On the Applicability of Sensitivity Analysis Formulas for Traffic Equilibrium Models

Summary

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.