A geographic information model for 3-D environmental suitability analysis in railway alignment optimization

Abstract

Railway alignment design is a complicated problem affected by intricate environmental factors. Although numerous alignment optimization methods have been proposed, a general limitation among them is the lack of a spatial environmental suitability analysis to guide the subsequent alignment search. Consequently, many unfavorable regions in the study area are still searched, which significantly degrades optimization efficiency. To solve this problem, a geographic information model is proposed for evaluating the environmental suitability of railways. Initially, the study area is abstracted as a spatial voxel set and the 3-D reachable ranges of railways are determined. Then, a geographic information model is devised which considers topographic influencing factors (including those affecting structural cost and stability) as well as geologic influencing factors (including landslides and seismic impacts) for different railway structures. Afterward, a 3-D environmental suitability map can be generated using a multi-criteria decision-making approach to combine the considered factors. The map is further integrated into the alignment optimization process based on a 3-D distance transform algorithm. The proposed model and method are applied to two complex realistic railway cases. The results demonstrate that they can considerably improve the search efficiency and also find better alignments compared to the best alternatives obtained manually by experienced human designers and produced by a previous distance transform algorithm as well as a genetic algorithm.

Keywords

Railway design alignment optimization geographic information analysis spatial environmental suitability multi-criteria decision-making

1. Introduction

1.1 Problem statement

Alignment development is a crucial part of the railway design process, which fundamentally determines the safety and economy of railway construction and operation [1]. The main tasks of alignment determination are to find the best railway locations [2, 3], geometric configurations [4] and structural components [5, 6]. This is complex work that should consider multiple constraints [7], hard-to-quantify objectives [8] and nearly infinite numbers of possible alternatives [9] in a large study area [10]. However, in present engineering practice railway alignment design depends heavily on manual work. Even for experienced designers, this process can take many months with great efforts but still overlook many potentially promising alternatives.

Figure 1.

Sketch maps of (a) a horizontal alignment with its horizontal points of intersection (HPIs), and (b) a vertical alignment with its vertical points of intersection (VPIs).

To solve the above problems, numerous researchers have proposed various alignment optimization approaches. A spatial alignment can be divided into horizontal and vertical alignments, as shown in Fig. 1. Therefore, the alignment optimization of railways and highways can be classified into three categories: horizontal, vertical and 3-D alignment optimizations.

1.2 Literature review

For horizontal alignment optimization: Audet and Dennis [11] proposed a derivative-free optimization approach named NOMAD which was further extended in Mondal et al. [12] by combining a parallel search package to produce horizontal alignments within a predefined corridor. Casal et al. [13] customized a sequential quadratic programming (SQP) for horizontal alignment optimization, considering the avoidance of obstacles. Lee et al. [14] presented a two-stage heuristic approach by combining a neighborhood search and a mixed integer program to optimize costs of horizontal alignments. Sushma and Maji [15] designed a modified motion planning algorithm incorporating low-discrepancy sampling and sequential quadratic techniques to yield horizontal alignments.

For vertical alignment optimization: Hare et al. [16] devised a mixed integer linear programming method considering earthwork allocation. Then, Beiranvand et al. [17] further incorporated a multi-haul quasi network model into this method to improve its accuracy and reliability. Babapour et al. [18] applied a genetic algorithm (GA) and particle swarm optimization (PSO) to minimize the costs of vertical alignments for forest roads. Ghanizadeh et al. [19] implemented a GA to optimize four kinds of objective functions in a vertical alignment optimization. Sushma et al. [20] presented an ant colony optimization (ACO) for dynamically exploring and exploiting feasible search spaces in vertical alignment development.

For 3-D alignment optimization: an influential GA [21, 22, 23, 24] was pioneered in Jong and Schonfeld [25] for highway alignment optimization. This GA has afterward been widely applied to alignment optimization [26, 27, 28]. As an alternative to GA, Shafahi and Bagherian [29] customized a PSO to solve 3-D highway alignment optimization problems. Vázquez-Méndez et al. [30] developed a two-stage method for optimizing 3-D road alignments, where a GA, PSO, and NOMAD were used in the first stage and an SQP was applied in the second stage. For mountainous regions with drastically-undulating terrain, Li et al. [5, 31] proposed a bidirectional scanning strategy based on a 2-D distance transform (2D-DT), which was first introduced in alignment optimization by de Smith [32] to generate alignment alternatives. The 2D-DT was further improved to a 3-D distance transform (3D-DT) in Pu et al. [33] by enlarging its search spaces and finally parallelized in Song et al. [34] to improve its search efficiency.

Figure 2.

Comparison of previous alignment optimization methods with the present method.

Figure 3.

Presentations of (a) ground grids and (b) their corresponding voxels in z-direction.

1.3 Focus of this study

The above approaches have been successfully applied in many real-world cases. Their common search strategy is to first generate a large number of alignment alternatives and then evaluate the fitness (e.g., cost and risk) of these alternatives based on their specific environmental inputs (e.g., terrain and geo-hazard data) [35, 36, 37]. Although many local regions in the study area are unsuitable for traversing alignments, they are still identified and evaluated during the search process. Thus, considerable computational resources and time are wasted for this computationally-intensive problem. Actually, in actual design processes, alignment engineers should first prescreen environmentally-unfavorable regions for railways or highways according to local topographic and geologic conditions, to save valuable time and resources in the subsequent alignment design stage. This is called a proactive evaluation in this paper (Fig. 2). Regrettably, this proactive evaluation is generally done according to designers’ qualitative experiences rather than through computerized analysis of spatial data.

For automated alignment optimization, such a proactive evaluation regarding environmental factors is also desirable before the detailed alignment search. However, few relevant studies can be found in this field. In the literature, Song et al. [7] developed a Geographic Information System (GIS)-based multi-criteria decision-making (MCDM) model for environmental suitability analysis of railway alignments, but mainly focused on geographic analysis of earthwork, i.e., cuts and fills. However, since railway alignments can also traverse the search space through tunnels and bridges, a 3-D environmental suitability analysis is needed.

Therefore, in this paper, a proactive environmental suitability model is presented to assist the railway design process, using geographic information techniques. In this model, the search space for alignment optimization is first abstracted as a voxel set and then processed with a 3-D reachability analysis for railways to determine their horizontally reachable boundaries and vertically reachable ranges. The obtained railway reachable ranges are afterward classified according to the railway structures’ characteristics. Based on this, topographic and geologic factors affecting railway design are considered using geographic information analyses. Specifically, the topographic influences on the railway structures’ cost and stability are formulated. From the geologic perspective, landslide and seismic hazards are combined. The above factors are integrated with an MCDM method called CRITIC to generate a complete 3-D environmental suitability map of the study area. The map is finally incorporated into a previous 3D-DT algorithm to guide the alignment search process. Through applications to two complex real-world examples, the effectiveness of the above model and method is verified and discussed in detail.

The remainder of the paper is organized as follows: Section 2 shows the data preprocessing of environmental inputs. Section 3 provides the 3-D environmental suitability evaluation model for railway alignments. Sections 4 and 5 present two case studies that assess the efficiency and effectiveness of the devised model. Section 6 summarizes this research.

2. Preprocessing of environmental inputs

2.1 Presentation of the search space

In this paper, the search space for alignment optimization is abstracted as a voxel set composed of ground grids and their corresponding voxels in the z-direction, as shown in Fig. 3. The size of every voxel is defined as in Pu et al. [33]. Each voxel stores data required for alignment design, such as the ground elevation, landslides distribution (including that of debris flows and rockfalls according to Cruden and Varnes, [38]), peak ground acceleration (PGA) and lithologic type.

Figure 4.

Diagrams of (a) horizontally reachable range and (b) vertically reachable range.

2.2 Determination of railway-reachable ranges

Generally, to overcome the elevation difference between the start and end points as well as to avoid obstacles, the alignment must necessarily be circuitous while satisfying the maximum gradient constraint ( $G_{\max}$ ). However, the circuity of alignments should not exceed a maximum threshold $\gamma_{\max}$ for economy and safety reasons [7]. Thus, the grids whose horizontal locations satisfy the following relation can be deemed as horizontally reachable for railway alignments:

$\displaystyle\frac{L_{S}+L_{E}}{L_{SE}}<\gamma_{\max}=\max\left\{{\frac{\left|% {H_{S}-H_{E}}\right|}{L_{SE}\cdot G_{\max}},1}\right\}+\varepsilon$ (1)

Figure 5.

Spatially reachable range of railways.

Figure 6.

Dividing the reachable range according to structural characteristics.

where $L_{S}$ or $L_{E}=$ the linear distance between every horizontal grid and the start or end points, respectively, as shown in Fig. 4a; $L_{SE}=$ the linear distance between the start and end points; $H_{S}$ and $H_{E}=$ the design elevations of the start and end points, respectively; $\varepsilon=$ an empirical parameter input by model users, whose range is generally between 0–0.5 according to [7]. Generally, due to the increased resistance in curve and tunnel sections, the adopted railway gradient should be adjusted downward from $G_{\max}$ . Thus, $\varepsilon$ is needed to account for such “gradient losses” in determining a maximum threshold of alignment circuity based on the theoretically minimum circuity factor (i.e., the first item on the right-hand side of Eq. (1)) that is needed to overcome the elevation difference between the start and end points.

Afterward, as shown in Fig. 4b, the vertically reachable range ( $\Omega$ ) corresponding to each horizontal grid can be computed with Eqs (2) and (2.2):

$\displaystyle\Omega=(h_{s\min,}h_{s\max})\cap(h_{e\min,}h_{e\max})$ (2) $\displaystyle\left\{{{\begin{array}[]{*{20}c}{h_{s\min}=H_{S}-G_{\max}\cdot L_% {S}\cdot\gamma_{\max}}\\ {h_{s\max}=H_{S}+G_{\max}\cdot L_{S}\cdot\gamma_{\max}}\\ \end{array}}}\right.$ (3) $\displaystyle\left\{{{\begin{array}[]{*{20}c}{h_{e\min}=H_{E}-G_{\max}\cdot L_% {E}\cdot\gamma_{\max}}\\ {h_{e\max}=H_{E}+G_{\max}\cdot L_{E}\cdot\gamma_{\max}}\\ \end{array}}}\right.$

Thus, the spatially-reachable range of the search space for a railway can be illustrated as in Fig. 5.

Finally, considering different railway structures’ characteristics, the reachable ranges can be divided according to elevation difference ( $\Delta h$ ) between every voxel and its corresponding ground elevation. Specifically, using the threshold values for fill-bridge height ( $h_{fb}$ ), general bridge-high bridge height ( $h_{b}$ ), maximum allowable bridge height ( $h_{b\max}$ ), cut-tunnel depth ( $d_{ct}$ ), shallow buried-normal buried tunnel depth ( $d_{sn}$ ) and normal buried-deep buried tunnel depth ( $d_{nd}$ ), the reachable ranges can be divided as shown in Fig. 6. The threshold values can be found in Pu et al. [39] and will be explained in Section 3. Hence, different environmental suitability analysis methods can be proposed for different parts in the reachable ranges, with details provided in the following section.

3. Environmental suitability evaluation

In this study, environmental suitability indicates how well an alignment can be constructed and operated within specific surrounding environmental conditions. Thus, higher environmental suitability yields lower costs of construction, operation and other impacts for traversing alignments. After obtaining the spatially reachable ranges of railway alignments, the environmental suitability evaluation model can be developed. However, this is a complicated problem since:

(1)
Numerous environmental factors, such as topography and geology, should be considered in a railway alignment design process.
(2)
Some evaluations of these factors, such as regarding geologic hazards, are difficult to conduct quantitatively and depend heavily on experts’ experience in current real-world designs, which involves great subjectivity.
(3)
Moreover, the environmental impacts of alignments must be evaluated in combination with specific railway structures (as seen in Fig. 6). However, different structures are determined dynamically during the search process and have different characteristics with respect to environmental inputs. (For example, tunnels have significantly different vulnerability to landslide hazards compared to bridges, cuts and fills.) Thus, it is challenging to proactively assess railway structures’ environmental suitability.

To tackle the above problems, a 3-D environmental suitability analysis program is developed step-by-step while considering topographic and geologic impacts of railway alignments.
3.1 Topographic factors

For topography, cost and stability are two dominating factors that are closely related to terrain conditions. Usually, to overcome the complex terrain, expensive bridges and tunnels are required along an alignment and, hence, increase its cost. Thus, cost control is a major factor for railway alignment design influenced by topography. In addition, structural stability is also affected by topographic factors, such as the slope angle, slope direction, terrain relief and roughness, as previously discussed in Song et al. [7]. Therefore, the topographic influences on alignments are evaluated by considering these two objectives: cost $F_{c}$ , which is computed considering railway structures’ construction investments, and stability $F_{s}$ , which is converted into a proxy related to the design elevations of structures. Formulations of these two objectives will be determined for different structures.

3.1.1 Cut and fill

As shown in Fig. 7, the cost and stability of an earthwork section on the surface can be estimated as follows:

Figure 7.

A fill section.

(1) Cost

$\displaystyle F_{c}=U_{E}\cdot V_{cf}$ (4)

where $U_{E}=$ unit cost of cut or fill volumes ( $\yen$ m ${}^{3}$ ); $V_{cf}=$ cut or fill volume (m ${}^{3}$ ), which is computed as:

$\displaystyle V_{cf}=w_{v}\cdot S_{c}$ (5) $\displaystyle S_{c}=\left({l_{w}+\tan\varphi\cdot\left|{\Delta h}\right|}% \right)\cdot\left|{\Delta h}\right|$ (6) $\displaystyle\quad\qquad\!\!\!\!-\sqrt{p\cdot\left({p-a}\right)\cdot\left({p-b% }\right)\cdot\left({p-c}\right)}$ $\displaystyle\left\{{{\begin{array}[]{*{20}c}{p=\frac{1}{2}\cdot\left({a+b+c}% \right)}\\ {\frac{a}{\sin\varphi}=\frac{b}{\sin\beta}=\frac{c}{\sin\left({\pi-\varphi-% \beta}\right)}}\\ {c=l_{w}+2\cdot\tan\varphi\cdot\left|{\Delta h}\right|}\\ \end{array}}}\right.$ (7)

where $w_{v}=$ grid width (m); $S_{c}=$ cross section area (m ${}^{2}$ ); $l_{w}=$ earthwork section width (m); $\varphi=$ slope ratio angle ( ${}^{\circ}$ ); $\Delta h=$ elevation difference between railway design elevation and ground elevation (m); $a$ , $b$ , $c$ and $p$ are length-related parameters that can be computed with Eq. (7); $\beta=$ grid slope gradient ( ${}^{\circ}$ ). It is noteworthy that since detailed alignment locations are unknown at the environmental suitability analysis stage, the above cost-related analysis involves many uncertainties. In this condition, a worst-case philosophy is usually adopted for simplifying the problem [40]. Thus, in this analysis, the railway direction is assumed to be perpendicular to the slope aspect since that may maximize $\Delta h$ and, hence, earthwork volumes, as discussed in Song et al. [7].

(2) Converted stability

It is important to note that, at a macro alignment design stage, detailed engineering information of railway structures, such as the embankment material, bridge beam form and tunnel lining construction, are often unavailable. The available data at this stage mainly include the structures’ locations, design elevations and distributions along the alignment. Thus, it is reasonable to simplify the stability analysis by using these available parameters to conduct a proactive environmental suitability before the alignment search.

Figure 8.

Influences of (a) elevation difference $\Delta h$ and (b) slope gradient $\beta$ on an earthwork section’s stability.

There are two main factors influencing the stability of earthwork sections, i.e., $\Delta h$ and $\beta$ (Fig. 8). As $\Delta h$ and $\beta$ increase, the stability of a cut or fill decreases [7]. Thus, Eq. (8) is proposed as a proxy to estimate earthwork sections’ stability. The structural stability is enhanced as this proxy increases.

$\displaystyle F_{s}=\frac{\cos\beta}{|{\Delta h}|}$ (8)

3.1.2 Tunnel

(1) Cost

According to Section 2.2, tunnels can be divided into three types, that is, shallow buried, normal buried and deep buried sections. Different cost functions are used for these three types of tunnel sections:

$\displaystyle F_{c}=\left\{{\begin{array}[]{l}c_{t1}\cdot w_{v}\\ \quad d_{ct}<|{\Delta h}|\leqslant d_{sn}\\ \left[{\frac{c_{t2}-c_{t1}}{d_{nd}-d_{sn}}\cdot({|{\Delta h}|-d_{sn}})+c_{t1}}% \right]\cdot w_{v}\\ \quad d_{sn}<|{\Delta h}|<d_{nd}\\ c_{t2}\cdot w_{v}\\ \quad|{\Delta h}|>d_{nd}\\ \end{array}}\right.$ (9)

where $c_{t1}$ , $c_{t2}=$ unit tunnel costs for shallow buried and deep buried sections, respectively ( $\yen$ /m).

(2) Converted stability

The stability of shallow buried tunnel is evaluated with the uniformly distributed loosening pressure $q$ . Based on the Code for Design of Railway Tunnel [41], the formula of $q$ is as follows:

$\displaystyle q=\left\{{\begin{array}[]{l}\gamma\cdot|{\Delta h}|\\ \quad|{\Delta h}|<h_{q}\\ \gamma\cdot|{\Delta h}|\cdot\left({1-\frac{\lambda\cdot\tan\theta}{B}\cdot|{% \Delta h}|}\right)\\ \quad h_{q}<|{\Delta h}|<d_{sn}\\ \end{array}}\right.$ (10)

where $\gamma$ , $\lambda$ , and $\theta=$ geotechnical parameters; $B$ and $h_{q}=$ tunnel structural parameters. These can be directly determined according to the grade of surrounding rocks ( $R$ ) and CDRT [41].

Thus, the proxy of a shallow tunnel’s stability can be formulated as follows:

$\displaystyle F_{s}=R\cdot\Delta h$ (11)

The stability of deep buried tunnel is estimated with Eq. (12), according to Rajinder et al. [42]. The tunnel stability is enhanced as the result of this equation decreases.

$\displaystyle\frac{\sigma_{\theta}}{\sigma_{cm}}=\frac{2\cdot P}{\sigma_{cm}}=% \frac{2\cdot\gamma\cdot\Delta h}{0.7\cdot\gamma\cdot R^{\raise 3.01pt\hbox{$1$% }\!\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}\!\lower 3.01pt\hbox{$3$}}}$ (12)

Thus, the proxy of a deep tunnel’s stability can be formulated as:

$\displaystyle F_{s}=\frac{R^{\raise 3.01pt\hbox{$1$}\!\mathord{\left/{% \vphantom{13}}\right.\kern-1.2pt}\!\lower 3.01pt\hbox{$3$}}}{\left|{\Delta h}% \right|}$ (13)

Overall, the proxy of a tunnel’s stability is:

$\displaystyle F_{s}=\left\{{{\begin{array}[]{ll}R\cdot\left|{\Delta h}\right|&% \left|{\Delta h}\right|<d_{sn}\\ \frac{R^{\raise 3.01pt\hbox{$1$}\!\mathord{\left/{\vphantom{13}}\right.\kern-1% .2pt}\!\lower 3.01pt\hbox{$3$}}}{\left|{\Delta h}\right|}&\left|{\Delta h}% \right|>d_{sn}\\ \end{array}}}\right.$ (14)

3.1.3 Bridge

(1) Cost

Based on Section 2.2, three conditions should be considered for bridges according to $\Delta h$ , i.e. general bridges, high bridges and bridge-unreachable ranges. The different cost functions for these three sections are as follows:

$\displaystyle F_{c}=\left\{{{\begin{array}[]{l}c_{b1}\cdot w_{v}\\ \quad h_{fb}<\Delta h<h_{b}\\ \left[{\frac{c_{b2}-c_{b1}}{h_{b\max}-h_{b}}\cdot({\Delta h-h_{b}})+c_{b1}}% \right]\cdot w_{v}\\ \quad h_{b}<\Delta h<h_{b\max}\\ \infty\\ \quad\Delta h>h_{b\max}\\ \end{array}}}\right.$ (15)

where $c_{b1}$ , $c_{b2}=$ unit costs of general and high bridges, respectively ( $\yen$ /m).

(2) Converted stability

For bridge stability analysis, the displacement-based failure criterion is widely used in practice [43, 44]. Thus, in this paper, the drift ratio capacity $d$ is used to measure bridge stability:

$\displaystyle d=\frac{H^{2}}{\varsigma}$ (16)

where $H=$ pier height (m); $\varsigma=$ a mechanical parameter related to structural details and assumed to be a constant at the alignment design stage.

Through the above analysis, the proxy of a bridge’s stability is defined as:

$\displaystyle F_{s}=\frac{1}{\Delta h^{2}}$ (17)

Overall, the preliminary cost and stability functions are derived below. They will be further modified in Section 3.3.

$\displaystyle F_{c}=\left\{{\begin{array}[]{l}\infty\\ \quad\Delta h>h_{b\max}\\ \left[{\frac{c_{b2}-c_{b1}}{h_{b\max}-h_{b}}\cdot\left({\Delta h-h_{b}}\right)% +c_{b1}}\right]\cdot w_{v}\\ \quad h_{b}<\Delta h<h_{b\max}\\ c_{b1}\cdot w_{v}\\ \quad h_{fb}<\Delta h<h_{b}\\ U_{E}\cdot w_{v}\cdot S_{c}\\ \quad-d_{ct}<\Delta h<h_{fb}\\ c_{t1}\cdot w_{v}\\ \quad-d_{sn}<\Delta h<-d_{ct}\\ \left[{\frac{c_{t2}-c_{t1}}{d_{nd}-d_{sn}}\cdot\left({\left|{\Delta h}\right|-% d_{sn}}\right)+c_{t1}}\right]\cdot w_{v}\\ \quad-d_{nd}<\Delta h<-d_{sn}\\ c_{t2}\cdot w_{v}\\ \quad\Delta h<-d_{nd}\\ \end{array}}\right.$ (18) $\displaystyle F_{s}=\left\{{{\begin{array}[]{ll}\frac{1}{\Delta h^{2}}&\Delta h% >h_{fb}\\ \frac{\cos\beta}{\left|{\Delta h}\right|}&-d_{\mbox{ct}}<\Delta h<h_{fb}\\ R\cdot\left|{\Delta h}\right|&-d_{nd}<\Delta h<-d_{{ct}}\\ \frac{R^{\raise 3.01pt\hbox{$1$}\!\mathord{\left/{\vphantom{13}}\right.\kern-1% .2pt}\!\lower 3.01pt\hbox{$3$}}}{\left|{\Delta h}\right|}&\Delta h<-d_{nd}\\ \end{array}}}\right.$ (19)

3.2 Geologic factors

Alignments may inevitably traverse geologic hazard regions, which creates considerable potential risks [45, 46]. However, conducting geologic assessments is complicated due to the uncertainties of alignment optimization [4, 40]. In this paper, landslide hazard $F_{h}$ and seismic risk $F_{r}$ are considered to evaluate the geologic impacts of railways. These two factors are selected because they are the most typical geo-hazards that usually cause great losses during the construction and operation of a railway.

3.2.1 Landslide impacts

Studies relevant to landslide hazard are rare in the literature. In this paper, two models proposed in Song et al. [45] and Pu et al. [39] are employed to evaluate landslide impacts. Specifically, Pu et al. [39] established a landslide zonation model to classify the study area into three kinds of subregions, namely landslide outbreak regions, buffer regions and fuzzy regions. For grids in different kinds of subregions, different equations have been proposed to estimate landslide hazards, as shown below:

$\displaystyle F_{O}=\omega_{S}\cdot\omega_{V}$ (20) $\displaystyle F_{B}=\frac{1}{2}\cdot m\cdot v^{2}$ (21) $\displaystyle F_{IV}=\sum{IV}$ (22)

Figure 9.

Diagrams for normalized factors of (a) cost; (b) stability; (c) landslide hazard; (d) seismic risk and (e) remoteness of the point $P$ in Fig. 14.

where $F_{O}=$ a grid’s landslide hazard value in outbreak regions, obtained by multiplying a geohazard susceptibility weight ( $\omega_{S}$ ) and a structural vulnerability weight ( $\omega_{V}$ ), with details provided in Song et al. [45]; $F_{B}=$ a grid’s landslide hazard value in buffer regions that computed based on kinetic energy theory by considering a landslide’s mass ( $m$ ) and moving velocity ( $v$ ), with details provided in Pu et al. [7]; $F_{IV}=$ a grid’s landslide hazard value in fuzzy regions, which is the sum of so-called information values (IVs, [39]) considering the contributions of different environmental parameters to landslide occurrences, such as those in the Normalized Difference Vegetation Index (NDVI, which reflects the degree of land vegetation coverage, as explained in Zhang et al. [8]) and lithology. After obtaining $F_{O}$ , $F_{B}$ and $F_{IV}$ , they are normalized for combination. It should be noted that, during the above analysis, different structures are evaluated with different vulnerability functions with respect to landslide hazards. Thus, the landslide hazards for different structures are computed separately as discussed in Pu et al. [39] and explained in Section 3.3.

3.2.2 Seismic impacts

Currently, many railway projects are located in earthquake-prone regions and hence are greatly threatened by seismic activity. In this field, Song et al. [47] developed the first known probabilistic seismic risk analysis model for railway alignment optimization. Inspired by that model, a structural damage probability $F_{r}$ is derived in this work to measure the voxels’ seismic risks:

$\displaystyle F_{r}=\sum\limits_{i=1}^{2}\{P_{f}[{DS_{i}({h,\textit{PGA}})}]{}% \cdot R[{DS_{i}({h,\textit{PGA}})}]\}$ (23)

where $DS_{i}(\cdot)=$ damage state function ( $i=$ 1: moderate; $i=$ 2: extensive), which represents structural damage degree under the influences of railway design elevation $h$ and peak ground acceleration PGA; $P_{f}(\cdot)=$ damage probability function which is obtained by using fragile curves that are differently defined for different structures, with more details provided in Song et al. [47]; $R(\cdot)=$ damage ratio function defined in HAZUS [48] for different structures based on their specific damage states.

3.3 Multi-criteria decision-making analysis

Besides the above topographic and geologic factors, another engineering factor, i.e., the degree of remoteness ( $F_{d}$ ) which reflects the perpendicular distance of every spatial voxel to the straight line between the start and end points, is also considered for environmental suitability of railway alignments. This factor is considered because railways should generally avoid excessive circuity in order to reduce cost and facilitate operations. Thus, a total of five factors, namely $F_{c}$ , $F_{s}$ , $F_{h}$ , $F_{r}$ and $F_{d}$ are involved in the proposed model.

To combine these factors for a comprehensive environmental suitability analysis, they should first be nondimensionalized [8], as shown below:

$\displaystyle x^{\prime}=\gamma\cdot\frac{x-x_{\min}}{x_{\max}-x_{\min}}$ (24)

where $x^{\prime}=$ nondimensionalized value; $\gamma=$ structural coefficient. It is set at 1, 2 and 3, respectively, for earthwork sections, bridges and (normal & deep buried) tunnels, based on the importance of different structures according to the empirical results in Song et al. [45] and Pu et al. [39]. The stability of shallow buried tunnels is enhanced with greater buried depth and, thus, $\gamma$ is computed as $1+2\cdot\Delta h/d_{sn}$ ; $x=$ a voxel’s environmental suitability value. The nondimensionalized factors of a real grid in the case study section (i.e., point $P$ in Fig. 14) are illustrated in Fig. 9. Afterward, the five factors can be integrated to obtain the comprehensive environmental suitability through a multi-criteria decision-making (MCDM) analysis [49, 50, 51, 52]. Since different factors affect environmental suitability of railways in varied degrees, criteria weights must be determined for different factors. In this study, the CRITIC (criteria importance through inter-criteria correlation [53]) method is adopted to determine such criteria weights.

CRITIC is a widely used method to determine the weights of different criteria during an MCDM process [54, 55]. This method includes two indicators, i.e., fluctuation and correlation. They are combined to obtain the criteria weight $\omega_{c}$ :

$\displaystyle\omega_{c}=\sigma_{f}\cdot({1-\rho})$ (25)

where $\sigma_{f}=$ each criterion’s standard deviation to reflect the fluctuation of data; $\rho=$ a correlation coefficient between a criterion and every other one, which represents a criterion’s differentiation among other criteria.

Then, the comprehensive environmental suitability value of each voxel can be computed as:

$\displaystyle V_{ES}=\sum\limits_{i=1}^{5}{\omega_{ci}}\cdot x_{i}^{\prime}$ (26)

After obtaining the environmental suitability values of all voxels in the search space, these voxels can be classified into three grades (i.e., I, II and III) based on the actual railway design requirements. It should be noted that, under difficult circumstances, some local regions in the study area may be entirely comprised of Grade III voxels that are unsuitable for railway alignments to traverse, if the global search space is directly classified into three grades. Consequently, all these low-grade voxels are inevitably checked during the search, which wastes considerable computational time and resources. Hence, in order to avoid an indiscriminate search in such a situation, the reachable range is further divided into several equal sub-parts along the straight line between start and end points (as shown in Fig. 10). Afterward, instead of classifying the global search space into three grades, the voxels in each sub-part are divided into three grades.

Figure 10.

Boundaries of sub-parts.

Figure 11.

Example environmental suitability curves of points (a) $P$ and (b) $Q$ in Fig. 14.

Hence, a 3-D graded suitability map can be generated and employed to assist the alignment search process. To demonstrate the relation between a voxel’s suitability grade and $\Delta h$ , Fig. 11 is provided based on the information for two real grids (i.e., points $P$ and $Q$ in Fig. 14).

3.4 Alignment optimization

Alignment design is essentially a shortest path problem. In this regard, the 3D-DT (3-dimensional distance transform) proposed in Pu et al. [33] is a typical shortest path algorithm that has been successfully applied to solve alignment optimization problems. However, its efficiency is sensitive to the numbers of voxels, especially for the previous alignment optimization mode illustrated in Fig. 2. Thus, it is desirable to improve its search efficiency with the proposed model. It is also worth noting that there are also various nature-inspired optimization approaches [56] such as the neural dynamic algorithm [57], evolutionary algorithm [58, 60], and bacteria foraging algorithm [59] that have potentials for solving alignment optimization problems. However, to apply these method, considerable efforts are required to customize them with specific constraint-handling and solution evaluation operators. In this study, the major focus is to develop an environmental suitability model for alignment optimization and, thus, a previous 3D-DT is adopted to avoid repeated and time-consuming works.

Figure 12.

Global paths generated by 3D-DT through a bi-directional scan.

Figure 13.

Topographic map of the study area.

Figure 14.

Geologic map of the study area.

Figure 15.

Lithologic map of the study area.

The basic flow of the 3D-DT is to propagate and accumulate local paths to approximate the global paths by using a bidirectional scanning strategy (Fig. 12). Specifically, the start and end points in the landscape are defined as target points. Then, the bidirectional scan is conducted for all voxels from the start to end and inversely, from end to start for generating two sets of local paths to the two targets respectively. Finally, for the voxels that can reach the start and end points concurrently, their corresponding local paths are accumulated as global paths to connect start and end points. All the global paths are thus defined as a path set $\Phi$ . Traditionally, all voxels, including the alignment-unfavorable ones, in the study area must be scanned by the 3D-DT, which is very computationally-intensive. Therefore, in this study, the 3-D environmental suitability map is integrated into the previous 3D-DT to form an ES-3D-DT algorithm to improve search quality and efficiency.

The scan priority of ES-3D-DT is first Grade I voxels, and then Grade II. Only when no Grades I and II voxels are found are Grade III voxels scanned. During optimization, cost, structural stability, landslide hazard, seismic risk and remoteness are considered as objective functions. Hence, a multi-criteria tournament decision (MTD) method is also used to evaluate the generated paths.

The MTD is a multi-criteria analysis method devised by Parreiras and Vasconcelos [61]. In this method, a tournament function $T_{j}(\cdot)$ and a ranking function $R_{a}(\cdot)$ are required. In Eq. (27) $T_{j}(\cdot)$ presents the superior and inferior relations between path $a$ and every other path in the path set $\Phi$ regarding objective $j$ . $t_{j}(a,b)$ is the pair-wise comparison which equals 1 (or 0) when $a$ is superior (or inferior) to another solution regarding objective $j$ . According to Eq. (28), $R_{a}(\cdot)$ computes a ranking index by summing solution $a$ ’s $T_{j}(\cdot)$ values based on the criteria weights computed by CRITIC. Then the best path (with highest rank) can be selected. It should be noted that the best path is comprised of piecewise linear tangents without curves, which should be refined into an alignment with detailed curve and tangent sections [31]. However, the fitted alignment will deviate from the original path, which may decrease solution quality or even make the resulting alignment infeasible especially in complex mountainous regions. Thus, a customized PSO [62] is employed to further optimize the fitted alignment.

$\displaystyle\left\{{\begin{array}[]{l}T_{j}({a,\Phi})=\sum\limits_{b\in\Phi% \cap b\neq a}{\frac{t_{j}({a,b})}{\|\Phi\|-1}}\\ t_{j}({a,b})=\left\{{\begin{array}[]{ll}1&f_{j}(a)>f_{j}(b)\\ 0&f_{j}(a)<f_{j}(b)\\ \end{array}}\right.\\ \end{array}}\right.$ (27) $\displaystyle R_{a}=\left({\prod\limits_{j=1}^{5}{T_{j}^{\omega_{j}}}}\right)^% {\raise 3.01pt\hbox{$1$}\!\mathord{\left/{\vphantom{15}}\right.\kern-1.2pt}\!% \lower 3.01pt\hbox{$5$}}$ (28)

The above 3-D environmental suitability analysis model and ES-3D-DT solution method will be applied to the following complex real-world railway cases to assess their effectiveness.

4. Case study I

4.1 Case profile

The Kangding Section of the well-known Sichuan-Tibet Railway, which is one of the most complicated railway projects in the world, is used here as a real-life case example. The linear distance and elevation difference between the start and end points are 41,996 m and 741 m, respectively. In the study area there are 63 existing landslide zones and 14 active faults. The PGA value varies from 0.2 g to 0.4 g in this region. The needed data for railway design are provided by the China Railway Eryuan Survey and Design Group Co. Ltd. The topographic, geo-hazard, lithologic and NDVI maps of the study area are shown in Figs 13–16, respectively. The unit costs are shown in Table 1.

Table 1
Unit costs of case I

Item	Value	Item	Value
Rail track ( $\yen$ /m)	4,000	Right-of-way ( $\yen$ /m ${}^{2}$ )	70
Fill earthwork ( $\yen$ /m ${}^{3}$ )	20	Cut earthwork ( $\yen$ /m ${}^{3}$ )	22
(L $<$ 500 m) Bridge (10 ${}^{4}$ $\yen$ /m)	4.35	(L $>$ 500 m) Bridge (10 ${}^{4}$ $\yen$ /m)	7.5
(L $<$ 10 km) Tunnel (10 ${}^{4}$ $\yen$ /m)	7.33	(L $>$ 25 km) Tunnel (10 ${}^{4}$ $\yen$ /m)	17
One bridge abutment (10 ${}^{4}$ $\yen$ )	20	One tunnel portal (10 ${}^{4}$ $\yen$ )	20

Table 2

Data interpretations for Fig. 17b

Value	Color	Interpretation	Fraction
0	Blue	Unreachable area	N/A
1	Green	Can only be traversed by tunnels	90.9%
2	Orange	Can be traversed by tunnels and bridges	1.5%
3	Red	Can be traversed by tunnels, bridges and earthwork sections	7.6%

Figure 16.

NDVI map of the study area.

Figure 17.

Illustrations of (a) The stereo terrain map and (b) its corresponding horizontally reachable range.

4.2 Results

4.2.1 Railway spatially-reachable ranges

According to Eq. (1), $\gamma_{\max}$ is set at 1.05. The stereo terrain condition and its horizontally reachable range are shown in Fig. 17. Figure 17b shows the possible distributions of different structures in the horizontally reachable range. The detailed interpretations can be found in Table 2. It can be observed that, due to the pervasiveness of high mountains, 90.9% of the study areas can only be traversed by tunnels. The exceptions are a few areas near the start and end points. Then, according to Eqs (2) and (2.2), the vertically reachable range can be computed and the final spatially reachable region is provided in Fig. 18.

4.2.2 Spatial environmental suitability

The spatial distribution of the voxels’ environmental suitability values is shown in Fig. 19. It can be found that the values are clustered in a small range, i.e., 2.5–3.0, in several sub-parts. Thus, alignment search in these regions may be limited by low efficiency. Therefore, these values need to be classified into Grade I, II and III according to a specific ratio, as explained in Section 3.3. Subsequently, the railway reachable range is separated into four sub-parts along the straight line between the start and end points. In each sub-part, the ratio of Grade I, II and III voxels is set at 4 : 4 : 2. The produced 3-D environmental suitability map is shown in Fig. 20.

Furthermore, as shown in Table 3, it can be found that Grade I voxels are largely concentrated in the elevation ranges of 3,100–3,450 m and 3,450–3,800 m, accounting for 50.7% and 59.7% of total voxels, respectively. However, Grade I voxels merely account for 10.4% of the total voxels in the range between 2,400 m and 2,750 m. Therefore, in this railway case, the elevation ranges of 3,100–3,450 m and 3,450–3,800 m are the most suitable ranges for alignments to traverse while the elevation range of 2,400–2,750 m is the worst. Based on this 3-D environmental suitability map, the search space of 3D-DT can be effectively narrowed by screening out low-grade voxels.

Table 3
Fractions of Grade I, II and III voxels in different elevation ranges

Elevation range (m)	Grade I	Grade II	Grade III
2,400–2,750	10.4%	40.8%	48.8%
2,750–3,100	29.6%	49.6%	20.8%
3,100–3,450	50.7%	37.5%	11.8%
3,450–3,800	59.7%	18.5%	21.8%

Figure 18.

Spatially reachable region.

Figure 19.

Distribution of all voxels’ environmental suitability values.

Figure 20.

3-D environmental suitability map.

Figure 21.

Search time of 3D-DT and ES-3D-DT at resolutions of 30 m, 60 m and 90 m.

Table 4

Running time and generated paths of 3D-DT and ES-3D-DT based on different grid resolutions

Resolution (m)	Method	Suitability analysis	Search time	Speed-up ratio	Solution numbers
90	3D-DT	N/A	469 s	N/A	68
	ES-3D-DT	24.6 s	108 s	4.3	62
60	3D-DT	N/A	1,581 s	N/A	72
	ES-3D-DT	90.0 s	394 s	4.0	67
30	3D-DT	N/A	13,693 s	N/A	73
	ES-3D-DT	240.2 s	4,621 s	3.0	69

Table 5

Detailed comparisons among A ${}_{\text{M}}$ , A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$

Item	A ${}_{\text{M}}$	A ${}_{\text{3D-DT}}$	A ${}_{\text{ES-3D-DT}}$
Railway length (m)	43,600	42,711	41,441
Right-of-way area (m ${}^{2}$ )	210,663	240,354	236,723
Embankment/excavation volume (m ${}^{3}$ )	27,905/99,374	25,254/131,636	36,492/129,934
Number of bridges-Total length (m)	5–1,600	1–40	1–62
Number of tunnels-Total length (m)	5–40,735	3–40,621	4–39,349
Total construction cost (million $\yen$ )	6,442	6,137	6,071
Improvement	N/A	4.7%	5.8%
Landslide hazard	30.2	25.6	24.9
Improvement	N/A	15.5%	17.7%
Seismic risk	115.6	75.9	80.9
Improvement	N/A	34.4%	30.0%
Stability	250.5	259.6	262.3
Improvement	N/A	3.6%	4.7%
Remoteness	69.4	60.1	53.6
Improvement	N/A	13.4%	22.8%

4.2.3 Improvements of search efficiency

The grid resolution is a key factor affecting the execution efficiency of 3D-DT [33]. Referring to the Quality Requirement for Digital Surveying and Mapping Achievements (State Bureau of Surveying and Mapping of China [63]), the grid resolution between 25 and 100 m for 3D-DT search can satisfy the requirement of alignment design. Thus, three groups of experiments based on grid resolutions of 90 m, 60 m and 30 m are conducted to compare the performances of 3D-DT and ES-3D-DT. All programs are run on a personal computer with Intel Core i7-7700 CPU @ 3.60 GHz and RAM of 16 G. The comparison results are provided in Table 4 and Fig. 21.

It can be found that the search time of 3D-DT increases exponentially from 469 s to 13,693 s with the increase of the grid resolution from 90 m to 30 m. Thus, the search time increases by a factor of 29.2 due to the increase of scanned voxels. Compared to 3D-DT, with the implementation of environmental suitability evaluation to screen out low-grade voxels, the speed-up ratios of ES-3D-DT reach 4.3, 4.0 and 3.0, respectively, for grid resolutions of 90 m, 60 m and 30 m. In addition, the suitability analysis time is only 24.6 s, 90.0 s and 240.2 s, respectively, for grid resolutions of 90 m, 60 m and 30 m. This is acceptable compared with the time saved by ES-3D-DT search. Moreover, compared to the previous 3D-DT, the numbers of solutions found by ES-3D-DT are only reduced by 6, 5 and 4, respectively, for grid resolutions of 90 m, 60 m and 30 m. This shows that the automated alignment optimization is not significantly degraded by skipping low-grade voxels in the search space. Thus, the above analysis reveals the effectiveness of the proactive environmental suitability model in improving performances of alignment search algorithms.

4.2.4 Alignment solutions

Lastly, the best alternatives produced respectively by experienced designers in the China Railway Eryuan Engineering Group Co. Ltd (A ${}_{\text{M}}$ ), 3D-DT (A ${}_{\text{3D-DT}}$ ) and ES-3D-DT (A ${}_{\text{ES-3D-DT}}$ ) are compared. The detailed results are listed in Table 5. The horizontal and vertical alignments of A ${}_{\text{M}}$ , A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ are shown in Figs 22–25.

Figure 22.

Horizontal alignments of A ${}_{\text{M}}$ , A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ .

Figure 23.

The vertical alignment of A ${}_{\text{M}}$ .

Figure 24.

The vertical alignment of A ${}_{\text{3D-DT}}$ .

Figure 25.

The vertical alignment of A ${}_{\text{ES-3D-DT}}$ .

Figure 26.

Fractions of A ${}_{\text{M}}$ , A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ in Grade I, II and III voxels. The horizontal background is the highest suitability grade of every grid considering its corresponding spatial voxels.

Table 6

Alignment fractions in different grades of voxels

Fraction	A ${}_{\text{M}}$	A ${}_{\text{3D-DT}}$	A ${}_{\text{ES-3D-DT}}$
I	80.3%	83.2%	84.7%
II	14.2%	10.9%	12.2%
III	5.5%	5.9%	3.1%

Compared with A ${}_{\text{M}}$ the railway lengths of A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ decrease by (43,600 $-$ 42,711) $=$ 889 m and (43,600 $-$ 41,441) $=$ 2,159 m, respectively. Moreover, the total tunnel lengths are 40,735 m, 40,621 m and 39,349 m, respectively, for A ${}_{\text{M}}$ , A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ . Compared with A ${}_{\text{M}}$ , the total bridge lengths also decrease by 1,560 m and 1,538 m, respectively, for A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ . Thus, by shortening the expensive tunnels and bridges, the construction cost of A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ can be reduced by 4.7% and 5.8%, respectively.

Regarding landslide hazard, by traversing regions less affected by landslides, the corresponding values are decreased by 15.5% and 17.7%, respectively, for A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ , compared to the manual solution. For seismic risk, the values of A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ are decreased by 34.4% and 30.0%, respectively, compared to A ${}_{\text{M}}$ . This occurs mainly due to the reduced railway lengths in severe seismic zones (i.e., PGA $\geqslant$ 0.4 g) for A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ . Specifically, the railway lengths in such zones are 9,819 m, 8,784 m and 9,103 m, respectively, for A ${}_{\text{M}}$ , A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ .

Also, the remoteness values of A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ are improved by 13.4% and 22.8%, respectively, compared to A ${}_{\text{M}}$ . This occurs because A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ are closer to the straight line between the start and end points. The corresponding values for stability are improved by 3.6% and 4.7%, respectively, for A ${}_{\text{3D-DT}}$ and A ${}_{\text{ES-3D-DT}}$ compared to A ${}_{\text{M}}$ . This occurs because the high bridges, high fills and deep cuts are effectively reduced by the computer-generated alternatives.

Ultimately, the fractions of different alignments in Grade I, II and III regions are computed, as shown in Table 6 and Fig. 26. It can be found that A ${}_{\text{ES-3D-DT}}$ traverses the highest fraction of Grade I voxels and the lowest fraction of Grade III voxels compared to both A ${}_{\text{M}}$ and A ${}_{\text{3D-DT}}$ .

Therefore, the above results confirm that the developed model and method are practicable and effective for solving real alignment optimization problems.

5. Case study II

5.1 Case profile

Here, the Yingxiu-Wolong Railway Section which is located in a complicated mountainous region with drastically undulating terrain, is used as the second case study. The linear distance between the start and end points is 28,204 m. There are also 25 existing landslide zones and 4 active faults in the study area. The PGA value varies from 0.15 g to 0.2 g in this region. The stereo terrain map is shown in Fig. 27. The cost information is provided in Table 7.

Figure 27.

The stereo terrain map.

Table 7

Unit costs of case II

Item	Value	Item	Value
Rail track ( $\yen$ /m)	4,000	Right-of-way ( $\yen$ /m ${}^{2}$ )	90
Fill earthwork ( $\yen$ /m ${}^{3}$ )	18	Cut earthwork ( $\yen$ /m ${}^{3}$ )	20
(L $<$ 500 m) Bridge (10 ${}^{4}$ $\yen$ /m)	4.8	(L $>$ 500 m) Bridge (10 ${}^{4}$ $\yen$ /m)	5.8
(L $<$ 5 km) Tunnel (10 ${}^{4}$ $\yen$ /m)	6.5	(L $>$ 5 km) Tunnel (10 ${}^{4}$ $\yen$ /m)	7.33
One bridge abutment (10 ${}^{4}$ $\yen$ )	20	One tunnel portal (10 ${}^{4}$ $\yen$ )	20

Table 8

Results on changing the grade ratios

Group	Ratio	Search time	Solution number
Group I	4 : 4 : 2	474 s	46
Group II	3 : 4 : 3	383 s	45
Group III	2 : 4 : 4	372 s	34

Figure 28.

Environmental suitability maps generated in groups I (a), II (b) and III (c).

5.2 Results

5.2.1 Determining environmental suitability grades

The ratio of Grade I, II and III is a value that should be analyzed in specific cases. To guarantee the solution quantity and quality, the fraction of Grade I voxels should generally be large. However, that may degrade search efficiency. To test detailed impacts, three groups of ratio settings (labelled as Group I, II and III) of 4 : 4 : 2, 3 : 4 : 3 and 2 : 4 : 4 respectively, are used for Grade I, II and III voxels in this experiment to generate three environmental suitability maps (Fig. 28). Then, based on these maps, the ES-3D-DT is run separately three times. The detailed experimental results are listed in Table 8.

It can be observed that the search time of ES-3D-DT increases slightly from 372 s to 383 s for Groups III and II whereas the number of solutions is significantly reduced by 11 in that condition. Thus, although the decrease of Grade I’s ratio accelerates the search efficiency, that also reduces the number of solutions. Moreover, the number of solutions is merely increased by 1 from Group II to Group I with notably increased search time. That means Group II is the most appropriate for this case.

Table 9
Detailed comparisons between A ${}_{\text{ES-GA}}$ and A ${}_{\text{ES-3D-DT}}$

Item	A ${}_{\text{ES-GA}}$	A ${}_{\text{R-ES-3D-DT}}$
Railway length (m)	33,517	32,011
Right-of-way area (m ${}^{2}$ )	185,476	175,493
Embankment/excavation volume (m ${}^{3}$ )	102,766/	190,816/
	128,056	207,382
Number of bridges-Total length (m)	3–2,733	3–3,903
Number of tunnels-Total length (m)	6–26,801	6–23,867
Total construction cost (million $\yen$ )	2,191	2,047
Improvement	N/A	6.6%
Landslide hazard	16.2	12.7
Improvement	N/A	21.6%
Seismic risk	45.4	42.1
Improvement	N/A	7.3%
Stability	182.8	201.7
Improvement	N/A	10.3%
Remoteness	65.1	60.8
Improvement	N/A	6.6%

Figure 29.

Horizontal alignments of A ${}_{\text{ES-GA}}$ and A ${}_{\text{R-3D-DT}}$ .

Figure 30.

The vertical alignment of A ${}_{\text{ES-GA}}$ .

Figure 31.

The vertical alignment of A ${}_{\text{ES-3D-DT}}$ .

Moreover it is noteworthy that the three groups of ES-3D-DT runs produce the same best alignment. This occurs because the ES-3D-DT is a deterministic shortest path algorithm and, thus, despite the changes in the number of solutions, the best alignment remains the same. To evaluate the quality of this alignment (denoted as A ${}_{\text{R-ES-3D-DT}}$ ), a GA developed in Li et al. [31] is adopted to produce another alignment (denoted as A ${}_{\text{ES-GA}}$ ) for comparison. The results are demonstrated in Table 9. The horizontal and vertical alignments of A ${}_{\text{R-ES-3D-DT}}$ and A ${}_{\text{ES-GA}}$ are shown in Figs 29–31. Detailed data interpretations are provided in the next section. Alignment solutions.

Compared with A ${}_{\text{ES-GA}}$ , the railway length of A ${}_{\text{R-ES-3D-DT}}$ is significantly reduced by 1,506 m. Moreover, A ${}_{\text{R-ES-3D-DT}}$ ’s total tunnel length is also reduced by 2,934 m. Thus, mainly by reducing these expensive tunnels, the construction cost of A ${}_{\text{R-ES-3D-DT}}$ is 6.6% below that of A ${}_{\text{ES-GA}}$ . Meanwhile, by shortening the alignment sections in landslide and earthquake regions, A ${}_{\text{R-ES-3D-DT}}$ ’s landslide hazard and seismic risk are also decreased by 21.6% and 7.3%, respectively, compared to A ${}_{\text{ES-GA}}$ . Also, A ${}_{\text{R-ES-3D-DT}}$ ’s remoteness value and stability are improved by 6.6% and 10.3%, respectively compared to those of A ${}_{\text{ES-GA}}$ . These occur because the ES-3D-DT found an alignment that is close to the straight line between the start and end points as well as reduces the deep-buried tunnel’s length to improve stability, as measured with Eq. (13). These results indicate that the deterministic shortest path algorithm (i.e., ES-3D-DT) performs better than the stochastic heuristic GA since the ES-3D-DT scans every feasible voxel in Fig. 28 and, hence, has larger search spaces than the GA.

Therefore, this case study further confirms that the developed model and method are effective and practical for assisting real-world design.

6. Conclusion

Railway alignment design is a complex problem affected by multiple environmental factors. Although numerous alignment optimization methods have been devised and successfully applied to many realistic cases, a common limitation among them is the lack of a proactive environmental suitability analysis to guide the alignment search process. Thus, regions unfavorable for railways may still be found and evaluated, which wastes considerable computational resources and also influences the optimization efficiency. To solve this problem, a 3-D environmental suitability model is developed to guide the railway design process using geographic information analyses.

In this model, the search space is initially abstracted as a voxel set. Then, the 3-D reachable ranges of railways are determined and divided based on different railway structures’ characteristics. Afterward, a geographic information analysis is designed to evaluate the environmental suitability of these reachable ranges by mainly considering topographic and geologic impacts. The topographic influences on the railway structures’ cost and stability are formulated. Regarding geology, landslide hazard and seismic risk are combined. Moreover, the degree of remoteness, which reflects spatial voxels’ perpendicular distance to the straight line between the start and end points, is also considered. To integrate these five factors, a multi-criteria decision-making (MCDM) method called CRITIC is adopted to generate a comprehensive environmental suitability map. The map is further incorporated into a previous 3D-DT algorithm to form an ES-3D-DT for automated alignment search. Finally, the above model and solution method are applied to two complicated real-world case. Through the experimental results of case I, it is found that:

(1)

Due to the ubiquity of high mountains, 90.9% of the study areas can only be traversed by tunnels in the spatially-reachable range. Furthermore, the elevation ranges of 3,100–3,450 m and 3,450-3,800 m are the most suitable ranges for alignments to traverse with largely-concentrated Grade I voxels.

(2)

Compared to 3D-DT, the speed-up ratios of ES-3D-DT are 4.3, 4.0 and 3.0, respectively, for grid resolutions of 90 m, 60 m and 30 m. Moreover, it is observed that, by narrowing the search space using the environmental suitability map, the number of solutions found by ES-3D-DT is not significantly influenced. Thus, the devised proactive environmental suitability model can improve the performance of alignment search algorithms.

(3)

Through comparison with the best manually-design solution, it is verified that both 3D-DT and ES-3D-DT can generate better alignments regarding the considered five environmental factors. Furthermore, the alternative produced by ES-3D-DT also outperforms the one yielded by 3D-DT. This occurs mainly because the alignment found by ES-3D-DT traverses the highest fraction of high-grade voxels in the search space.

Case II further confirms the practicability of the devised model by analyzing the impacts of the ratio of different environmental suitability grades on model performance. Besides, the ES-3D-DT is compared with a previous GA. It is found that the ratio of different environmental suitability grades is closely related to search efficiency and solution quantity but has no significant impact on the best computer-generated solution. Also, it is observed that the ES-3D-DT can generate better solutions than a GA.

Thus, the above analyses demonstrate the effectiveness of the proposed model and method in solving real-life railway alignment optimization problems.

In future studies, the devised environmental suitability model may be further improved by adding more geological factors (e.g., snow-slide and karst) and ecologic factors (e.g., climate, carbon emissions, floodplains [27] and wetlands [64]). Moreover, other nature-inspired approaches, such as spiral dynamics algorithm [65], water drop algorithms [66] and harmony search algorithm [67], may also be pursued as alternative solution methods for alignment optimization

Footnotes

Acknowledgments

This work is partially funded by the National Key R&D Program of China with award number 2021YFB2600403, the National Science Foundation of China (NSFC) with award number 52078497, the Central South University Special Scholarship for Study Abroad.

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A geographic information model for 3-D environmental suitability analysis in railway alignment optimization

Abstract

Keywords

1. Introduction

1.1 Problem statement

2. Preprocessing of environmental inputs

2.1 Presentation of the search space

3.1.1 Cut and fill

(1) Cost

(2) Converted stability

(1) Cost

(2) Converted stability

(1) Cost

(2) Converted stability

3.2.1 Landslide impacts

4.1 Case profile

Table 1 Unit costs of case I

4.2.1 Railway spatially-reachable ranges

4.2.2 Spatial environmental suitability

Table 3 Fractions of Grade I, II and III voxels in different elevation ranges

4.2.4 Alignment solutions

5.1 Case profile

5.2.1 Determining environmental suitability grades

Table 9 Detailed comparisons between A ES-GA and A ES-3D-DT

Footnotes

Acknowledgments

References

Table 1
Unit costs of case I

Table 3
Fractions of Grade I, II and III voxels in different elevation ranges

Table 9
Detailed comparisons between A ${}_{\text{ES-GA}}$ and A ${}_{\text{ES-3D-DT}}$