Method Of Successive Averages Research Articles

Static Parking Choice Model with Consideration of Parking Duration

This study extended existing parking equilibrium studies by analytically considering the impact of parking duration on both the demand and the supply sides. After comparisons with route choice equilibrium studies, the parking search function is modified and a user equilibrium model is proposed for general parking systems. Then a solution algorithm based on the method of successive averages is presented and it is noted that the equilibrium model can be equivalently converted into an optimization problem. Two numerical studies assuming simplified (fixed value) and realistic (random variable) parking durations are presented to demonstrate the effectiveness of the methods. Sensitivity analyses and numerical comparisons are also conducted in which the thresholds of parking demand and parking duration are identified at the saturation point of parking facilities. In addition, numerical results show that the heterogeneity (in terms of value of time and parking duration) of travelers significantly influences their parking location choices; the result is higher turnover rates at parking lots with higher parking charges.

Nonadditive Public Transit Fare Pricing Under Congestion with Policy Lessons from a Case Study in Toronto, Ontario, Canada

With increasing urbanization and the development of technologies that support automated fare collection, policy makers need decision-support tools to evaluate differentiated public transit fare pricing policies. However, the state-of-the-art tools that consider congestion effects account only for additive fares. A stochastic user equilibrium model with elastic demand was extended to handle nonadditive station-to-station–based fares and was solved by using a method of successive averages. In this paper, an illustrative example is used to show how simple price elasticities alone are not enough to predict the effects of a fare on demand within even a simple eight-node congested network. The first case study of a fare pricing policy was conducted in Toronto, Ontario, Canada; in this case, a distance-based policy was used for the Toronto Transit Commission subway system with respect to downtown and nondowntown subpopulations. The analysis found that compared with the base scenario of a Can$3 fixed fare, there are Pareto-improving fare policies (e.g., fixed rate of Can$2 and variable rate of Can$0.06/km), but the same policy might not be Pareto-improving for all subpopulations. These findings call for more sophisticated fare pricing policies for Toronto (e.g., zone-based) that can cater to specific needs of subpopulations.

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.

Integrated Mode Choice and Dynamic Traveler Assignment in Multimodal Transit Networks: Mathematical Formulation, Solution Procedure, and Large-Scale Application

This paper introduces an integrated mode choice–multimodal transit assignment model and solution procedure intended for large-scale urban applications. The cross-nested logit mode choice model assigns travelers to car, transit, or park-and-ride. The dynamic multimodal transit assignment–simulation model determines minimum hyperpaths and assigns and simulates transit and park-and-ride travelers iteratively until the network approaches a state of equilibrium. After a given number of iterations, the updated transit network travel times are fed into the mode choice model and the model reassigns travelers to transit, car, or park-and-ride. The outer feedback loop between the mode choice model and the transit assignment model continues until the mode probabilities for each traveler do not change between iterations. A unique contribution of the method presented in this paper is that it reaches mode choice convergence with the use of disaggregate agents (travelers) instead of aggregate modal flows at the origin–destination level. The integrated model is successfully implemented on the Chicago Transit Agency’s bus and train network in Illinois. Different procedures for reaching convergence are tested; the results suggest that a gap-based formulation is more efficient than the method of successive averages.

Comparison study of the reactive and predictive dynamic models for pedestrian flow

This paper formulates the reactive and predictive dynamic models for pedestrian flow and presents a comparison of the two models. The path-choice behavior of pedestrians in the reactive dynamic model is described that pedestrians tend to walk along a path with the lowest instantaneous cost. The desired walking direction of pedestrians in the predictive dynamic model is chosen to minimize the actual cost based on predictive traffic conditions. An algorithm used to solve the two models encompasses a cell-centered finite volume method for a hyperbolic system of conservation laws and a time-dependent Hamilton–Jacobi equation, a fast sweeping method for an Eikonal-type equation, and a self-adaptive method of successive averages for an arisen discrete fixed point problem. The two models and their algorithm are applied to investigate the spatio-temporal patterns of flux or density and path-choice behaviors of pedestrian flow marching in a facility scattered with an obstacle. Numerical results show that the two models are able to capture macroscopic features of pedestrian flow, traffic instability and other complex nonlinear phenomena in pedestrian traffic, such as the formation of stop-and-go waves and clogging at bottlenecks. Different path-choice strategies of pedestrians cause different spatial distributions of pedestrian density specially in the high-density regions (near the obstacle and exits).

Research on Urban Road Congestion Pricing Strategy Considering Carbon Dioxide Emissions

Congestion pricing strategy has been recognized as an effective countermeasure in the practical field of urban traffic congestion mitigation. In this paper, a bi-level programming model considering carbon dioxide emission is proposed to mitigate traffic congestion and reduce carbon dioxide emissions. The objective function of the upper level model is to minimize the sum of travel costs and the carbon dioxide emissions costs. The lower level is a multi-modal transportation network equilibrium model. To solve the model, the method of successive averages (MSA) and the shuffled frog leaping algorithm (SFLA) are introduced. The proposed method and algorithm are tested through the numerical example. The results show that the proposed congestion pricing strategy can mitigate traffic congestion and reduce carbon emissions effectively.

On the Convergence of the Method of Successive Averages for Calculating Equilibrium in Traffic Networks

The traffic assignment problem aims to calculate an equilibrium route flow vector, generally by seeking a zero of an appropriate objective function. If a continuous dynamical system follows a descent direction for this objective function at each nonequilibrium route flow vector, the system converges to equilibrium. It is shown that when this dynamical system is discretized with a fixed step length, the system eventually approaches close to equilibrium provided that the objective function is continuously differentiable and that the rate of descent is bounded below. The method of successive averages is widely used in traffic assignment; it has a decreasing step size at each iteration. With the same conditions as above, it is shown that the resulting dynamical system converges to equilibrium. In the steady-state model, the necessary conditions are shown to be satisfied, provided that the route cost vector is a continuously differentiable monotone function of the route flow vector. However, continuous differentiability of the cost function is shown not to hold in the dynamic queueing model.

Activity-based market equilibrium for capacitated multimodal transport systems

Empirical studies have shown that demand for multimodal transport systems is highly correlated with activity schedules of individuals. Nonetheless, existing analytical equilibrium models of multimodal systems have only considered trip-based demand. We propose a new market equilibrium model that is sensitive to traveler activity schedules and system capacities. The model is based on a constrained mixed logit model of activity schedule choice, where each schedule in the choice set is generated with a multimodal extension of the household activity pattern problem. The extension explicitly accounts for both passenger choices of activity participation and multimodal choices like public transit, walking, and vehicle parking. The market equilibrium is achieved with Lagrangian relaxation to determine the optimal dual price of the capacity constraint, and a method of successive averages with column generation finds an efficient choice set of activity schedules to assign flows over the dynamic network load capacities. An example illustrates the model and algorithm, effects similar to Vickrey’s morning commute model can be observed as a special case. A case study of the Oakville Go Transit station access “last mile” problem in the Greater Toronto Area is conducted with 166 survey samples reflecting 3680 individuals. Results suggest that a $10 fixed parking fee at Oakville station would lead to a reduction of access auto share from 54.8% to 49.5%, an increase in access transit share from 20.7% to 25.9%, and a disutility increase of 11% for the of single-activity residents of Oakville.

Effect of speed limits in degradable transport networks

This paper studies how link-specific speed limits influence the performance of degradable transport networks, in which the capacity of each link is a degradable random variable. The distribution and cumulative distribution of link travel time have been presented with the effect of speed limits taken into account. The mean and variance of link and route travel time are formulated. Three link states have been classified, and their physical meanings have been discussed. The relationship between critical capacity, travel time and speed limit has been elaborated. We have proposed a Speed Limit- and Reliability-based User Equilibrium (SLRUE), adopting travel time budget as the principle of travelers’ route choice. A heuristic method employing the method of successive averages is developed to solve the SLRUE in degradable networks. Through numerical studies, we find that for some networks both the mean and standard deviation of the total travel time could be reduced simultaneously by imposing some speed limits. The speed limit design problem has been studied, and it is found that imposing speed limits cannot always reduce the total travel time budget of a network.

A bi-level programming for bus lane network design

This paper proposes a bi-level programming model to solve the design problem for bus lane distribution in multi-modal transport networks. The upper level model aims at minimizing the average travel time of travelers, as well as minimizing the difference of passengers’ comfort among all the bus lines by optimizing bus frequencies. The lower level model is a multi-modal transport network equilibrium model for the joint modal split/traffic assignment problem. The column generation algorithm, the branch-and-bound algorithm and the method of successive averages are comprehensively applied in this paper for the solution of the bi-level model. A simple numerical test and an empirical test based on Dalian economic zone are employed to validate the proposed model. The results show that the bi-level model performs well with regard to the objective of reducing travel time costs for all travelers and balancing transit service level among all bus lines.

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).

System convergence in transport models: algorithms efficiency and output uncertainty

Transport models most often involve separate models for traffic assignment and demand. As a result, two different equilibrium mechanisms are involved, (i) the internal traffic assignment equilibrium, and (ii) the external equilibrium between the assignment model and the demand model. The objective of this paper is to analyse convergence performance for the external loop and to illustrate how an improper linkage between the converging parts can lead to substantial uncertainty in the final output. Although this loop is crucial for the performance of large-scale transport models it has not been analysed much in the literature. The paper first investigates several variants of the Method of Successive Averages (MSA) by simulation experiments on a toy-network. It is found that the simulation experiments produce support for a weighted MSA approach. The weighted MSA approach is then analysed on large-scale in the Danish National Transport Model (DNTM). It is revealed that system convergence requires that either demand or supply is without random noise but not both. In that case, if MSA is applied to the model output with random noise, it will converge effectively as the random effects are gradually dampened in the MSA process. In connection to DNTM it is shown that MSA works well when applied to travel-time averaging, whereas trip averaging is generally infected by random noise resulting from the assignment model. The latter implies that the minimum uncertainty in the final model output is dictated by the random noise in the assignment model.

Activity-based Market Equilibrium for Capacitated Multimodal Transport Systems

Empirical studies have shown that demand for multimodal transport systems is highly correlated with activity schedules of individuals. Nonetheless, existing analytical equilibrium models of multimodal systems have only considered trip-based demand. We propose a new market equilibrium model that is sensitive to traveler activity schedules and system capacities. The model is based on a constrained mixed logit model of activity schedule choice, where each schedule in the choice set is generated with a multimodal extension of the Household Activity Pattern Problem. The extension explicitly accounts for both passenger choices of activity participation and multimodal choices like public transit, walking, and vehicle parking. The market equilibrium is achieved with Lagrangian relaxation to determine the optimal dual price of the capacity constraint, and a method of successive averages with column generation finds an efficient choice set of activity schedules to assign flows over the dynamic network load capacities. An example illustrates the model and algorithm, effects similar to Vickrey's morning commute model can be observed as a special case. A case study of the Oakville Go Transit station access “last mile” problem in the Greater Toronto Area is conducted with 166 survey samples reflecting 3,680 individuals. Results suggest that a $10 fixed parking fee at Oakville station would lead to a reduction of access auto share from 54.8% to 49.5%, an increase in access transit share from 20.7% to 25.9%, and a disutility increase of 11% for the of single-activity residents of Oakville.

Multiclass Stochastic User Equilibrium Model with Elastic Demand

This paper proposes a multiclass, multidelay stochastic user equilibrium model with elastic demand (MC-MDSUE-ED) that explicitly considers both systematic delay and accidental delay in the route choice decision process. This new model hypothesizes that for each user class and each origin–destination (O-D) pair, no traveler can reduce his or her “multi-delay travel time,” defined as the systematic delay plus accidental delay, by unilaterally changing paths. The actual travel demand for each user class between each O-D pair satisfies its elastic demand function. Travel time budget and robust optimization theory are used to compute the systematic delay and accidental delay, respectively. The MC-MDSUE-ED conditions compose a cobweb model. A self-adaptive bisection algorithm that converts the problem with elastic demand to a series of problems with fixed demand was developed for solving the cobweb model. A path-based quasi method of successive averages was used to solve the MC-MDSUE model with fixed demand. Numerical examples illustrate the essential ideas of the proposed model and the applicability of the proposed solution algorithm.

Approaches for solving the stochastic equilibrium assignment with variable demand: internal vs. external solution algorithms

This paper proposes and compares different approaches within the general fixed-point framework that allows to deal with multi-user (stochastic) equilibrium assignment with variable demand (VD). The aim was threefold: (i) compare the efficiency and the effectiveness of the internal and the external approaches to stochastic equilibrium assignment with VD; (ii) investigate the efficiency and the effectiveness of different algorithms based on the method of successive averages and its extensions; (iii) investigate the effects of different averaging schemes, different convergence criteria and different path choice models, such as Multinomial Logit model, C-Logit model and Multinomial Probit model. Analyses were carried out with respect to a real network and considering different indicators of both efficiency and effectiveness.

Modeling the Effects of Public Bicycle Schemes in a Congested Multi-Modal Road Network

With increasing concerns about environmental and energy issues in many large Chinese cities, local authorities are introducing public bicycle schemes to promote the use of green transportation modes. This paper proposes a novel model for investigating the effects of the public bicycle schemes in a congested multi-modal road network with auto, bus, and public bicycle travel. The decision-making process of travelers regarding travel modes and route choices is assumed to follow a hierarchical choice structure. The effects of pollution emissions by motorized vehicles (i.e., autos and buses), crowding discomfort in buses, and riding fatigue on bicycles are considered in the proposed model. The multi-modal travel choice equilibrium problem is formulated as an equivalent variational inequality problem. The existence and uniqueness of the solution of the proposed model are examined. A heuristic solution algorithm that combines a diagonalization approach and the method of successive averages is adapted to solve the proposed model. A numerical example is given to illustrate the application of the proposed model and solution algorithm. Findings are reported on the effects of the public bicycle schemes and emission tax policy on the multi-modal transportation system. The optimal public bicycle rental price and emission tax for maximization of social welfare can also be determined by the proposed model.

Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams

Real-time coordinated traffic management strategies that benefit from parsimonious models with aggregated network dynamics, provide a new generation of smart hierarchical strategies to improve network capacity and performance. However, this raises the question of route choice behavior in case of heterogeneous urban networks, where different parts of the city are subject to different types of control. Traffic equilibrium phenomena have not been thoroughly investigated in these models. Approximate traffic equilibrium conditions can be integrated within the parsimonious traffic models to develop regional routing strategies, while detailed route choice strategies can be incorporated at a later stage in a hierarchical framework. In this study, we develop an aggregated and approximate dynamic traffic assignment (DTA) procedure to be incorporated in the macroscopic fundamental diagram (MFD) dynamics, and establish dynamic stochastic user equilibrium (DSUE) conditions. The methodology consists of two main components; stochastic network loading and a fixed-point solution method. Loading procedure is designed to handle stochastic components in the model such as trip length uncertainty, variation of speeds across the links, perception error of travelers. The results taken from this procedure are averaged through the well-known method of successive averages (MSA) to reach fixed-point solution for the system. Real-time route guidance strategies can be revisited towards a “system of systems” approach.

Stochastic equilibrium assignment with variable demand: Theoretical and implementation issues

Recently, it has been pointed out that transport models should reflect all significant traveler choice behavior. In particular, trip generation, trip distribution, modal split as well as route choice should be modeled in a consistent process based on the equilibrium between transport supply and travel demand. In this paper a general fixed-point approach that allows dealing with multi-user stochastic equilibrium assignment with variable demand is presented. The main focus was on investigating the effectiveness of internal and external approaches and of different algorithmic specifications based on the method of successive averages within the internal approach. The vector demand function was assumed non-separable, non-symmetric cost functions were adopted and implementation issues, such updating step and convergence criterion, were investigated. In particular the aim was threefold: (i) compare the internal and the external approaches; (ii) investigate the effectiveness of different algorithmic specifications to solve the variable demand equilibrium assignment problem through the internal approach; (iii) investigate the incidence of the number of the links with non-separable and/or asymmetrical cost functions. The proposed analyses were carried out with respect to two real-scale urban networks regarding medium-size urban contexts in Italy.

A traffic assignment model for passenger transit on a capacitated network: Bi-layer framework, line sub-models and large-scale application

In the urban setting, the roadway and railway modes of mass transit are basically purported to carry large flows of passengers. Thus the issue of flowing capacity is crucial in the design and planning of a transit network. As a transit system involves two types of traffic units, respectively passengers and vehicles, there is a broad range of capacity phenomena: (i) as a vehicle has given seat capacity, additional riders have to stand which is less comfortable and more exposed to in-vehicle crowding, (ii) the total capacity in a vehicle, including sitting and standing places, influences the wait time on platform if it is exceeded by the number of candidate riders, (iii) the exchange capacity at vehicle doors influences the vehicle dwell time at a station, (iv) from the station dwell times stems the run time of vehicles – hence of passengers – and in turn the service frequency, (v) vehicle traffic is constrained by dwell time and operating margins, which may reduce the frequency delivered, etc. The paper provides a static, macroscopic model of traffic assignment to a transit network, in which these capacity phenomena are captured. A key feature is the line sub-model that deals with a line of operations, comprised of one or several service routes, by using the topological order of stations. From a matrix of flows by pair of access-egress stations, the sub-model derives the matrix of average passenger costs by access-egress pair, as well as local passenger wait time and the apparent frequency of each leg. At the network level, passenger route choice is modeled by optimal hyperpaths that are route-based (as in De Cea and Fernandez, 1989). It is shown that there exists a state of traffic equilibrium. A Method of Successive Averages is put forward to compute equilibrium. A large scale application to the whole transit network of greater Paris is presented, with focus on capacity issues.

A user equilibrium, traffic assignment model of network route and parking lot choice, with search circuits and cruising flows

The paper provides a novel network model of parking and route choice. Parking supply is represented by parking type, management strategy including the fare, capacity and occupancy rate of parking lot, and network location, in relation to access routes along the roadway network. Trip demand is segmented according to origin–destination pair, the disposal of private parking facilities and the individual preferences for parking quality of service. Each traveller is assumed to make a two stage choice of, first, network route on the basis of the expected cost of route and parking and, second, local diversion on the basis of a discrete choice model. Search circuits are explicitly considered on the basis of the success probability to get a slot at a given lot and of the transition probabilities between lots in case of failure.The basic endogenous model variables are the route flows, the lot success probabilities and the transition probabilities between lots. These give rise to the cost of a travel route up to a target lot and to the expected cost of search and park from that lot to the destination. Traffic equilibrium is defined in a static setting. It is characterized by a mixed problem of variational inequality and fixed point. Equilibrium is shown to exist under mild conditions and a Method of Successive Averages is put forward to solve for it. Lastly, a planning instance is given to illustrate the effects of insufficient parking capacity on travel costs and network flows.