Sensitivity analysis of the variable demand probit stochastic user equilibrium with multiple user classes

This paper presents a formulation of the multiple user class, variable demand, probit stochastic user equilibrium model. Sufficient conditions are stated for differentiability of the equilibrium flows of this model. This justifies the derivation of sensitivity expressions for the equilibrium flows, which are presented in a format that can be implemented in commercially available software. A numerical example verifies the sensitivity expressions, and that this formulation is applicable to large networks

A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows

This paper deals with two mathematically similar problems in transport network analysis: trip matrix estimation and traffic signal optimisation on congested road networks. These two problems are formulated as bi-level programming problems with stochastic user equilibrium assignment as the second-level programming problem. We differentiate two types of solutions in the combined matrix estimation and stochastic user equilibrium assignment problem (or, the combined signal optimisation and stochastic user equilibrium assignment problem): one is the solution to the bi-level programming problem and the other the mutually consistent solution where the two sub-problems in the combined problem are solved simultaneously. In this paper, we shall concentrate on the bi-level programming approach although we shall also consider mutually consistent solutions so as to contrast the two types of solutions. The purpose of the paper is to present a solution algorithm for the two bi-level programming problems and to test the algorithm on several networks

Data-driven Estimation of Origin-Destination Demand and User Cost Functions for the Optimization of Transportation Networks

In earlier work (Zhang et al., 2016) we used actual traffic data from the Eastern Massachusetts transportation network in the form of spatial average speeds and road segment flow capacities in order to estimate Origin-Destination (OD) flow demand matrices for the network. Based on a Traffic Assignment Problem (TAP) formulation (termed "forward problem"), in this paper we use a scheme similar to our earlier work to estimate initial OD demand matrices and then propose a new inverse problem formulation in order to estimate user cost functions. This new formulation allows us to efficiently overcome numerical difficulties that limited our prior work to relatively small subnetworks and, assuming the travel latency cost functions are available, to adjust the values of the OD demands accordingly so that the flow observations are as close as possible to the solutions of the forward problem. We also derive sensitivity analysis results for the total user latency cost with respect to important parameters such as road capacities and minimum travel times. Finally, using the same actual traffic data from the Eastern Massachusetts transportation network, we quantify the Price of Anarchy (POA) for a much larger network than that in Zhang et al. (2016).Comment: Preprint submitted to The 20th World Congress of the International Federation of Automatic Control, July 9-14, 2017, Toulouse, Franc

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

Applications of sensitivity analysis for probit stochastic network equilibrium

Network equilibrium models are widely used by traffic practitioners to aid them in making decisions concerning the operation and management of traffic networks. The common practice is to test a prescribed range of hypothetical changes or policy measures through adjustments to the input data, namely the trip demands, the arc performance (travel time) functions, and policy variables such as tolls or signal timings. Relatively little use is, however, made of the full implicit relationship between model inputs and outputs inherent in these models. By exploiting the representation of such models as an equivalent optimisation problem, classical results on the sensitivity analysis of non-linear programs may be applied, to produce linear relationships between input data perturbations and model outputs. We specifically focus on recent results relating to the probit Stochastic User Equilibrium (PSUE) model, which has the advantage of greater behavioural realism and flexibility relative to the conventional Wardrop user equilibrium and logit SUE models. The paper goes on to explore four applications of these sensitivity expressions in gaining insight into the operation of road traffic networks. These applications are namely: identification of sensitive, ‘critical’ parameters; computation of approximate, re-equilibrated solutions following a change (post-optimisation); robustness analysis of model forecasts to input data errors, in the form of confidence interval estimation; and the solution of problems of the bi-level, optimal network design variety. Finally, numerical experiments applying these methods are reported