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Sensitivity Analysis of Traffic Equilibria

Department of Mathematics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
mipat{at}math.chalmers.se

The contribution of the paper is a complete analysis of the sensitivity of elastic demand traffic (Wardrop) equilibria. The existence of a directional derivative of the equilibrium solution (link flow, least travel cost, demand) in any direction is given a characterization, and the same is done for its gradient. The gradient, if it exists, is further interpreted as a limiting case of the gradient of the logit-based SUE solution, as the dispersion parameter tends to infinity. In the absence of the gradient, we show how to compute a subgradient. All these computations (directional derivative, (sub)gradient) are performed by solving similar traffic equilibrium problems with affine link cost and demand functions, and they can be performed by the same tool as (or one similar to) the one used for the original traffic equilibrium model; this fact is of clear advantage when applying sensitivity analysis within a bilevel (or mathematical program with equilibrium constraints, MPEC) application, such as for congestion pricing, OD estimation, or network design. A small example illustrates the possible nonexistence of a gradient and the computation of a subgradient.

Key Words: traffic equilibrium; stochastic user equilibrium; sensitivity analysis; directional derivative; bilevel optimization
History: Received: October 2001; revised: August 2002; accepted: December 2002.

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