Equilibrium Assignment Problem Research Articles

Perturbed Utility Stochastic Traffic Assignment

This paper develops a fast algorithm for computing the equilibrium assignment with the perturbed utility route choice (PURC) model. Without compromise, this allows the significant advantages of the PURC model to be used in large-scale applications. We formulate the PURC equilibrium assignment problem as a convex minimization problem and find a closed-form stochastic network loading expression that allows us to formulate the Lagrangian dual of the assignment problem as an unconstrained optimization problem. To solve this dual problem, we formulate a quasi-Newton accelerated gradient descent algorithm (qN-AGD*). Our numerical evidence shows that qN-AGD* clearly outperforms a conventional primal algorithm and a plain accelerated gradient descent algorithm. qN-AGD* is fast with a runtime that scales about linearly with the problem size, indicating that solving the perturbed utility assignment problem is feasible also with very large networks. Funding: This work has been financed by the European Union—NextGenerationEU.

An efficient hyperpath-based algorithm for the capacitated transit equilibrium assignment problem

This paper studied the capacitated transit equilibrium assignment problem (CTEAP), which particularly accounts for the capacity effect that impacts passengers' route choices in a transit network. Specifically, we impose strict capacity constraints on line segments to ensure that the passenger flow does not exceed the line capacity. By introducing associated Lagrange multipliers, we derived the generalised hyperpath cost function and established an equivalent variational inequality formulation. A solution framework is developed to solve CTEAP based on the Method of Multipliers. To handle destroyed Cartesian product structures, we transformed the master problem into a sequence of uncapacitated subproblems, which can be tackled by two modified Newton-type hyperpath-based algorithms. Numerical analyses were performed for a small Gentile network and a large Shenzhen transit network to evaluate the impacts of the capacity constraint on passenger flows and computation costs. Our results demonstrate the superiority of the proposed CTEAP model to prevent over-loaded flows, and show the promise of applying the hyperpath-based algorithm in the implementation of real-world networks.

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).

Sensitivity analysis for transit equilibrium assignment and applications to uncertainty analysis

Systematic uncertainty analysis can be used to quantitatively evaluate variation in model outputs and identify the critical sources of uncertainty to improve the reliability and stability of a system. To analyze the effects of uncertainties in transit networks that may be caused by probabilistic travel demand, congestion, or vehicle frequencies, this study develops a sensitivity-based uncertainty analysis approach as a post-analysis tool for equilibrium transit systems. The congestion effect is considered in the waiting time and in-vehicle travel time of a passengers’ route-choice model. The hyperpath concept is used to manage passengers’ riding strategies due to the common-line problem at transit stops. A hyperpath-based gradient projection (GP) solution algorithm is developed for the solution of the variational inequality formulation of the transit equilibrium assignment problem (TEAP). A restricted sensitivity analysis approach originally developed for road networks is re-developed for the TEAP in transit networks. An analytical sensitivity-based approach is derived to conduct uncertainty analysis for the TEAP, which enables the simultaneous propagation of uncertainties from different input sources to the model outputs. Numerical examples are provided for the following purposes. (1) To demonstrate three applications of the sensitivity analysis of the TEAP, namely the perturbed solution estimation problem, the critical parameter identification problem, and the paradox analysis problem. (2) To illustrate the use of uncertainty analysis of the TEAP, such as estimating the variance and confidence level of model outputs with respect to various model inputs/parameters as random variables, and ranking the importance of arcs using sensitivity-based uncertainty analysis. (3) To demonstrate the applicability of the proposed approach to real transit networks. The findings demonstrate not only the importance of the analytical sensitivity analysis development for the TEAP, but also for the practical applications of sensitivity and uncertainty analyses.

Hyperpath-based algorithms for the transit equilibrium assignment problem

The transit equilibrium assignment problem (TEAP) aims to predict the distributions of passenger flows on lines or line segments in a transit network. Compared to the traffic assignment problem (TAP) for highway networks, the TEAP is much less studied, especially in terms of solution algorithms. This paper proposes two Newton-type hyperpath-based algorithms for a frequency-based TEAP formulation that considers both the congestion effect related to crowding and the queuing effect related to boarding. These newly developed algorithms, as well as two benchmark algorithms from the literature, are tested and compared on a number of transit networks, including two constructed using real-world data. The results show the proposed hyperpath-based algorithms significantly outperform the benchmark algorithms in large networks.

Traffic assignment problem under tradable credit scheme in a bi-modal stochastic transportation network: A cumulative prospect theory approach

The traffic equilibrium assignment problem under tradable credit scheme (TCS) in a bi-modal stochastic transportation network is investigated in this paper. To describe traveler’s risk-taking behaviors under uncertainty, the cumulative prospect theory (CPT) is adopted. Travelers are assumed to choose the paths with the minimum perceived generalized path costs, consisting of time prospect value (PV) and monetary cost. At equilibrium with a given TCS, the endogenous reference points and credit price remain constant, and are consistent with the equilibrium flow pattern and the corresponding travel time distributions of road sub-network. To describe such an equilibrium state, the CPT-based stochastic user equilibrium (SUE) conditions can be formulated under TCS. An equivalent variational inequality (VI) model embedding a parameterized fixed point (FP) model is then established, with its properties analyzed theoretically. A heuristic solution algorithm is developed to solve the model, which contains two-layer iterations. The outer iteration is a bisection-based contraction method to find the equilibrium credit price, and the inner iteration is essentially the method of successive averages (MSA) to determine the corresponding CPT-based SUE network flow pattern. Numerical experiments are provided to validate the model and algorithm.

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.

Instability of departure time choice problem: A case with replicator dynamics

Stability is an important condition for equilibrium solutions to be realised in a real transport system. If the solution is not stable, it is vulnerable to a small perturbation onto the system, and consequently equilibrium would not be observed in the real world. While it has been known that an equilibrium solution of a static equilibrium assignment problem is stable in a various types of evolution dynamics with mild conditions, the stability of solutions in dynamic user equilibrium (DUE) assignments has rarely been investigated. This study proves that an equilibrium solution of the departure time choice problem is not Lyapunov stable when the replicator dynamics is employed. As the uniqueness of the equilibrium solution has been proven, it can be concluded that there is no stable equilibrium solution in the departure time choice problem when the replicator dynamics is assumed as an evolution dynamics.

Fixed-point fast sweeping weighted essentially non-oscillatory method for multi-commodity continuum traffic equilibrium assignment problem

This work presents a fixed-point fast sweeping weighted essentially non-oscillatory method for the multi-commodity continuum traffic equilibrium assignment problem with elastic travel demand. The commuters’ origins (i.e. home locations) are continuously dispersed over the whole city with several highly compact central business districts. The traffic flows from origins to the same central business district are considered as one commodity. The continuum traffic equilibrium assignment model is formulated as a static conservation law equation coupled with an Eikonal equation for each commodity. To solve the model, a pseudo-time-marching approach and a third order finite volume weighted essentially non-oscillatory scheme with Lax–Friedrichs flux splitting are adopted to solve the conservation law equation, coupled with a third order fast sweeping numerical method for the Eikonal equation on rectangular grids. A fixed-point fast sweeping method that utilizes Gauss–Seidel iterations and alternating sweeping strategy is designed to improve the convergence for steady state computations of the problem. A numerical example is given to show the feasibility of the model and the effectiveness of the solution algorithm.

Urban Growth, Transport Planning, Air Quality and Health: A Multi-Objective Spatial Analysis Framework for a Linear Monocentric City

In this paper, we propose a multi-objective spatial analysis framework to evaluate the economic, environmental and health impacts of transport investment strategies under different urban growth scenarios. We consider a linear monocentric city (LMC) wherein residents are distributed continuously along an urban corridor and commute daily to a common destination, the central business district (CBD), represented by one end of the linear city. Two modes are available: car and rail. Users can travel from their residence to the CBD by car on a congestible highway with stochastic travel time, or by rail from the most convenient nearby station, which they reach by walking or cycling. Travellers throughout the city have a distribution of reliability preferences. Individual mode choice is determined using the notion of travel time budget surplus to take into consideration the travel time, travel time reliability and monetary cost associated with each mode. We assume users would like to minimise travel time, monetary cost, and maximise travel time reliability. The resulting formulation is a three-objective user equilibrium model (TBSmaxTUE). For the continuous monocentric city model, TBSmaxTUE can be formulated as a fixed point problem. To admit a numerical solution the continuum is discretised, allowing it to be expressed as a standard network equilibrium assignment problem. The performance of this LMC model can then be analysed against multiple objectives. We consider the economic objective to minimise total system travel time; the environmental objective to minimise total tailpipe emissions from car trips; and health objectives to minimise pollutant uptake while also maximising the level of physical activity during the journey to work.

Ecology‐based coexistence scheme for heterogeneous cognitive radio networks over TV white space

Heterogeneous cognitive radio networks (H-CRNs) operating in the same TV white space (TVWS) spectrum cause severe mutual interferences, which heavily degrades the network performance of coexisting H-CRNs. Inspired by the Lotka–Volterra competition model of different species in a biological ecosystem, this study formulates the coexistence of H-CRNs over TVWS as an equilibrium assignment problem of a discrete non-linear control system (DNCS). By using the local linearisation method, the authors can find an approximate linear control system (ALCS) model to the DNCS. Further, they obtain an effective feedback control to the equilibrium assignment of the ALCS via solving a sufficient condition in the form of linear matrix inequalities. Third, they propose a novel ecology-inspired heterogeneous coexistence algorithm (EHCA) based on the obtained feedback control. It is shown in simulations that the proposed EHCA could guarantee any desired but feasible spectrum share among H-CRNs.

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.

A modified Physarum-inspired model for the user equilibrium traffic assignment problem

• A modified Physarum -inspired model for the user equilibrium problem is proposed. • Two-way traffic characteristics and multiple OD pairs are considered. • The proposed Physarum -inspired model is very efficient in terms of both the accuracy and the execution time. The user equilibrium traffic assignment principle is very important in the traffic assignment problem. Mathematical programming models are designed to solve the user equilibrium problem in traditional algorithms. Recently, the Physarum shows the ability to address the user equilibrium and system optimization traffic assignment problems. However, the Physarum model are not efficient in real traffic networks with two-way traffic characteristics and multiple origin–destination pairs. In this article, a modified Physarum -inspired model for the user equilibrium problem is proposed. By decomposing traffic flux based on origin nodes, the traffic flux from different origin–destination pairs can be distinguished in the proposed model. The Physarum can obtain the equilibrium traffic flux when no shorter path can be discovered between each origin–destination pair. Finally, numerical examples demonstrate the rationality and convergence properties of the proposed model.

A Bio-Inspired Approach to Traffic Network Equilibrium Assignment Problem.

Finding an equilibrium state of the traffic assignment plays a significant role in the design of transportation networks. We adapt the path finding mathematical model of slime mold Physarum polycephalum to solve the traffic equilibrium assignment problem. We make three contributions in this paper. First, we propose a generalized Physarum model to solve the shortest path problem in directed and asymmetric graphs. Second, we extend it further to resolve the network design problem with multiple source nodes and sink nodes. At last, we demonstrate that the Physarum solver converges to the user-optimized (Wardrop) equilibrium by dynamically updating the costs of links in the network. In addition, convergence of the developed algorithm is proved. Numerical examples are used to demonstrate the efficiency of the proposed algorithm. The superiority of the proposed algorithm is demonstrated in comparison with several other algorithms, including the Frank-Wolfe algorithm, conjugate Frank-Wolfe algorithm, biconjugate Frank-Wolfe algorithm, and gradient projection algorithm.

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.

Network flow assignment as a fixed point problem

This paper deals with the user equilibrium problem (flow assignment with equal journey time by alternative routes) and system optimum (flow assignment with minimal average journey time) in a network consisting of parallel routes with a single origin-destination pair. The travel time is simulated by arbitrary smooth nondecreasing functions. We prove that the equilibrium and optimal assignment problems for such a network can be reduced to the fixed point problem expressed explicitly. A simple iterative method of finding equilibriumand optimal flow assignment is developed. The method is proved to converge geometrically; under some fairly natural conditions the method is proved to converge quadratically.

Acceleration of Double-Projection Method in Asymmetrically Formulated Traffic Assignment

AbstractIn this paper, a methodology for accelerating the double-projection method designed for the asymmetric traffic equilibrium assignment problem is proposed. One of the factors that determine the efficiency of the projection method is a properly selected step size. This study customized the step size for each origin-destination (O-D) pair by formulating the assignment problem on the space of path flows. In addition, even if the transportation network is decomposed into each O-D pair, the step size based on the entire path flows is also available. The study developed a step-size selection strategy with two different available step sizes obtained from different spaces of path flows vectors. During the projections of path flows for each O-D pair, the strategy compares the step size customized for that O-D pair and the one from the entire path flow vectors and selected a bigger step size. Several numerical tests have been conducted on various sized networks and convergence criteria. The methodology consi...

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).