An operation-agnostic stochastic user equilibrium model for mobility-on-demand networks with congestible capacities

Evaluating the impact of privately-owned Mobility-on-Demand (MoD) services is important from a regulatory perspective. There is a need to model multimodal equilibria with MoD to support policymaking. While there exists a large body of literature on MoD services focusing on service design under equilibrium modeling, these studies commonly adopt assumptions of MoD operational policies. However, such policies might not be shared with regulatory agencies due to commercial privacy concerns of private operators. We model multimodal equilibrium with MoD systems in an operation-agnostic manner based on empirical observations of flow and capacity. This is done with a Flow-Capacity Interaction (FC) matrix that captures systematic effect of congestible capacities, a phenomenon in MoD systems where capacities are affected by flows. The FC matrix encapsulates the operation and demand patterns by capturing the empirical equilibrium relationship between flows and capacities. An operation-agnostic logit-based stochastic user equilibrium (SUE) formulation is proposed and proof of equivalence of the SUE formulation is derived. The proof shows that, unlike static capacities, path delays are not just the sum of the Lagrange multipliers of the links on the paths, but dependent on the whole network. We name this phenomenon as “non-separable link delays”. A solution algorithm that finds SUE with a bounded path set is proposed, with a custom Frank-Wolfe algorithm to solve the non-linear SUE formulation. Since the FC matrix cannot be directly observed, an inverse optimization problem is introduced to estimate it with observed flow and capacity data. Two numerical examples are provided with sensitivity tests. An empirical example with yellow taxi data of downtown Manhattan, NY is provided to demonstrate effectiveness of estimating the FC matrix from real data, and for determining the equilibrium that captures the underlying flow-capacity dynamics.