Balancing Fairness and Efficiency in Traffic Routing via Interpolated Traffic Assignment

System optimum (SO) routing, wherein the total travel time of all users is minimized, is a holy grail for transportation authorities. However, SO routing may discriminate against users who incur much larger travel times than others to achieve high system efficiency, i.e., low total travel times. To address the inherent unfairness of SO routing, we study the $\beta$-fair SO problem whose goal is to minimize the total travel time while guaranteeing a $\beta\geq 1$ level of unfairness, which specifies the maximal ratio between the travel times of different users with shared origins and destinations. To obtain feasible solutions to the $\beta$-fair SO problem while achieving high system efficiency, we develop a new convex program, the Interpolated Traffic Assignment Problem (I-TAP), which interpolates between a fair and an efficient traffic-assignment objective. We then leverage the structure of I-TAP to develop two pricing mechanisms to collectively enforce the I-TAP solution in the presence of selfish homogeneous and heterogeneous users, respectively, that independently choose routes to minimize their own travel costs. We mention that this is the first study of pricing in the context of fair routing. Finally, we use origin-destination demand data for a range of transportation networks to numerically evaluate the performance of I-TAP as compared to a state-of-the-art algorithm. The numerical results indicate that our approach is faster by several orders of magnitude, while achieving higher system efficiency for most levels of unfairness.