Optimal measurement of traffic hysteresis under traffic oscillations: A binary integer programming approach

Traffic hysteresis, a phenomenon observed daily in traffic characterized by flow and density loops on the fundamental diagram, is attributed to intra/inter-driver heterogeneity. Despite extensive analysis, a systematic method to measure hysteresis accounting for these heterogeneities, ensuring the traffic state over each measurement time window remains within an approximate speed, is still missing. This gap becomes more pronounced with the advent of mixed traffic, including automated vehicles (AVs) and human-driven vehicles (HDVs), whose behaviors can significantly differ. This paper investigates an optimal method for measuring traffic hysteresis loops, applicable to both AVs and HDVs, without requiring detailed knowledge of car-following behaviors. Specifically, by representing vehicle trajectories as a directed graph, finding the optimal hysteresis measurement window involves finding an optimal matching that accommodates intra/inter-driver heterogeneity and minimizes speed differences within response time boundaries. We further formulate the matching problem as a variant of the minimum-cost network flow problem, commonly encountered in the logistics domain. A binary integer programming model is proposed, which can be efficiently solved using state-of-the-art solvers. We prove the feasibility and boundedness of our framework to ensure its broad applicability. To validate our method, we compared it with the prevailing parallelogram measurement method in a pure AV scenario under single-frequency oscillation. Our method proved to be more effective, providing more accurate estimates close to theoretical derivations. Subsequently, we applied our method to general traffic conditions, demonstrating that it yields a lower speed variation within measurement regions. Through a series of quantitative analyses on hysteresis, our framework also suggests that the prevailing method may overestimate hysteresis.