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The standard model of traffic flow used in the analysis of urban traffic is the Wardrop equilibrium. The existence of traffic flows that reduce costs for some travelers without increasing the costs for any other travelers when compared to the equilibrium defines a Generalized Braess Paradox. We provide a practical methodology for detecting such flows and report the existence of such a flow in the Sioux Falls study network when links with equilibrium flows in the free-flow range are regarded as constant cost.

Congestion or toll pricing problems in [ HeR98 ] require a solution to the system problem (the traffic assignment problem that minimizes the total travel delay) to define the set of all valid tolls or the toll set. For practical problems, it may not be possible to obtain an exact solution to the system problem and the inaccuracy in an approximate system solution may render the toll set empty. When this occurs, this paper offers alternative toll sets based on relaxed optimality conditions. With carefully chosen parameters, tolls from the relaxed toll sets are shown theoretically and empirically (using four transportation networks in the literature) to induce route choices that are nearly system optimal.

We present an optimization approach for jointly learning the demand as a function of price, and dynamically setting prices of products in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, but rather assume parametric families of demand functions that are learned over time. In the first part of the paper, we consider the noncompetitive case and present dynamic programming algorithms of increasing computational intensity with incomplete state information for jointly estimating the demand and setting prices as time evolves. Our computational results suggest that dynamic programming based methods outperform myopic policies often significantly. In the second part of the paper, we consider a competitive oligopolistic environment. We introduce a more sophisticated model of demand learning, in which the price elasticities are slowly varying functions of time, and allows for increased flexibility in the modeling of the demand. We propose methods based on optimization for jointly estimating the Firm’s own demand, its competitors’ demands, and setting prices. In preliminary computational work, we found that optimization based pricing methods offer increased expected revenue for a firm independently of the policy the competitor firm is following.

We study congestion pricing of road networks with users differing only in their time values. In particular, we analyze the marginal social cost (MSC) pricing, a tolling scheme that charges each user a penalty corresponding to the value of the delays inflicted on other users, as well as its implementation through fixed tolls. We show that the variational inequalities characterizing the corresponding equilibria can be stated in symmetric or nonsymmetric forms. The symmetric forms correspond to optimization problems, convex in the fixed-toll case and nonconvex in the MSC case, which hence may have multiple equilibria. The objective of the latter problem is the total value of travel time, which thus is minimized at the global optima of that problem. Implementing close-to-optimal MSC tolls as fixed tolls leads to equilibria with possibly non-unique class specific flows, but with identical close-to-optimal values of the total value of travel time. Finally we give an adaptation, to the MSC setting, of the Frank-Wolfe algorithm, which is further applied to some test cases, including Stockholm.

The construction of toll highways by concessions awarded to private companies leads to the need of forecasting their usage in order to estimate the future stream of revenues. Two main modeling approaches for this problem that result in variants of multiclass network equilibrium models, are presented and commented upon.

The paper by Tobin and Friesz [ ToF88 ] brought the classic nonlinear programming subject of sensitivity analysis to transportation science. It is still the most widely used device by which “gradients” of traffic equilibrium solutions (that is, flows and/or demands) are calculated, for use in bilevel transportation planning applications such as network design, origin-destination (OD) matrix estimation and problems where link tolls are imposed on the users in order to reach a traffic management objective. However, it is not widely understood that the regularity conditions proposed by them are stronger than necessary. Also, users of their method sometimes misunderstand its limitations and are not aware of the computational advantages offered by more recent methods. In fact, a more often applicable formula was proposed already in 1989 by Qiu and Magnanti [ QiM89 ], and Bell and Iida [ BeI97 ] describe one of the cases in practice in which the formula by Tobin and Friesz [ ToF88 ] would not be able to generate sensitivity information, because one of their regularity conditions fails to hold. This paper provides a short overview of a sensitivity formula that provides directional derivatives of traffic equilibrium flows, route and link costs, and demands, exactly when they exist, and which are found in [ PaR03 ] and [ Pat04 ], For the simplicity of the presentation, we provide the analysis for the simplest cases, where the link travel cost and demand functions are separable, so that we can work with optimization formulations; this specialization was first given in [ JoP04 ]. The connection between directional derivatives and the gradient is that exactly when the directional derivative mapping of the traffic equilibrium solution is linear in the parameter, the solution is differentiable. The paper then provides an overview of the formula of Tobin and Friesz [ ToF88 ], and illustrates by means of examples that there are several cases where it is not applicable: First, the requirement that the equilibrium solution be strictly complementary is too strong—differentiate points may not be strictly complementary. Second, the special matrix invertibility condition implies a strong requirement on the topology of the traffic network being analyzed and which may not hold in practice, as noted by Bell and Iida [ BeI97 ](page 97); moreover, the matrix condition may fail to hold at differentiable points. The findings of this paper are hoped to motivate replacing the previous approach with the more often applicable one, not only because of this fact but equally importantly because it is intuitive and also can be much more efficiently utilized: the sensitivity problem that provides the directional derivative is a linearized traffic equilibrium problem, and the sensitivity information can be generated efficiently by only slightly modifying a state-of-the-art traffic equilibrium solver. This is essential for bringing the use of sensitivity analysis in transportation planning beyond the solution of only small problems.

We propose an application of a new set of models referred to as “Stable Dynamics” which provide a flexible yet rigorous way to model traffic congestion in large urban areas. Data requirements are extremely low, since supply and demand data can be given by GIS systems. This approach is based on the requirements that (1) the maximum entering flow for each link is given and that (2) Wardrop principle holds. In this paper, we supplement this basic model by parking choice. We focus on the case where the commuters use private and public transportation from the origin to the destination (and back to the origin). We propose a consistent model of both for the day commuting and the morning-evening commuting and show that such extension can be formulated as standard convex mathematical problems.

We suppose given a variable demand model with some control parameters to represent prices, a smooth function V which measures departure from equilibrium and a smooth function Z which measures overall disbenefit. We suppose that we wish to minimise Z subject to the constraint that the disequilibrium function V is no more than ε , where we think of ε as a small positive number. The paper suggests a simultaneous descent direction to solve this bilevel optimisation problem; such a direction reduces Z and V simultaneously and may often be computed by simply bisecting the angle between −∇ Z and −∇ V . The paper shows that following a direction Δ which employs the simultaneous descent direction as its central element leads, under natural conditions which preclude edge effects (where a flow may be zero or a price may be maximum), to the set of those approximate equilibria (where V ≤ ε ) at which Z is stationary. Then the method is extended on the one hand to deal with edge effects (allowing a route flow to be zero or a price to be the maximum permitted), by ensuring that the direction Δ followed anticipates nearby edges of the feasible region, using reduced gradients instead of gradients, and on the other hand to deal with signal controls. Within the optimisation procedure proposed here, optimisation and equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.

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.

This paper considers the optimal toll design problem that uses the Probit model to determine travellers’ route-choices. Under probit, the route flow solution to the resulting stochastic user equilibrium (SUE) is unique and can be stated implicitly as a function of tolls. This reduces the toll design problem to an optimization problem with only nonnegativity constraints. Additionally, the gradient of the objective function can be approximated using the chain rule and the first order Taylor approximation of the equilibrium condition. To determine SUE, this paper considers two techniques. One uses Monte-Carlo simulation to estimate route choice probabilities and the method of successive averages with its prescribed step length. The other relies on the Clark approximation and computes an optimal step length. Although both are effective at solving the toll design problem, numerical experiments show that the technique with the Clark approximation is more robust on a small network.