Railway alignment optimization is considered one of the most complicated and time‐consuming problems in railway planning and design. It requires searching among the infinite potential alternatives in huge three‐dimensional (3D) search spaces for a near‐optimal alignment, while considering complex constraints and a nonlinear objective function. In mountainous regions, the complex terrain and constructions require additional and more complex constraints than in topographically simpler regions. In this paper, the authors solve this problem with an algorithm based on a 3D distance transform (3D‐DT). Compared with previous two‐dimensional distance transform (2D‐DT) methods developed in this field, the feasible search spaces of 3D‐DT are greatly increased. Consequently, this new method can find more alternatives with higher qualities. In this approach, an erythrocyte‐shaped 3D neighboring mask is developed to narrow local search spaces and speed up the search process. Besides, a stepwise‐backstepping strategy is designed to dynamically determine feasible 3D search spaces and efficiently search the study area. During the 3D‐DT search process, multiple constraints, including geometric, construction, and location constraints, are effectively handled. After the 3D‐DT search, a genetic algorithm is employed to optimize the 3D‐DT paths into final alignments. Finally, this novel approach is applied to an actual case in a complex mountainous region. The comprehensive cost of the best solution generated by 3D‐DT is 16% below a manual solution produced by very experienced human designers. Furthermore, the total number of feasible alternatives found by 3D‐DT is 4.3 times greater than by 2D‐DT. The comprehensive cost of the best 3D‐DT solution is 10% below the best one generated by 2D‐DT.

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Railways are greatly threatened by geological hazards whose disastrous effects include severe economic losses as well as serious casualties. It is vital to properly account for such geological hazardous impacts during a railway alignment optimization process. However, geological factors are complex, especially in mountainous regions. Besides, economic factors are also crucial in railway alignment design. Therefore, railway alignment optimization can be termed as a cost-hazard bi-objective decision-making process. So far, least-cost railway alignment optimization has been studied quite thoroughly while the complicated geological hazard factors have received relatively little attention. In this study, a bi-objective alignment optimization model considering cost and geological hazard is developed. A novel geological railway alignment optimization model is proposed, which includes spatial geological constraints and geological hazard evaluations, after geological railway alignment design criteria are presented and analyzed for three kinds of typical geological hazards: debris flows, landslides, and rockfalls. The geological hazard evaluation includes geological susceptibility and vulnerability assessments. Then, this model is integrated with a previous least-cost alignment optimization model to construct a cost-hazard bi-objective model. The alignment searching processes are also improved to solve the proposed model by integrating geological-constraints-handling and bi-objective alignment optimization approaches. Finally, the effectiveness of the proposed method is verified by applying it to a complicated real-world case. The results show that the proposed method can produce less expensive and safer solutions than the best alignment designed by experienced human designers while satisfying all required design standards. Moreover, the method's applicability for solving actual problems is further demonstrated through the sensitivity analysis. 1 INTRODUCTION Geological hazards greatly threaten the construction and operation of railways. Their disastrous effects include not

Mountain railway alignment optimization is known as a very complex engineering problem that should consider many factors, such as drastically undulating terrain, geological hazard impacts, and additional constraints. Moreover, many mountain railway projects are located in earthquake‐prone regions and hence are greatly threatened by seismic activity. Thus far, most alignment optimization studies aim at finding the least‐cost solutions within budget but slight attention has been paid to reducing the complex seismic risk through optimization. In this paper, the first known quantitative seismic risk assessment model for railway alignment optimization is presented, which combines probabilistic seismic fragility analysis and probabilistic seismic loss analysis. Three methods for fragility analysis of bridge, tunnel, and earthwork sections are designed and a specific event tree is developed for seismic loss analysis. Moreover, multiple preliminary constraints are specified for alignments traversing active faults. Afterwards, the seismic risk assessment model is combined with a least‐cost model to formulate a bi‐objective optimization model. To solve it, a particle swarm optimization algorithm is improved by blending the crowding distance computation (CDC) and, especially, a novel marginal benefit analysis (MBA) to search for pareto‐optimal solutions during optimization. A prescreening and repairing operator is also designed to handle the fault constraints. Finally, when applying the proposed procedure to a complex realistic railway case, the results show that the hybrid CDC+MBA bi‐objective solver can find better pareto‐optimal solutions than the generic CDC method. Besides, detailed data analysis shows that the present method can produce less expensive as well as safer solutions than the best alignment designed by experienced human engineers.

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