support authorities who should usually make a choice between two conflicting issues, namely, reserve capacity maximization and delay minimization.

Passenger flow control with multi-station coordination in subway networks: algorithm development and real-world case study

ABSTRACTThis paper addresses the problem of passenger flow control in a multi-line subway network during peak hours. A new multi-station coordinated passenger flow control model is proposed to simultaneously adjust the number of inbound and transfer passengers entering multiple stations or lines. The implementation of passenger flow control governs the redundant passengers to queue at given facilities, which is similar to ramp metering that regulates the number of vehicles entering a highway segment. The proposed model is a bi-level programming, whose upper level aims to optimize the system performance with different passenger flow control strategies, while the lower level is a logit-based stochastic user equilibrium assignment problem under a given strategy, considering passenger flow evolution, dynamic path cost, and route choice. An algorithm, integrating the method of successive averages with genetic algorithm, is developed to solve the model. A real-world case study is conducted to examine the validity.

Bi-level Programming Model for Exclusive Bus Lanes Configuration in Multimodal Traffic Network

Abstract Setting exclusive bus lanes (EBLs) on existing roads is a major measure to practise bus priority. However, the capacities for different vehicles will be changed with such strategies and then the travel times of different modes will be also changed. Accordingly, the presence of EBLs may be associated with significant impacts on the travellers’ mode choices and the performance of the whole traffic network, especially in a congested condition. This paper attempts to propose a EBLs design model for multimodal traffic network, in which the total travel cost of all travellers in network is regarded as the optimization objective. Firstly, this paper analyses the effects of EBLs on the capacities for transit vehicles and private cars and formulates the corresponding impedance functions based on the travel demand. The complex travellers’ choice behaviours (including mode choice and route choice) are analysed and a variational inequality model is then proposed to describe user equilibrium assignment problem in multimodal network. Further, a bi-level programming model is proposed to describe the EBLs design problem. The branch and bound algorithm is then given for solving the proposed bi-level problem. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of such model and algorithm.

A stochastic user equilibrium assignment problem in discrete network design problem

A stochastic user equilibrium assignment problem in discrete network design problem

A stochastic user equilibrium assignment problem in discrete network design problem

We formulate the transportation discrete network design problem (DNDP) as a mixed-integer bi-level mathematical problem, based on the concept reserve capacity. The upper level goal programme maximises the reserve capacity by designing the direction of street and increasing the street capacity through lane addition. The lower level problem is stochastic user equilibrium traffic assignment problem within a probit-based path choice decision framework which generates user equilibrium flow patterns. Because of non-convexity nature of the model, meta-heuristic methods used to solve the problem and we used Monte Carlo simulation approach to compute path choice probabilities and stochastic user equilibrium solved by method of successive averages. A hybrid genetic algorithm with simulated annealing and an evolutionary simulated annealing algorithm are proposed. Numerical examples are presented to verify the proposed model and algorithm.

Combined Gravity Model Trip Distribution and Paired Combinatorial Logit Stochastic User Equilibrium Problem

The equivalent mathematical formulation of the combined doubly-constrained gravity-based trip distribution and paired-combinatorial-logit stochastic user equilibrium assignment problem (CDA-PCL-SUE) is proposed. Its first order conditions are shown to be equal to the gravity equations and PCL formula. The proposed solution method is a path-based partial linearization algorithm to approximately solve the restricted CDA-PCL-SUE. The proposed algorithm is a three-phase iterative process. Phase 1 is an entropy maximization problem on O-D flow space that can be solved by Bregman’s balancing algorithm. Phase 2 is a PCL SUE problem that can be solved by PCL formula. Phase 3 is line search. CDA-PCL-SUE is solved on a small network and a real network, the city of Winnipeg network. The proposed algorithms with the six line search methods, namely, golden section (GS), bisection (BS), Armijo’s rule (AR), method of successive averages (MSA), self-regulated averaging (SRA) scheme, and quadratic interpolation (QI) scheme, are compared in terms of various convergence characteristics: root mean square error, step size, KKT-based mean square error and objective function. In terms of computational efficiency, under different path set sizes, dispersion parameters, impedance parameters and demand levels, the following line search methods are ordered from best to worst: SRA, GS, AR, QI, BS and MSA. The performances of Armijo’s rule and QI have greater variances. The performance of QI is worse with the increase of the path set size. Given all other factors being the same, the increase of dispersion parameter, path set size or demand level yields the increase of CPU time, whereas the change of impedance parameter does not influence CPU time. In addition, CDA-PCL-SUE is compared with its multinomial-logit counterpart (CDA-MNL-SUE).

Optimizing capacity of signalized road network with reversible lanes

This paper studies the network capacity problem on signalized road network with reversible lanes. A Mixed Network Design Problem (MDNP) is formulated to describe the problem where the upper-level problem is a mixed integer non-linear program designed to maximize the network capacity by optimizing the input parameters (e.g. the signal splits, circles, reassigned number of lanes and O–D demands), while the lower-level problem is the common Deterministic User Equilibrium (DUE) assignment problem formulated to model the drivers’ route choices. According to whether one way strategy is permitted in practice, two strategies for implementing reversible roadway are considered. In the first strategy, not all lanes are reversible and the reversible roadways always hold its ability to accommodate the two-way traffic flow. In the second strategy, one-way road is allowed, which means that all the lanes are reversible and could be assigned to one flow direction if the traffic flow in both directions is severally unsymmetrical. Genetic Algorithm (GA) is detailedly presented to solve the bi-level network capacity problem. The application of the proposed method on a numerical example denotes that Strategy 2 can make more use of the physical capacity of key links (signal controlled links), thus, the corresponding network capacity outperforms it is of Strategy 1 considerably.

Topological Effects and Performance Optimization in Transportation Continuous Network Design

Because of the limitation of budget, in the planning of road works, increased efforts should be made on links that are more critical to the whole traffic system. Therefore, it would be helpful to model and evaluate the vulnerability and reliability of the transportation network when the network design is processing. This paper proposes a bilevel transportation network design model, in which the upper level is to minimize the performance of the network under the given budgets, while the lower level is a typical user equilibrium assignment problem. A new solution approach based on particle swarm optimization (PSO) method is presented. The topological effects on the performance of transportation networks are studied with the consideration of three typical networks, regular lattice, random graph, and small-world network. Numerical examples and simulations are presented to demonstrate the proposed model.

A new one-level convex optimization approach for estimating origin–destination demand

Accurately estimating Origin–Destination (OD) trip tables based on traffic data has become crucial in many real-time traffic applications. The problem of OD estimation is traditionally modeled as a bilevel network design problem (NDP), which is challenging to solve in large-scale networks. In this paper, we propose a new one-level convex optimization formulation to reasonably approximate the bilevel structure, thus allowing the development of more efficient solution algorithms. This one-level approach is consistent with user equilibrium conditions, and improves previous one-level relaxed OD estimation formulations in the literature by ‘equilibrating’ path flows using external path cost parameters. Our new formulation can, in fact, be viewed as a special case of the user equilibrium assignment problem with elastic demand, and hence can be solved efficiently by standard path-based traffic assignment algorithms with an iterative parameter updating scheme. Numerical experiments indicate that this new one-level approach performs very well. Estimation results are robust to network topology, sensor coverage, and observation error, and can achieve further improvements when additional data sources are included.

Suitable Cost Functions for Signalized Arterials and Freeways, in the User Equilibrium Assignment Problem

The aim of this paper is to design and prove new link travel time functions (TTF), which are combinations of known delay functions. The proposed TTF are useful for modeling two types of arc: light signalized arcs and limited access arcs. These TTF are used for formulating and solving the User Equilibrium Assignment (UEA) problem by means the Frank-Wolfe algorithm, obtaining more accurate results. The process for solving the UEA problem was improved because usually it is solved for only one travel time function or by flow-speed curves, or without considering overflow periods. An application on a Mexico City sub-network is presented.

Computational Experience on Advanced Algorithms for User Equilibrium Traffic Assignment Problem and Its Convergence Error

Many algorithms have been investigated extensively for decades to solve the user equilibrium (UE) assignment problem, and new algorithms are actively proposed even in this new century. The first objective of this study is to compare the performance of the proposed algorithms on several practical networks and demonstrate their characteristics empirically. At earlier stages of iteration, every algorithm shows a fairly similar performance of convergence with regard to the calculation time, but at later stages of iteration, advanced algorithms exhibit faster performance. The use of some of these fast algorithms results in the convergence error reaching the maximum limit of arithmetic precision of the computer, which means that a virtually exact solution can be achieved. The second objective of this study is to investigate the convergence error and seek an appropriate convergence criterion for the UE assignment in practice. We found that the difference between the temporal and exact solutions for the link flow (i.e., convergence error of the link flow) is nearly proportional to the duality gap of the mathematical optimization problem equivalent to the UE traffic assignment problem. This means that the convergence error of a link flow can possibly be estimated from the duality gap.

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.

Bilevel programming model and solution method for mixed transportation network design problem

By handling the travel cost function artfully, the authors formulate the transportation mixed network design problem (MNDP) as a mixed-integer, nonlinear bilevel programming problem, in which the lower-level problem, comparing with that of conventional bilevel DNDP models, is not a side constrained user equilibrium assignment problem, but a standard user equilibrium assignment problem. Then, the bilevel programming model for MNDP is reformulated as a continuous version of bilevel programming problem by the continuation method. By virtue of the optimal-value function, the lower-level assignment problem can be expressed as a nonlinear equality constraint. Therefore, the bilevel programming model for MNDP can be transformed into an equivalent single-level optimization problem. By exploring the inherent nature of the MNDP, the optimal-value function for the lower-level equilibrium assignment problem is proved to be continuously differentiable and its functional value and gradient can be obtained efficiently. Thus, a continuously differentiable but still nonconvex optimization formulation of the MNDP is created, and then a locally convergent algorithm is proposed by applying penalty function method. The inner loop of solving the subproblem is mainly to implement an all-or-nothing assignment. Finally, a small-scale transportation network and a large-scale network are presented to verify the proposed model and algorithm.

Model and Solution Algorithm for Multiclass User Equilibrium Assignment Problem in Degradable Road Network

This paper proposes a multiclass user equilibrium assignment model for ascertaining the difference in travel choice behavior of different user classes in a degradable road network. In the proposed model, on the basis of travelers' risk attitudes toward network uncertainty, all travelers are categorized into three classes: risk-averse travelers, risk-prone travelers, and risk-neutral travelers. The multiclass user network equilibrium conditions are presented and formulated as a variational inequality (VI) model, and a heuristic solution algorithm for solving the VI model is given. An example is used to illustrate the application of the proposed model and solution algorithm. The study in this paper will help to understand heterogeneous travelers' route choice behavior and can serve as a useful tool for evaluating the effects of various transport policies on different user classes at the strategic level.

Transit passenger origin-destination estimation in congested transit networks with elastic line frequencies

This paper deals with the transit passenger origin-destination (O-D) estimation problem by using updated passenger counts in congested transit networks and outdated prior O-D matrix. A bilevel programming approach is extended for the transit passenger O-D updating problem where the upper-level problem seeks to minimize the sum of error measurements in passenger counts and O-D matrices, while the lower level is the stochastic user equilibrium assignment problem for congested transit networks. The transit assignment framework is based on a frequency-adaptive transit network model in this paper, which can help determine transit line frequencies and the network flow pattern simultaneously in congested transit networks. A heuristic solution algorithm is adapted for solving the transit passenger O-D estimation problem. Finally, a numerical example is used to illustrate the applications of the proposed model and solution algorithm.

Dynamic departure time and stochastic user equilibrium assignment

This paper investigates a logit-based combined departure time and dynamic stochastic user equilibrium assignment (DDSUE) problem and shows the existence of DDSUE condition as well as exploring its properties: a novel invariant quantity of DDSUE assignments is established. The relations between daily volume of traffic and cost incurred from travelling are also investigated. In order to calculate the equilibrium, we propose a solution algorithm that can solve the problem directly, and without appealing to an equivalent mathematical programme. Since the model is based on paths where all feasible paths are enumerated in advance, we propose an efficient method that works within a reasonable path set. Through two numerical examples, we show the DDSUE and some valuable results from the model such as relation between departure flow (or path inflow) and departure cost (or path cost). Sensitivity analysis of travel volumes with respect to travel costs and model parameters is also undertaken.

A dynamic schedule-based model for congested transit networks

In this paper we propose a model and algorithm for solving the equilibrium assignment problem in a congested, dynamic and schedule-based transit network. We assume that the time varying origin–destination trip demand is given. All travelers have full predictive information (that have been gained through past experience) about present and future network conditions and select paths that minimize a generalized cost function that encompasses four components: (a) in-vehicle time; (b) waiting time; (c) walking time; and (d) a line change penalty. All transit vehicles have a fixed capacity and operate precisely as specified in pre-set timetables. Passengers queue at platforms according to the single channel first-in-first-out discipline. By using time-increment simulation, the passenger demand is loaded onto the network and the available capacity of each vehicle is updated dynamically. After each simulation run, the passenger arrival and departure profiles at all stations are recorded and these are used to predict dynamic queuing delays. From such delays, minimum paths are updated and used for the next simulation run. The user equilibrium assignment problem is solved iteratively by the method of successive averages.

A Multi-Class Dynamic User Equilibrium Model for Queuing Networks with Advanced Traveler Information Systems

This paper presents a formulation and solution algorithm for a composite dynamic user-equilibrium assignment problem with multi-user classes, in order to assess the impacts of Advanced Traveler Information Systems (ATIS) in general networks with queues. Suppose that users equipped with ATIS will receive complete information and hence be able to choose the best departure times and routes in a deterministic manner, while users not equipped with ATIS will have incomplete information and hence may make decisions on departure times and routes in a stochastic manner. This paper proposes a discrete-time, finite-dimensional variational inequality formulation that involves two criteria regarding the route and departure time choice behaviors, i.e., the deterministic dynamic user equilibrium and the nested logit-based stochastic dynamic user equilibrium. The formulation is then converted to an equivalent “zero-extreme value” minimization problem. A heuristic algorithm based on route/time-swapping process is proposed, which iteratively adjusts the route and departure time choices to reach closely to an extreme point of the minimization problem. A numerical example is used to demonstrate the effectiveness of the proposed approach for assessing the ATIS impacts such as changes in individual travel costs, departure times, route inflows, queuing peaks and total network travel cost.

Dynamic traffic modelling and dynamic stochastic user equilibrium assignment for general road networks

This paper investigates the requirements of dynamic traffic modelling, and proposes the deterministic queuing model as a plausible link performance function to describe the relationship between inflows, outflows, and link travel costs in time-varying condition. Then, it explains how we can perform logit-based stochastic network loading for general road networks in the dynamic case. In particular, this paper shows how to perform dynamic stochastic network loadings for many-to-many origin–destination pairs, and what should be considered to maintain correct flow propagation in the network loading process. Next, this paper shows how the stochastic dynamic user equilibrium (SDUE) assignment problem can be solved without direct evaluation of the objective function. For this purpose, a quadratic interpolation, the method of successive average, and the pure network loading method are adopted at the line-search step in the solution algorithm. Numerical examples show that the present SDUE assignment model with a quadratic interpolation gives rise to a convergent solution with good quality whilst needing less computation time. Furthermore, it is found that the predictive cost-flow association (Proceedings of the European Transport Conferences, Seminar F, P434, 1999, p. 79) is preferable to the reactive one because the former can produce consistent assignment patterns regardless of the size of dispersion parameter θ in the logit model for route choice.