New formulations of the stochastic user equilibrium with logit route choice as an extension of the deterministic model

This paper addresses the stochastic user equilibrium (SUE) in the case where the route choice is the multinomial logit model (MNL). Our main finding is that MNL SUE can be formulated and solved as an immediate extension of the deterministic user equilibrium (DUE) through a particular application of Wardrop’s first principle. The latter states, in general, that at equilibrium, the cost of all used routes is equal and not higher than those of unused routes. The extension is achieved by applying this statement to the “perceived cost” of a choice alternative, which is defined here as its generalized cost plus the logarithm of its choice probability multiplied by the logit parameter. Thus, substituting in DUE models the generalized costs with the perceived costs allows to easily adapt to MNL SUE the existing formulations and algorithms for DUE, as well as to manage a smooth transition of the route choice model from stochastic to deterministic by reducing the logit parameter down to zero. Particular consideration is devoted to the interpretation of the numerical solution as a restricted logit model, where only sufficiently good alternatives receive a positive probability. A family of MNL SUE models is then presented ranging from nonlinear optimization to variational inequalities and fixed-point problems, with both explicit and implicit path enumeration. A range of numerical tests is presented with the aim of assessing the continuity of the model results for decreasing logit parameter and proving the applicability of the proposed approach to real size networks, with particular emphasis on the performance and convergence of the methods.