Mike Maher

A probit-based stochastic user equilibrium assignment model

Stochastic methods of traffic assignment have received much less attention in the literature than those based on deterministic user equilibrium (UE). The two best known methods for stochastic assignment are those of Burrell and Dial, both of which have certain weaknesses which have limited their usefulness. Burrells is a Monte Carlo method, whilst Dials logit method takes no account of the correlation, or overlap,between alternative routes. This paper describes, firstly, a probit stochastic method (SAM) which does not suffer from these weaknesses and which does not require path enumeration. While SAM has a different route-finding methodology to Burrell, it is shown that assigned flows are similar. The paper then goes on to show how, by incorporating capacity restraint (in the form of link-based cost-flow functions) into this stochastic loading method, a new stochastic user equilibrium (SUE) model can be developed. The SUE problem can be expressed as a mathematical programming problem, and its solution found by an iterative search procedure similar to that of the Frank-Wolfe algorithm commonly used to solve the UE problem. The method is made practicable because quantities calculated during the stochastic loading process make the SUE objective function easy to compute. As a consequence, at each iteration, the optimal step length along the search direction can be estimated using a simple interpolation method. The algorithm is demonstrated by applying it successfully to a number of test problems, in which the algorithm shows good behaviour. It is shown that, as the values of parameters describing the variability and degree of capacity restraint are varied, the SUE solution moves smoothly between the UE and pure stochastic solutions.

THE INFLUENCE OF TREND ON ESTIMATES OF ACCIDENTS AT JUNCTIONS

While reliable estimates of expected accidents can be achieved by combining observed accidents and accident model predictions using an empirical Bayes approach, there are a number of obstacles to the widespread adoption of the method. This paper concentrates on problems associated with the available predictive models. Of particular concern is the effect on model predictions of accident trends over time resulting from, for instance, traffic growth or national road safety programmes. Since accident models invariably include traffic flow as an explanatory variable, the effects of flow changes can be included provided that account is taken of the nonlinear relationship between accidents and exposure. It is, however, common to assume that accident risk per unit of exposure is constant over time, whereas national data imply that accident risk is declining. In addition, there is a need, in practice, to rank and evaluate remedial sites in terms of the specific accident types or severities which might be targeted by treatment (for example, wet road accidents in the case of surface treatment). This then raises the question of whether the proportions of accidents of various types varies over time or with traffic flow and site characteristics. Generalized linear modelling was used to develop regression estimates of expected junction accidents (both in total and disaggregated by severity, road surface condition and lighting condition) which allow for the possibility of accident risk varying over time. Accident risk at the sample of some 500 junctions was shown to be declining annually by an average of 6%, with no significant difference in the value of trend between accident types. The factors which affected the proportions of accidents of various types included the method of junction control, speed limit and traffic flow.

The evaluation and application of a fully disaggregate method for trip matrix estimation with platoon dispersion

This paper addresses the problem of estimating an Origin-Destination (O-D) matrix with platoon dispersion from fully disaggregate data: that is, the passage times of vehicles at the entries and exits or the origins and destinations of a network. Given a list of entry times and a list of exiting times, a fully disaggregate method or a matching method tries to match each pair of entry-exiting times such that the resultant journey times for each O-D pair fit the given distributions best. Bell et al., 1991 formulated the problem of finding the most likely set of matches as a 0-1 assignment problem. In this paper, two estimators will be considered: maximum likelihood estimator and the matching-Furness technique. It will be shown that the maximum likelihood estimation can be formulated more generally as a transportation problem. The matching-Furness method is investigated by use of simulated traffic data and is applied to a set of real data collected on the Brescia-Padova motorway network in Italy. The matching-Furness method gives consistently better estimates than the least squares estimator (Cremer and Keller, 1987) and the linked static-dynamic correlation model (Keller and Ploss, 1987). The method is therefore suitable for off-line estimations for general networks.

Minimal Revenue Network Tolling: System Optimisation Under Stochastic Assignment

The classical road tolling problem is to toll network links such that, under the principles of Wardropian User Equilibrium (UE) assignment, a System Optimising (SO) flow pattern is obtained. Such toll sets are however non-unique, and further optimisation is possible: for example, minimal revenue tolls create the desired SO flow pattern at minimal additional cost to the users. In the case of deterministic assignment, the minimal revenue toll problem is capable of solution by various methods, such as linear programming [BHR97] and heuristically by reduction to a multi-commodity max-flow problem [Dia00]. However, it is generally accepted that deterministic models are less realistic than stochastic, and thus it is of interest to investigate the principles of tolling under stochastic modelling conditions. This paper develops methodologies to examine the minimal revenue toll problem in the case of Stochastic User Equilibrium. Tolling solutions for both ‘true’ System Optimum and Stochastic System Optimum under SUE are derived, using both logit and probit assignment methods.

RISK ANALYSIS IN CONSTRUCTION NETWORKS USING A MODIFIED STOCHASTIC ASSIGNMENT MODEL

Abstract A review of construction network analysis indicates that new methods are needed for quantifying risks in project evaluation. The paper proposes a new analytical method, the Modified Stochastic Assignment Model (MSAM), for the prediction of project duration. The proposed method is inspired by a previous method used solely in traffic networks, the Stochastic Assignment Model (SAM). The MSAM method applies Clarks approximation to find the longest project duration. Two cases are used to demonstrate the validity and application of the MSAM in construction project evaluations. The accuracy of the MSAM is assessed by comparing it with the Monte Carlo Simulation (MCS). A comparison of the MSAM with other methods, such as PERT and PNET, has also been presented. It is found that the new method is an analytical counterpart of the MCS and is very efficient in saving computational time whilst taking full account of the correlations between paths.

Algorithms for the Solution of the Combined Traffic Signal Optimisation and Equilibrium Assignment Problem

The combined traffic signal optimisation and equilibrium assignment problem is one in which a traffic engineer tries to optimise the performance of traffic signals while road users choose their routes so as to minimises their travel costs. Two types of solutions can be defined in the combined problem: the mutually consistent solution and the global optimal solution. The former is a solution at which the two sub-problems are solved simultaneously while the latter is a solution to the bi-level programming formulation of the combined problem. In this paper, we consider the combined signal optimisation and stochastic user equilibrium assignment problem. We present two types of algorithms for the mutually consistent solution and one type of algorithm for the bi-level solution to the problem. The algorithms are tested on a small network to examine their convergence and efficiency.