Equitable transportation network design for signal-free smart intersections

Intersections often become bottlenecks, leading to delays due to stop-and-go operations for navigating conflicting traffic movements. Connected and autonomous vehicles (CAVs) are expected to alleviate this issue by coordinating their movement to navigate intersections smoothly without traffic signals. However, it may take time for human-driven vehicles (HVs) to be replaced by CAVs. During this transition period, we aim to develop a hybrid intersection design (HID) that strategically integrates signal-free smart intersections with traditional signal-based ones by optimizing the locations of smart intersections and setting appropriate signal timings for conventional intersections. This HID approach may result in distributional welfare effects across different road users, with HV users potentially facing disadvantages because they have no access to smart intersections and their connecting links. To facilitate equitable HIDs, we develop four bi-level programming models that address the inequity issue by incorporating considerations of ethical principles, including utilitarian, sufficient, difference, and maximax principles. For each bi-level program, the transportation planner determines HID decisions, incorporating equity into the objectives and/or constraints as guided by the underlying ethical principle, at the upper level, whereas travelers make their user optimal routing choices with the given equitable HID at the lower level. We formulate the lower-level problem as signal-free smart intersections embedded network equilibrium with mixed traffic and derive its equivalent variational inequality (VI) problem, and prove the existence of VI solutions. Besides, we prove that no traveler will be worse off for HID under the difference principle compared to the signal-based control, and establish the relationship of total travel times for HIDs under four ethical principles. To solve these bi-level programs, we first reformulate them into single-level mathematical programs with equilibrium constraints (MPECs). These MPECs are approximated by the corresponding mixed-integer linear programs (MILPs), which enables existing algorithms for their approximated global optimum. We further generalize a non-uniform breakpoint selection technique with a proven minimal number of breakpoints to significantly reduce the problem size without compromising its computation accuracy. Besides, we develop a domain resizing technique to further reduce the problem size and enhance computational efficiency. Furthermore, since solving MILPs provides a lower bound for the original MPECs, we propose a modified augmented Lagrangian multiplier (MALM) approach to evaluate MILPs’ solution quality, which generates feasible solutions that serve as upper bounds for the MPECs. The consistently small gap ratios (i.e., 1 %) across all tested cases strongly validate that the developed MILPs are highly effective in finding solutions close to the global optimum for the MPECs. We also develop two feasibility problems tailored to sufficient and maximax principles to recover ethically acceptable solutions when the MALM method fails to satisfy ethical-related constraints. Our numerical experiments demonstrate that substantial reductions in path travel times for both CAVs and HVs can be achieved for HIDs under different ethical principles when the CAV market penetration ratio is significantly high (i.e., 80 %). Moreover, signal-free smart intersections are preferably allocated to locations that experience severe signal and/or queue delays under signal-based intersection control.