This paper resolves some recent puzzles and criticisms of the gradient-based sensitivity analysis of network traffic equilibria originally proposed by Tobin and Friesz, and develops a simple but general approach for calculating the derivatives of equilibrium link flows. We show that, under the conditions that the link cost function vector is strongly monotone and there exists a set of strictly positive flows on the minimum cost or equilibriated paths, the classical implicit function theorem can always be applied for any set of a maximum number of linearly independent and equilibrated paths. As a result, the equilibrium link flows are differentiable under these mild conditions and their derivatives can always be obtained by the formula developed from the implicit function theorem, without dependence on the network topology. We show that non-differentiability of the equilibrium link flows may occur only at a degenerate equilibrium point where there exists at least one degenerate, equilibrated path on which flow must be zero. The gradient formula, when applied to a degenerate equilibrium point, can provide the left and right derivatives of link flows and the desired derivatives with respect to a perturbation parameter when appropriately including or not including the degenerate path(s). Analytical examples facilitate understanding of the simplicity and generality of the corrected gradient formula and the flaws inherent in the old sensitivity analysis approach. (A) Reprinted with permission from Elsevier. For the covering abstract see ITRD E134766.
Samenvatting