A three‐dimensional distance transform for optimizing constrained mountain railway alignments

Abstract

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.

Mountain railway alignment optimization considering geological impacts: A cost‐hazard bi‐objective model

“…To connect the two endpoints in the landscape, the locations of PIs, including horizontal points of intersection (HPIs) and vertical points of intersection (VPIs), are first determined and then connected with tangents. After that, curves are configured at each PI and essential constructions are set along the railway while satisfying multiple constraints (Pu, Song, Schonfeld, Li, Zhang, Wang et al., 2019). Thus, the tasks of alignment optimization are to find the optimal locations for PIs and configurations for curves under specific optimization objectives.…”

“…These location constraints can be denoted as L ( X,Y,R,K,H ) ≤ 0. Construction constraints: To reflect limits on construction techniques, to avoid construction delays and to reduce constrution costs, the maximum bridge height and maximum tunnel length should be constrained. Moreover, when there are existing stations at endpoints, their tangent connection trajectory of horizontal connection angle and vertical gradient should be constrained (Pu, Song, Schonfeld, Li, Zhang, Wang et al., 2019). This can be denoted as C ( X,Y,R,K,H ) ≤ 0.…”

“…The alignment elevation tendency is explained as follows: Generally, in complex mountainous regions, to overcome elevation differences between two endpoints, the alignment elevation should be successively increased or decreased to reach the destination elevation (Li et al., 2017; Pu, Song, Schonfeld, Li, Zhang, Wang et al., 2019). Thus, among a tunnel's two portals, the one located at higher elevation is more hazardous than the lower one because debris flows and rocks may enter the higher portal and then bury or flood the tunnel along the alignment gradient (Figure 6).…”

“…(2020) combined cost, vegetation loss, and soil erosion to build a tri‐objective model and then solved it using a particle swarm optimization. Meanwhile, despite their successes in finding optimized alignments with low costs in flat plain regions, few of the above algorithms perform satisfactorily in complex mountainous regions, as discussed in our previous studies (Li et al., 2017; Pu, Song, Schonfeld, Li, Zhang, Wang et al, 2019). In addition, Karlson et al.…”

“…To solve these problems, we have proposed a two‐stage optimization strategy (Li et al., 2017). In the first stage (Li et al., 2016; Pu et al., 2018; Pu, Song, Schonfeld, Li, Zhang, Wang et al, 2019; Song et al, 2020), we determined several promising railway paths from the broad study area using distance transform (DT) algorithms, which are shortest‐path algorithms. Then, these paths were used as centerlines to generate a set of railway corridors (Area‐corridor stage).…”