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Urban traffic congestion remains a persistent issue for cities worldwide. Recent macroscopic models have adopted a mathematically well-defined relation between network flow and density to characterize traffic states over an urban region. Despite advances in these models, capturing the complex dynamics of urban traffic congestion requires considering the heterogeneous characteristics of trips. Classic macroscopic models, e.g., bottleneck and bathtub models and their extensions, have attempted to account for these characteristics, such as trip-length distribution and desired arrival times. However, they often make assumptions that fall short of reflecting real-world conditions. To address this, generalized bathtub models were recently proposed, introducing a new state variable to capture any distribution of remaining trip lengths. This study builds upon this work to formulate and solve the social optimum, a solution minimizing the sum of all users' generalized (i.e., social and monetary) costs for a departure time choice model. The proposed framework can accommodate any distribution for desired arrival time and trip length, making it more adaptable to the diverse array of trip characteristics in an urban setting. In addition, the existence of the solution is proven, and the proposed solution method calculates the social optimum analytically. The numerical results show that the method is computationally efficient. The proposed methodology is validated on the real test case of Lyon North City, benchmarking with deterministic and stochastic user equilibria.
Ameli et al. (Wed,) studied this question.