Abstract
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.
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