In a railway network, making a line plan is a critical optimization problem that determines the daily transportation capacity of the network, aiming at aligning it with passenger demand while minimizing the operational cost. However, the inherent uncertainty in passenger demand often makes the determined line plan infeasible to cover. Meanwhile, additional adjustments to the line plan due to fluctuating demand in the complex railway system can cause unexpected costs. To obtain a robust line plan, we propose a distributionally robust chance-constrained (DRCC) line planning model based on a type-$\infty $Wasserstein ambiguity set, aiming to generate a line plan that remains feasible with a pre-specified probability while accommodating a given distribution deviation tolerance. We present an equivalent tractable reformulation for the proposed DRCC model by explicitly characterizing the worst-case probability distribution. Furthermore, we develop valid inequalities and a warm start strategy tailored to this model to enhance computational efficiency. The proposed model and solution method are validated through numerical experiments conducted on the Wuhan-Guangzhou high-speed railway corridor. Results demonstrate the effectiveness of the solution acceleration techniques and underscore the advantage of the DRCC model over the robust optimization model and conventional chance-constrained model in reducing operational costs. Note to Practitioners—The motivation for this paper stems from the need to provide a daily line plan for the railway to meet the uncertain passenger demand with a pre-specified probability. We assume that an estimated probability distribution of the demand is available and characterized by finite scenarios with weights, but we do not need the estimation to be perfect/accurate. We propose a distributionally robust chance-constrained line planning model to maintain robustness under the deviation/inaccurate probability distribution. To solve the model efficiently, we develop a warm start method to quickly find a partial initial solution and design valid inequalities where adding them as constraints can shorten the solving procedure by reducing the feasible region of the model’s continuous relaxation. The produced line plan requires less operational cost than the robust optimization model to cover the same amount of demand by the benefits of leveraging the distributional information.

Train schedule consists of two major phases, train timetable optimization (TTO) and train timetable rescheduling (TTR), which are interconnected with each other and aim to maintain the safety and punctuality of high-speed railway operations under ideal conditions and unexpected disturbances. However, current preparation and adjustment of train timetables face challenges in real-time responsiveness and poor performance on large-scale instances. To alleviate these problems, we propose a unified scheduling model based on deep reinforcement learning (DRL) for both TTO and TTR problems with similar formulations. The key components of our approach include a state representation utilizing the Markov decision process that captures global train and station characteristics, and a policy network that extracts information from this representation to sequentially construct the train departure order. The main benefits of our framework include adaptability to different stopping plans and delay scenarios, decoupling from the problem size, and ensuring the feasibility of generated schemes. Furthermore, to improve the solution quality, we integrate the learned decision policies with a local search method, enabling the scalability of the model with little additional computation cost. Experiments on extensive TTO and TTR instances of the Beijing-Shanghai high-speed railway line demonstrate the effectiveness and practicality of our approach. Our DRL-based method outperforms all the heuristic rules and commercial solvers without retraining the model on various problem sizes, especially on large-scale cases under limited calculation time.

Summary The foundation of urban landscape design optimization is the precise evaluation of the effectiveness. To address the issues of strong subjectivity, low efficiency, and poor accuracy in urban landscape design evaluation methods, an intelligent evaluation method combining improved genetic algorithm and error backpropagation neural network is proposed. First, based on Maslow's demand theory and questionnaire survey results, it selects indicators to construct an evaluation index system for urban landscape design. Second, in response to the performance defects of the error backpropagation neural network model, the moth flame algorithm is used to optimize it. Then, in response to the defect that the optimization effect of the moth flame algorithm is not ideal enough, a multiple strategy including improved genetic algorithm is adopted to optimize it. Finally, an urban landscape design evaluation model is constructed based on improved error backpropagation neural network. The experimental results show that the fitting coefficient of the model is 0.9523, with a minimum deviation of less than 1%. The above results indicate that the proposed model can effectively improve the accuracy and efficiency of urban landscape design evaluation, providing data support for urban landscape design optimization. The research on the intelligent development of urban landscape design is of reference significance and has to some extent promoted the development of urban landscape design.

Unpredictable disturbances that occur when the train is running often make the actual timetable deviate from the planned timetable and deteriorate service quality for passengers. To provide a robust planned timetable that is reliable in operation, this paper presents a two-stage distributionally robust railway timetabling (DRRT) model, where the decisions in the first and second stages are the planned and actual timetables, respectively. To deal with the uncertain travel time, the proposed DRRT model assumes the true distribution belongs to a Wasserstein ambiguity set centered at the empirical distribution constructed from finite samples and aims to minimize the sum of the total idle time, travel time, and worst-case expected delay over the ambiguity set. To solve this model, we design a decomposition algorithm with provable convergence, which iteratively solves a master problem and a set of subproblems. To enhance the algorithm efficiency, a blend of two popular decomposition methods is proposed to integrate their advantages, and the solution of the subproblems is accelerated by using an uncapacitated minimum-cost flow interpretation of the second-stage optimization problem. Experiments on real-world instances demonstrate the effectiveness of these enhancements. The results also show that our model prominently outperforms the sample average approximation model in terms of the out-of-sample performance, especially when only a small sample set is available. Moreover, at the expense of equal traffic efficiency, our model is more robust than those models that do not utilize distribution information. Note to Practitioners—This paper was motivated by the problem of using finite historical data to provide a robust planned railway timetable, aiming at minimizing the traffic efficiency losses and expected delays under the worst-case distribution. Existing studies on dealing with the uncertain travel time usually assumed that the distribution is exactly known, or the distribution is completely unknown but the travel time belongs to an uncertainty set. However, in practice, the exact distribution is hard to know but can only be coarsely estimated from data. This paper proposes a data-driven two-stage model by assuming that the true distribution belongs to a 1-Wasserstein ball centered at the empirical distribution with a bounded support set, which is a middle ground between the above two distribution assumptions. To solve this problem efficiently, we propose a customized algorithm with provable convergence and then enhance it by exploiting problem-specific properties. This distributionally robust model can be easily shifted to other timetabling problems, and the algorithm enhancement techniques can also be applied to efficiently solve other two-stage recourse models with similar properties.

Contextual optimization, also known as predict-then-optimize or prescriptive analytics, considers an optimization problem with the presence of covariates (context or side information). The goal is to learn a prediction model (from the training data) that predicts the objective function from the covariates, and then in the test phase, solve the optimization problem with the covariates but without the observation of the objective function. In this paper, we consider a risk-sensitive version of the problem and propose a generic algorithm design paradigm called predict-then-calibrate. The idea is to first develop a prediction model without concern for the downstream risk profile or robustness guarantee, and then utilize calibration (or recalibration) methods to quantify the uncertainty of the prediction. While the existing methods suffer from either a restricted choice of the prediction model or strong assumptions on the underlying data, we show the disentangling of the prediction model and the calibration/uncertainty quantification has several advantages. First, it imposes no restriction on the prediction model and thus fully unleashes the potential of off-the-shelf machine learning methods. Second, the derivation of the risk and robustness guarantee can be made independent of the choice of the prediction model through a data-splitting idea. Third, our paradigm of predict-then-calibrate applies to both (risk-sensitive) robust and (risk-neutral) distributionally robust optimization (DRO) formulations. Theoretically, it gives new generalization bounds for the contextual LP problem and sheds light on the existing results of DRO for contextual LP. Numerical experiments further reinforce the advantage of the predict-then-calibrate paradigm in that an improvement on either the prediction model or the calibration model will lead to a better final performance.