Day-to-day dynamic traffic evolution in the urban traffic system with ride-sharing

The introduction of ride-sharing services has significantly diversified commuting options for urban residents. In multi-mode cities, residents select travel modes based on their perception of the costs for various available options, while these perceptions are updated daily. As a result, residents’ travel mode choices and mode-split traffic flows vary day by day. To capture the nonlinear evolution phenomenon of their mode choice behaviors and mode-split traffic flows, we focus on a linear monocentric city with the introduction of ride-sharing services and develop a deterministic discrete-time day-to-day dynamic evolution model. Our model incorporates residents’ limited perceptions to better reflect real-world scenarios. Moreover, considering one ride-sharing driver is allowed to pick up multiple passengers, specific constraints on the number of ride-sharing drivers and passengers (i.e., side constraints) are added to the model formulation. Thus, the proposed model is a discrete-time day-to-day dynamic asymmetric stochastic user equilibrium model with side constraints, which have rarely been studied. The uniqueness of optimal Lagrange multipliers corresponding to side constraints in the day-to-day dynamic evolution model is demonstrated, which makes us successfully extend related literature on static ride-sharing equilibrium to the study of dynamic stochastic ride-sharing user equilibrium problems. Furthermore, we consider the stability issue of the equilibrium and provide sufficient and necessary conditions for its asymptotic stability. Finally, numerical examples are conducted to validate the properties and effectiveness of our dynamic model.