This paper studies an integrated train timetabling and rolling stock circulation planning problem with stochastic demand and flexible train composition (TRSF). A novel stochastic integer programming model, which is formulated on a space-time underlying network to simultaneously optimize the train timetable and rolling stock circulation plan with flexible train composition, is proposed by explicitly considering the random feature of passenger distribution on an urban rail transit line. To solve this problem efficiently, the proposed model is decomposed into a master problem and a series of sub-problems regarding different stochastic scenarios. We further prove that each sub-problem model is equivalent to its linear programming relaxation problem, by proving that the coefficient matrix of each linear programming relaxation model is totally unimodular. Then, the classical Benders decomposition algorithm is applied to the studied problem. Based on the model characteristics, both single-cut and multi-cut methods with some speed-up techniques are developed to solve the proposed model in a novel and effective way. Numerical experiments are conducted on small-scale cases and large-scale cases derived from Shanghai Metro Line 17, and the results show that solving the stochastic problem can extract gains in efficiency and the value of stochastic solution tends to be high.

中文翻译：

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具有随机需求的综合列车时刻表和机车车辆流通规划问题的新精确算法

本文研究了具有随机需求和灵活列车组合（TRSF）的综合列车时刻表和机车车辆流通规划问题。明确考虑城市轨道交通线路客流分布的随机特征，提出一种基于时空底层网络的新型随机整数规划模型，以灵活的列车编组同时优化列车时刻表和机车车辆流通计划。为了有效地解决这个问题，所提出的模型被分解为一个主问题和一系列关于不同随机场景的子问题。通过证明每个线性规划松弛模型的系数矩阵是完全幺模的，我们进一步证明每个子问题模型等价于其线性规划松弛问题。然后，将经典的Benders分解算法应用于所研究的问题。根据模型特点，开发了带有一些加速技术的单切割和多切割方法，以新颖有效的方式求解所提出的模型。对上海地铁17号线的小规模案例和大规模案例进行了数值实验，结果表明，解决随机问题可以提高效率，且随机解的价值往往较高。