Michael G. H. Bell

Models and algorithms for road network design: a review and some new developments

The Network Design Problem (NDP) has long been recognized to be one of the most difficult and challenging problems in transport. In the past two decades, we have witnessed the development of a vast, growing body of research focused on formulations and solution procedures for the NDPs, which deal with the selection of either link improvements or link additions to an existing road network, with given demand from each origin to each destination. The objective is to make an optimal investment decision in order to minimize the total travel cost in the network, while accounting for the route choice behaviour of network users. In this paper, we present a general survey of existing literature in this area, and present some new developments in model formulations. We incorporate the elasticity of travel demand into the NDP and seek the economic‐based objective function for optimization. We also pose the mixed network design problem involving simultaneous choice of link addition and capacity improvement which is con...

Traffic restraint, road pricing and network equilibrium

Road pricing is now being advocated as an efficient means of managing traffic demand and of meeting other objectives, such as reducing the environmental impact of road traffic and improving public transport. This paper shows how a network toll pattern could be determined so as to reduce network travel demand to a desirable level. The demand between each origin-destination pair is described as a function of the generalized travel cost. When there is no toll charge, higher values of potential demand might cause congestion and queuing at bottleneck links of the road network. Queuing delay at saturated links may grow to choke off enough potential demand to reduce realized demand to the capacity of the network, thus leading to a queuing equilibrium where travel demand and travel cost match each other. In this paper, we first show how an elastic-demand network equilibrium model with queue could be used to determine this demand-supply equilibrium. We then seek a link toll pattern to remove the wasteful queuing delay, and/or restrain the realized demand to a desirable level to satisfy environment capacity constraints. We also show that the link toll pattern that could hold the traffic demand to a desirable level is not unique, a bi-level programming method is developed to select the best toll pattern among the feasible solutions based on pre-specified criteria.

AN EQUIVALENT CONTINUOUSLY DIFFERENTIABLE MODEL AND A LOCALLY CONVERGENT ALGORITHM FOR THE CONTINUOUS NETWORK DESIGN PROBLEM

The continuous network design problem (CNDP) is characterized by a bilevel programming model and recognized to be one of the most difficult and challenging problems in transportation. The main difficulty stems from the fact that the bilevel formulation for the CNDP is nonconvex and nondifferentiable, and indeed only some heuristic methods have been so far proposed. In this paper, the bilevel programming model for CNDPs is transferred into a single level optimization problem by virtue of a marginal function tool. By exploring the inherent nature of the CNDP, the marginal function for the lower-level user equilibrium problem is proved to be continuously differentiable and its functional value and derivative in link capacity enhancement can be obtained efficiently by implementing a user equilibrium assignment subroutine. Thus a continuously differentiable but still nonconvex optimization formulation of the CNDP is created and a locally convergent augmented Lagrangian method is applied to solve this equivalent problem. The descent direction in each step of the inner loop of the solution method can be found by doing an all or nothing assignment. These favorable characteristics indicate the potential of the algorithm to solve large CNDPs. Numerical examples are presented to compare the proposed method with some existing algorithms.

The estimation of origin-destination matrices by constrained generalised least squares

The Generalised Least Squares (GLS) approach to the estimation of Origin-Destination (OD) matrices permits the combination of survey and traffic count data in a way that allows for the relative accuracy of the two data sources. However, the procedure may result in estimates that infringe non-negativity or other constraints on the fitted values. A simple algorithm to solve the GLS problem subject to inequality constraints is presented and its convergence is proven. Expressions are derived for the variances and covariances of the fitted values both without active constraints. It is demonstrated that the imposition of inequality constraints can improve the accuracy of those fitted values directly and indirectly affected them, by reducing their sensitivity to error in the inputs.

TRAFFIC SIGNAL TIMING OPTIMISATION BASED ON GENETIC ALGORITHM APPROACH, INCLUDING DRIVERS' ROUTING

The genetic algorithm approach to solve traffic signal control and traffic assignment problem is used to tackle the optimisation of signal timings with stochastic user equilibrium link flows. Signal timing is defined by the common network cycle time, the green time for each signal stage, and the offsets between the junctions. The system performance index is defined as the sum of a weighted linear combination of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model of TRANSYT [User guide to TRANSYT, version 8, TRRL Report LR888, Transport and Road Research Laboratory, Crowthorne, 1980]. Stochastic user equilibrium assignment is formulated as an equivalent minimisation problem and solved by way of the Path Flow Estimator (PFE). The objective function adopted is the network performance index (PI) and its use for the Genetic Algorithm (GA) is the inversion of the network PI, called the fitness function. By integrating the genetic algorithms, traffic assignment and traffic control, the GATRANSPFE (Genetic Algorithm, TRANSYT and the PFE), solves the equilibrium network design problem. The performance of the GATRANSPFE is illustrated and compared with mutually consistent (MC) solution using numerical example. The computation results show that the GA approach is efficient and much simpler than previous heuristic algorithm. Furthermore, results from the test road network have shown that the values of the performance index were significantly improved relative to the MC.

RISK-AVERSE USER EQUILIBRIUM TRAFFIC ASSIGNMENT: AN APPLICATION OF GAME THEORY

Abstract Transport network users frequently have to make route choices in the presence of uncertainty about route costs. Uncertainty about costs differs from variation in cost perception, since uncertainty requires network users to have a strategy toward risk. The conventional approach is to add a safety margin based on the standard deviation of link cost. However, this requires the specification of the cost distribution. An alternative approach is presented here whereby the network user “plays through” all the possible eventualities before selecting his best route. A deterministic user equilibrium traffic assignment is shown to be equivalent to the mixed-strategy Nash equilibrium of an n-player, non-cooperative game. Then an n+m-player, non-cooperative game is formulated, where n network users seek their best routes and m origin–destination (OD)-specific demons penalise the network users maximally by failing links. The mixed-strategy Nash equilibrium of this game is shown to describe a risk-averse user equilibrium traffic assignment. A simple solution procedure is presented, along with an illustrative example.

THE REAL TIME ESTIMATION OF ORIGIN-DESTINATION FLOWS IN THE PRESENCE OF PLATOON DISPERSION

A number of procedures for the estimation in real time of origin-destination flows in junctions or small networks are reviewed. These are based on the assumption that the time taken by vehicles to traverse the junction or network is either small in relation to the chosen time interval or equal to some fixed number of time intervals. In reality, there will be a distribution of travel times that may span a number of time intervals. This paper proposes two ways to allow for this in the estimation of origin-destination flows. The first, appropriate if travel times are approximately geometrically distributed, makes use of the recurrence model of platoon dispersion. The second makes no assumption about the form of the distribution of travel times, but in general requires the estimation of substantially more parameters. Simulation results for data containing geometrically distributed travel times suggests that fitting the correctly specified model produces better estimates of the turning proportions than does the conventional approach where travel times are restricted to a fixed number of time intervals.

Alternatives to Dial's logit assignment algorithm

Two logit assignment methods for transportation networks are proposed as alternatives to Dials algorithm. While retaining the absence of a need for the enumeration of paths, they dispense with both forward and backward passes. They, therefore, do not require minimum node-to-node cost information beforehand. Both methods admit loops and paths that are otherwise inefficient in the Dial sense, which can arise in practice as a result of driver searching behaviour. The first method considers a finite number of paths and the second method an infinite number of paths in the presence of loops. The absence of any efficiency constraint on the set of feasible paths makes the algorithms attractive for use in stochastic user equilibrium methods or in the approximation of a user equilibrium assignment through stochastic user equilibrium methods. The similarity of the structure of one of the proposed algorithms with that of the Floyd-Warshall shortest path algorithm would allow the two to be combined.

A stochastic user equilibrium path flow estimator

The paper sets out a path flow estimator suitable for use in conjunction with urban traffic monitoring, control and guidance. Travel time for each link in the network is partitioned into undelayed travel time and delay. The links are assumed to be of two types. For the first type of link, an external estimate of flow and travel time over the estimation interval is provided. The second type of link is characterised by a finite capacity, and delay is incurred where demand would otherwise be in excess of capacity. Demand is determined by a logit route choice model. An equivalent convex programming problem is formulated and an iterative solution procedure is set out. The estimation of the dispersion parameter in the logit model is discussed, and a column generation method to avoid path enumeration is proposed. Diagnostic procedures and a number of other practical enhancements to the procedure, in particular the incorporation of prior information on the relative magnitudes of origin-destination movements, are considered.

Stochastic user equilibrium assignment in networks with queues

A stochastic user equilibrium assignment algorithm is presented for steady state store-and-forward networks. The links of the network have constant travel times and the links or nodes have finite capacities. When capacity is reached, delay sufficient to match demand to the available capacity is generated. It has been shown by others that the equilibrium assignment in networks of this kind is the solution to a particular linear programming problem. By adding an entropy term to the objective function, a convex nonlinear programming problem is formed which yields a stochastic user equilibrium assignment. For the case of link constraints, it is proven that the Lagrange multipliers of both the linear and the non-linear programming problems give the equilibrium delays in the network. The requirements for uniqueness are investigated. Iterative algorithms are formulated for solving the nonlinear programming problem with either link or node constraints and convergence is proven. For networks where path enumeration is likely to be a problem, a column generation technique is proposed. An illustrative example is presented.