A Model for Optimizing Railway Alignment Considering Bridge Costs, Tunnel Costs, and Transition Curves
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Abstract
Owing to wideranging searches (there are various alignments between two points) as well as complex and nonlinear cost functions and a variety of geometric constraints, the problem of optimal railway alignment is classified as a complex problem. Thus, choosing an alignment between two points is usually done based on a limited number of alignments designed by experts. In recent years, the study of railway alignment optimization has shown the importance of optimization and the introduction of various algorithms and their usefulness in solving different problems. It is expected that applying metaheuristic optimization algorithms such as methods based on swarm intelligence can lead to better alignments. In this study, we tried to modify models based on previous studies in order to design and develop a model based on a single framework to provide threedimensional optimization of alignments applicable in the real world. To obtain this, the particle swarm algorithm is used and a geographic information system is incorporated as a means of search in threedimensional space. In particular, the cost function used in previous studies considering the costs related to structures (bridges and tunnels) are improved regarding hydraulic structure alignments. Furthermore, the transition curve of horizontal alignment and slope restrictions of curves are considered in this project by using the penalty function in order to obtain the most practical results possible. Finally, this study examines three problems for which the results are acceptable in cases of railway alignment geometry and its application in the real world.
Keywords
Optimization Railway alignment Particle swarm optimization (PSO) algorithm Transition curves Bridges and tunnels Geographic information systems1 Introduction
When solving problems in continuous search space, there are many alignment alternatives between two points, which are characterized by topography, land ownership, and environmental issues. The objective function of the problem depends on various cost factors, which include supplier costs, user costs and outofsystem costs [1].
There are a number of constraints in railway alignment design, which can be classified as Capacity and alignment traffic constraints, Horizontal alignment constraints and Vertical alignment constraints [2].
Given the existing complexities in railway alignment design, researchers have considered many simplifying assumptions for the model railway alignment problem to achieve the proper alignment. Generally, these simplifications have led to final model solutions that are not applicable or cannot approximate the optimal solution obtained from the actual models. This work aims to reduce the simplifying assumptions and to modify optimization methods for existing algorithms. Results show that the applicability of the solutions of the proposed model are improved.
The paper is organized as follows: After the introduction, a review of the literature on methods previously developed for optimizing railway and highway alignments is provided in Sect. 2. Major railway costs and constraints associated with railway construction are discussed in Sect. 3, and the methodology developed for the railway alignment optimization model is introduced. In Sect. 4, we describe how we represent the railway alignment optimization problem in particle swarm optimization algorithms (PSO), and we discuss our approach to modeling railway alignments and structures in Sect. 5. Section 6 describes a basic model formulation. In Sect. 7 we demonstrate the model’s capability and feasibility by applying it to realworld problems. Sections 8 and 9 provide a discussion and conclusions drawn from this study.
2 Literature Review
Originally, railway alignment design was a 3D optimization model. This model has a multivariate objective function, and there are many nonlinear constraints in it. Because of the behavioral complexity of the threedimensional profile of the track, it requires a multistep solution to the problem. The threedimensional solution seems to be a problem owing to a variety of design options and multiplicity of parameters involved. For this reason, in many studies, railway alignment design has been performed in two steps. In the first step, the horizontal alignment is determined, and the vertical alignment is then adjusted correspondingly. Because the horizontal and vertical alignments are interdependent, various studies have explored ways to solve these two problems at the same time [3].
 a.
Vertical alignment optimization models
 b.
Horizontal alignment optimization models
 c.
3D optimization models (simultaneous horizontal and vertical alignment optimization)
Horizontal alignment optimization models are more complex than vertical alignment models, primarily because of the various parameters that must be considered, such as socioeconomic and environmental issues. Shafahi and Shahbazi [1] studied the design of optimal railway alignment using a genetic algorithm. The Shahbazi optimization model is nonbacktracking. The vertical alignment is formed within an allowed range along the horizontal alignment for the maximum allowable gradient of the railway track. Turner and Miles [4] reported on a cost/benefit alignment optimization model using the shortest path in a raster grid. Other studies have used calculus of variations [5, 6, 7, 8] and network optimization [9, 10, 11, 12].
Many studies in the literature have examined the optimization of vertical alignment. The most important parameter in the vertical alignment cost function is the volume of earthwork. The design of the vertical alignment, the problem of generating backtracking alignment, and the possibility of disconnecting alignments has not been posed because the vertical alignment is designed along the horizontal alignment.
Fwa et al. [13] proposed a method for optimizing vertical alignment by assuming a given horizontal alignment. They considered various constraints on the design of the vertical alignment, such as longitudinal slope, vertical curve, fixed points, and limitations of vertical curve on horizontal curves in the proposed model. A genetic algorithm was used as an optimization method, and a penalty function with a finite fixed unit was used to avoid unacceptable solutions.
The study by Easa [14] was one of the first to focus on alignment optimization. In order to optimize the longitudinal profile of the route and reduce the cost of earthwork, he proposed a model that minimizes the cost of earthwork and also fulfills the necessary geometric characteristics. In his model, the project line creates the necessary balance between earthwork sections (cut and fill sections), and the longitudinal slope and vertical curvature are subject to a maximum limit, while the minimum distance between the successive vertical curves is taken into account.
The major cost associated with the design of the vertical profile of a route is the cost of earthwork, which was addressed in a study by Goktepe et al. [15] on optimizing project line points with regard to the cost of earthwork units (embankment and excavation). They used the concept of weighted ground equivalency (WGE), which was achieved by placing the total embankment and excavation (earthwork) vectors in each cross section.
Studies using dynamic programming algorithms [16, 17, 18, 19] and linear programming algorithms [20, 21] have been reported in the field of vertical alignment optimization.
Chew et al. [22] were the first researchers to solve the problem of simultaneous optimization of 3D alignment for horizontal and vertical profiles. They used a numerical search method in their study.
The use of a metaheuristic algorithm for routing began with Jong’s [23] research. Jong and Schonfeld [24] then tried to ameliorate some of the defects using genetic algorithms. They provided a solving order with genetic algorithm changes which are used in routing problems.
Jha and Schonfeld [25] used a geographic information system (GIS) and genetic algorithms to solve the problem of optimal alignment. Assuming that the search area is a rectangular region, they divide this area into a grid with similar square cells. The cells are considered small enough that an assumption of similarity of properties within them (e.g., land costs, height) is reasonable. Hasany and Shafahi [2] used an ant colony optimization (ACO) method for optimizing railroad alignment. The modeling study was done in continuous space and utilizing GIS. In 2006, De Smith [26] used a network optimization method with limitations on curve radii and alignment gradient for route, railway, and pipeline routing. In the same year, Cheng and Lee [27] provided a 3D route optimization model using the neighborhood search method to determine a horizontal path and mixed hybrid planning to optimize the vertical path. Shafahi and Bagherian [3] used a relatively new and effective method in 2013 to solve the path routing problem with a particle optimization algorithm (bird swarm algorithm). This model uses geospatial data as a search space and performs searches in a continuous environment. Considering the geometric constraints in the design of the route, this model features the ability to produce return routes and the possibility of constructing the bridge and tunnel along the path, aligning with the list table, improving the cost functions used in previous studies and global search capability, and thus finding a nearoptimal realworld path. To determine an alignment, first, a number of intersection points are indicated that make a broken and rugged alignment. Then, with the help of other algorithms, the horizontal and vertical curves are matched to achieve a smooth alignment. The intersection points from the optimization problem are variables. The range of these point changes are on lines parallel to one another.
Similarly, Lai and Schonfeld [28] proposed a methodology for concurrent optimization of station locations and rail transit alignment connecting those stations, by accommodating multiple system objectives, satisfying various design constraints, and integrating the analysis models with a GIS database. Li et al. [29, 30] used a genetic algorithm and bidirectional distance transform to optimize railway alignments in mountainous terrain. Concurrent optimization of mountain railway alignment and station locations using a distance transform algorithm was recently reported by Pu et al. [31].
Many studies have been carried out for threedimensional route optimization with genetic algorithms, including Kim [32], Jong and Schonfeld [24], Tat and Tao [33], Kim et al. [34, 35, 36, 37], and Kang et al. [38, 39, 40, 41].
3 Railway Costs and Constraints

Rightofwaydependent costs

Lengthdependent costs

Trafficdependent costs

Volumedependent costs

Hydrology and hydrologicdependent costs

Bridge and tunneldependent costs
3.1 RightofWayDependent Costs
The costs associated with a road or railway route, depending on the passageway, include land acquisition and other costs for crossing the track from one area. Because of this, the railway track passageway has a special significance. According to the accuracy requirements, the study area is divided into cells with specific dimensions. It is assumed that each cell in the study area represents an area with the same cost of production. Crossing the railways from different cells carries different costs, and the final costs for the location depend on the total costs of the path components in the cells. With this cost system, cells that are located in areas where the railroad does not have to cross them (such as historic sites, or marsh locations or other special locations) can be defined as cells with a very high unit cost. This routing process will cause the total cost of the transit route in this area to be increased, and the probability of choosing it will be reduced.
3.2 LengthDependent Costs
The hypothesized superstructure in this paper is a ballasted one. This type of track includes ballast, tie, fastener, and rail. The use of ballast is very common, as ballasted tracks have very good performance. Although the maintenance cost for this type of track is high, its construction cost is lower than for ballastless tracks. There have been no major changes in the principles of ballasted superstructures since railway tracks first emerged, but certain developments have occurred within the railway transport industry in order to promote greater safety and speed, such as continuous welded rail (CWR), the use of concrete ties and heavier rail sections, elastic fastenings, machining repair operations, and the development of advanced equipment for measuring various parameters of track components, maintenance management, and so on.
3.3 TrafficDependent Costs
 a.The cost of purchasing, maintaining, and replacing rolling stock

Locomotive purchases

Locomotive maintenance

Wagon purchases

Wagon maintenance

 b.
Track maintenance and reconstruction cost
 c.
Cost in terms of the value of cargo and passenger time
3.4 VolumeDependent Costs
In this paper, the average endarea method is used to calculate the volume of earthwork [42]. To calculate the volume of earthwork (excavation and embankment operations) on horizontal alignment at specified intervals, the area of transverse sections (stations) is determined, and the volume of soil operations between the two stations is then calculated. Accuracy in estimating the volume of earthwork is dependent on the distance between stations. When the distance is smaller, a more accurate estimate of earthwork will be achieved [43]. Here, for calculating the volume of earthwork, the distance between stations is considered to be identical and equal to 50 m.
3.5 Hydrology and HydrologicDependent Costs
Since moisture and pore water pressure play a decisive role in the durability of ballast aggregates, the strength of construction, and stability of slopes, surface and underground water control are considered as key factors in the design and maintenance of railways. Given the importance of this issue with regard to railways, in this paper we have tried to consider the costs of the alignment of hydrologic affairs as much as possible.
3.6 Bridge and TunnelDependent Costs

If the excavation height exceeds the permissible level according to the material, then a tunnel needs to be constructed.

If the height of the embankment exceeds the virtual limit, it is necessary to construct a bridge along the alignment.

The bridge should be constructed if an alignment passes through an environmentally sensitive, marshy or riverbound area with operational problems on the embankment.
3.7 Constraints in Railway Track Alignment
 1.
Minimum radius of horizontal curve
 2.
Minimum vertical curve length
 3.
Maximum longitudinal slope of the alignment
3.7.1 Penalty Function for Violating the Minimum Horizontal Curve Radius
As long as the minimum radius of the track can be met along the way, the curvature limits of the alignment have been fulfilled. However, due to the small distance between the two points of intersection, there may be a discontinuity in horizontal alignment which is solved this problem using the algorithm reported in [46]. If the radius of the existing alignment is less than the minimum allowed radius of the route (\(R_{\hbox{min} }\)) according to Formula (3) (288 code in Iran), for the penalty of the alignment, the method presented in [47] is used.
3.7.2 Penalty Function for Violating the Minimum Vertical Curve Length
In order to observe the minimum vertical curve length, an approach similar to that of the horizontal curve radius is used. To correct the discontinuity of the vertical curves, the method presented in [43] is used. If the minimum length of the vertical curve is less than the minimum allowed length of the vertical curve (\(L_{\hbox{min} }\)) according to Formula (5) [48], the penalty of the alignment presented in [47] is used.
3.7.3 Penalty Function for Violating the Maximum Gradient of the Alignment
For the penalty function in the case of nonobservance of the maximum gradient for successive intersections i and i+1, a similar approach was encountered with two previous modes. The maximum allowed gradient for this part is considered to be 1.25%.
4 Representation of Railway Alignment in PSO
In the proposed railway alignment optimization model, a horizontal alignment is defined by the tangents, circular curves, and the connecting transition curve sections. A vertical alignment is defined by the graded tangents connected with parabolic curves. This arrangement of elements depends on the points of intersection of the horizontal alignment (PIs), so the definition of generating alignments can be summarized in the choice of different points of intersection. In this paper, the PSO algorithm is used to find the optimal railway track alignment.
The particle swarm optimization (PSO) algorithm is a one of the wide category of swarm Intelligence methods for solving optimization problems. The basic PSO algorithm was developed by Kennedy and Eberhart [49] and Tu et al. [50]. Execution of the PSO algorithm to solve highway alignment optimization problems is convenient for two reasons. First, PSO is inherently designed to solve continuous problems, and its efficiency has been reported in previous studies. Second, unlike the other algorithms proposed for highway alignment optimization (e.g., genetic algorithms), the PSO implicitly considers the spatial relations during the search process. Therefore, this algorithm can be considered as a promising method for alignment optimization problems [3].
PSO can be easily implemented, and it is computationally inexpensive compared with genetic algorithms, since its memory and CPU speed requirements are lower [50].
In the basic PSO algorithm, the position (x) and velocity (v) of each particle in the swarm can be updated after each iteration, using the following equations:
The number of cutting planes that are used affects the shape and even the cost of the alignment generated. For urban areas or areas with more complex topography, increasing the number of cutting planes will increase the accuracy of the solution, and for areas with simple topography or that are outside the city, an optimal solution can be achieved with fewer cutting planes [46]. Therefore, in this paper, the number of cutting planes is selected depending on the studied region, which is investigated in the results section.
It is important to note that in this paper, the cutting plane method was used, and since the alignment is nonbacktracking, this method is not suitable for mountainous regions, as they have a length limit and should be backtracked.
5 Modeling Railway Alignments
5.1 Horizontal Alignment Model
The railway horizontal alignment is composed of a series of tangent and horizontal curves. Horizontal curves include transition and circular curves. Transition curves are typically used between the straight alignment (tangent) and the circular curve. This prevents a sudden change in the curve radius, and when travel from the straight alignment to the circular curve occurs, transition curves prevent an increase in lateral acceleration. In railways, the implementation of transition curves has particular importance.
5.2 Horizontal Alignment Model Formulation
Definition of the critical points used in the horizontal curve [46]
Points  Definition 

\(TS_{i}\)  The point where the tangent of the alignment connects to the transition curve (the beginning of the transition curve) \(\forall i = 1, \ldots ,n\) 
\(SC_{i}\)  The point where the transition curve connects to the circular curve (the end of the transition curve and the beginning of the circular curve) \(\forall i = 1, \ldots ,n\) 
\(CS_{i}\)  The point where the circular curve connects to the transition curve (the end of the circular curve and the beginning of the transition curve) \(\forall i = 1, \ldots ,n\) 
\(ST_{i}\)  The point where the transition curve connects to the tangent of the alignment (the end of the transition curve and the beginning of the alignment tangent) \(\forall i = 1, \ldots ,n\) 
If the deviation angle of a PI is zero, then that point contains all points \(TS_{i}\), \(SC_{i}\), \(CS_{i}\), \(ST_{i}\), and \(PI_{i}\).
In Fig. 3, AB and CD are connected by a transition curve (clothoid) and BC by a circular curve . The point A (\(TS_{i}\)) is the start of the clothoid and has an infinite radius (mean \(R_{{s_{i} }} = \infty\) in \(TS_{i}\)) and a curve degree of zero. The radius of the transition curve along its length (\(l_{{s_{i} }}\)) decreases slowly until it reaches the radius of the circular curve at point B, which is the end point of the clothoid and the beginning of the circular curve (mean \(R_{{s_{i} }} = R_{{c_{i} }}\) in \(SC_{i}\)). These rules are also applied on the other side of the curve. A complete calculation can be seen in [53].
The point \(TS_{i}\) is the starting point for the clothoid on the line between \(PI_{i  1}\) and \(PI_{i}\), and the distance between \(PI_{i}\) and \(TS_{i}\) is displayed by \(L_{{TS_{i} }}\). The point \(ST_{i}\) is the starting point of the tangent on the other side of the curve; it is between points \(PI_{i}\) and \(PI_{i + 1}\), and its distance from point \(PI_{i}\) is equal to the length \(L_{{TS_{i} }}\). The next point, which represents the end of the clothoid and the beginning of the circular curve, is represented by \(SC_{i}\). The point \(CS_{i}\) also connects the circular curve to the transition curve (clothoid) in the other direction. In accordance with the equations in the Appendix, the coordinates of the points can be calculated [46].
5.2.1 Condition for Establishing a Transition Curve
If this condition is not met, the curve cannot be fitted with geometric specifications. In this paper, the algorithm for horizontal curves is implemented in such a way that a circular curve with a large radius is used instead of the curve with the clothoidcircleclothoid properties. In such a situation where the alignment has a geometric problem, a circular curve with a radius greater than 3000 m is used, in which there is no need to apply the superelevation for the railway alignment. Because of the lack of cant, no further use of clothoid is required.
5.3 Determining the Vertical Cutting Plane
At points \(VPI_{i} , \ldots ,VPI_{n + 1}\) the vertical curve is braced to make the alignment smoother. The coordinates of points \(VPI_{0}\) and \(VPI_{n + 1}\) are the coordinates of the points at the beginning and the end of the alignment, respectively. Therefore, the length of the horizontal alignment to point \(VPI_{i}\) must be calculated for each point to determine the horizontal axis \(VPI_{i}\) in the longitudinal profile (vertical alignment).
6 Basic Model Formulation
The functions are fully described in Sect. 3.
7 Case Studies
In this part of the paper, three examples are discussed. The first example is hypothesized to validate the model. In the second and third examples, real cases using topographic data obtained from maps of 1/25,000 of a region between the cities of Isfahan and Shahreza in Isfahan province are considered, with two different points at the beginning and the end.
7.1 Design Parameters and Constraints Considered to Solve Examples
Railway design assumptions
Characteristic  Value 

Passenger train design speed (km/h)  160 
Freight train design speed (km/h)  80 
Ballast (cm)  45 
Tie type  B70 
Rail type  UIC 60 
Cut slope  1:2 
Fill slope  1:2 
The distance between the stations is considered 50 m to calculate the earthwork volume.
7.2 Parameters Considered for the ProblemSolving Algorithm
The constant parameters of the PSO algorithms
Characteristic  Value 

Accelerator coefficients  \(c_{1} = c_{2} = 2\) 
Inertial weight  \(\omega = 0.1\)–0.6 (linear) 
Maximum speed in each dimension  Equivalent to 10% of its range 
Number of primary particles (paths)  25 particles (path) 
Algorithm repetitions  100 repetitions 
7.3 Number of Cutting Planes
In the alignment optimization problem, the number of cutting planes [representing the number of points of intersection (PIs) pathway] can vary for different regions. For example, for urban areas that have many complexities (the urban train alignment optimization problem), an increased number of cutting planes should be considered in order to obtain better results, but for suburban regions, this number can be smaller given the lower complexity (for example, the problem of optimizing the alignment of suburban railways).
As the number of points of intersection (and hence the number of cutting planes) increases, the calculation time increases. In each of the three examples, the number of cutting planes was considered for several different conditions and the solutions were compared. In order to make the design more realistic, these unit costs are based on the price list for the technical structure of the road and railways in 2014. In the following sections, the results and outputs obtained from the hypothetical example are first examined, and then the results of the second and third examples are presented as case studies of the IsfahanShahreza area.
7.4 The First Example
Comparison of alignment cost in Example 1
Number of cutting planes  Alignment cost during the project period (\(\times 10^{6} \,\$ \))  Maximum longitudinal gradient (%)  Alignment length (m) 

2  164.36  0.75  11,883.7 
3  162.05  0.6  11,991.2 
4  161.03  0.9  12,361.5 
5  160.51  0.5  12,346.2 
7.4.1 Sensitivity Analysis of Model Parameters
In this section, sensitivity analysis of the parameters used in the objective function is carried out. In all of these analyses, the alignments are constructed by assuming five cutting planes.
Two scenarios were designed to analyze the model performance. In the first scenario, all costs except the cost of earthwork, bridge, and tunnel are considered, and in the second scenario, the earthwork cost is added to the model to be evaluated.
7.4.1.1 Scenario 1: Solving the Model Taking into Account the LengthDependent Costs (All Costs Except Costs Related to the Volume of Earthwork, Bridges, and Tunnels)
7.4.1.2 Scenario 2: Solving the Model with All Costs Without the Possibility of Having a Bridge or Tunnel
In this scenario, since the model lacks the exploration of the bridge and tunnel option, it is expected that the optimal alignment will make detours around hills and lakes to avoid a high volume of earthwork. It should be noted that in this model, due to the lack of technical structures of the bridge and tunnel, the maximum altitude for earthwork is not considered, and the model assumes that the alignment can pass through the river and the lake at the embankment. As expected, the alignment tries to cross an area that is smaller in width than the rest of the river, and it is assumed that the river connected to the lake can be filled with soil and at a single cost such as drought. This is just a matter of sensitivity analysis, and it is clear that the result will not be an acceptable alignment.
7.5 The Second Example
Comparison of alignment costs in Example 2
Number of cutting planes  Alignment cost during the project period (\(\times 10^{6} \,\$ \))  Maximum longitudinal gradient (%)  Alignment length (m) 

11  394.81  1.2  37,366.76 
13  371.78  1.15  39,532.12 
15  361.54  1.25  38,802.80 
17  353.83  1.22  38,993.15 
7.6 The Third Example
Comparison of alignment costs in Example 3
Number of cutting planes  Alignment cost during the project period (\(\times 10^{6} \,\$ \))  Maximum longitudinal gradient (%)  Alignment length (m) 

11  492.34  1  51,800 
13  497.42  1.15  52,685.2 
15  487.17  1.12  48,949 
17  482.05  1.22  48,895.7 
The track between the two cities of Isfahan and Shahreza is also shown in yellow. As we can see, the alignment obtained according to the proposed model differs from the existing alignment. Due to the inaccessibility of the longitudinal profile of the existing alignment and the impossibility of calculating the volume of earthwork or the number of bridges and tunnels along the alignment, a comparison of the alignment costs obtained in this paper with the existing alignment was limited to the length of the alignment and its corridor. According to Google Earth, the length of the existing alignment is approximately 50 km. Therefore, the present study is only able to compare the passing location and the alignment length with that discussed for this model. It should be noted that the final alignment in the model is shorter than the existing alignment.
8 Discussion
In this work, the first example was hypothetical and meant to validate the model. The second example was evaluated based on real topography, and finally, the third example was a case study of the region between Shahreza and Isfahan.

Processor: Intel^{®} Core™ i7 CPU

RAM: 8 GB

System type: 64bit operating system
A period of approximately 6 hours was needed to run the largest problems in this paper with the abovementioned computer setup.
In each example, a different number of cutting planes (with different values) was used in this model. As expected, it was shown that the model provided different solutions by changing the number of orthogonal cutting planes. Due to the high geometrical constraints of railway alignments, it is important to include bridges and tunnels in the model. Using transition curves for the railway alignment plays a very important role in avoiding abrupt changes in radii of alignments and increased lateral acceleration when trains approach curves from a straight alignment.
9 Conclusion
 1.
Threedimensional optimization considering the transition curve on the horizontal alignment
 2.
Considering the possibility of constructing bridges and tunnels on the railway alignment
 3.
Considering the different characteristics of the railway alignment design to find the alignment that is most applicable
 4.
Improving the cost function used in previous studies and modifying simplifications that reduced the accuracy of the calculations
 5.
Handling data in a geographic information system
 6.
Using particle swarm optimization in railway alignment optimization.
Notes
Open Access
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
References
 1.Shafahi Y, Shahbazi MJ (2012) Optimum railway alignment. Int J Commun Netw Syst Sci 5(9A)Google Scholar
 2.Hasany RM, Shafahi Y (2016) Ant colony optimisation for finding the optimal railroad path. Paper presented at the Proceedings of the institution of civil engineerstransportGoogle Scholar
 3.Shafahi Y, Bagherian M (2013) A customized particle swarm method to solve highway alignment optimization problem. Comput Aided Civ Infrastruct Eng 28(1):52–67CrossRefGoogle Scholar
 4.Turner AK, Miles RD (1971) The GCARS system: a computerassisted method of regional route location (No. 348)Google Scholar
 5.Howard BE, Bramnick Z, Shaw J (1969) Optimum curvature principle in highway routing. J Highw Div 94:61–82Google Scholar
 6.Shaw JF, Howard BE (1982) Expressway route optimization by OCP. J Transp Eng 108(TE3):227–243Google Scholar
 7.Thomson N, Sykes J (1988) Route selection through a dynamic ice field using the maximum principle. Transp Res Part B Methodol 22(5):339–356CrossRefGoogle Scholar
 8.Wan F (1995) Introduction to the calculus of variations and its applications. CRC Press, Boca RatonzbMATHGoogle Scholar
 9.OECD (1973) Optimization of road alignment by the use of computers. Organisation for Economic Cooperation and Development, ParisGoogle Scholar
 10.Parker NA (1977) Rural highway route corridor selection. Transp Plan Technol 3(4):247–256CrossRefGoogle Scholar
 11.Trietsch D (1987) Comprehensive design of highway networks. Transp Sci 21(1):26–35CrossRefGoogle Scholar
 12.Trietsch D (1987) A family of methods for preliminary highway alignment. Transp Sci 21(1):17–25CrossRefGoogle Scholar
 13.Fwa T, Chan W, Sim Y (2002) Optimal vertical alignment analysis for highway design. J Transp Eng 128(5):395–402CrossRefGoogle Scholar
 14.Easa SM (1988) Selection of roadway grades that minimize earthwork cost using linear programming. Transp Res Part A Gen 22(2):121–136CrossRefGoogle Scholar
 15.Goktepe AB, Lav AH, Altun S (2005) Dynamic optimization algorithm for vertical alignment of highways. Math Comput Appl 10(3):341–350zbMATHGoogle Scholar
 16.Fwa T (1989) Highway vertical alignment analysis by dynamic programming. Transp Res Rec 1239:2–3Google Scholar
 17.Goh C, Chew E, Fwa T (1988) Discrete and continuous models for computation of optimal vertical highway alignment. Transp Res Part B Methodol 22(6):399–409CrossRefGoogle Scholar
 18.Murchland J (1973) Methods of vertical profile optimisation for an improvement to an existing road. Paper presented at the PTRC seminar proceedings, cost models and optimization in highwaysGoogle Scholar
 19.Puy Huarte J (1973) OPYGAR: optimisation and automatic design of highway profiles. Paper presented at the PTRC seminar proceedings on cost models and optimization in highways, session LGoogle Scholar
 20.Chapra SC, Canale RP (1988) Numerical methods for engineers, vol 2. McGrawHill, New YorkGoogle Scholar
 21.Revelle CS, Whitlatch E, Wright J (1997) Civil and environmental systems engineering. Prentice Hall, New JerseyGoogle Scholar
 22.Chew E, Goh C, Fwa T (1989) Simultaneous optimization of horizontal and vertical alignments for highways. Transp Res Part B Methodol 23(5):315–329CrossRefGoogle Scholar
 23.Jong J (1998) Optimizing highway alignments with genetic algorithms. University of Maryland, College Park. Ph.D. dissertationGoogle Scholar
 24.Jong JC, Schonfeld P (2003) An evolutionary model for simultaneously optimizing threedimensional highway alignments. Transp Res Part B Methodol 37(2):107–128CrossRefGoogle Scholar
 25.Jha M, Schonfeld P (2000) Geographic information systembased analysis of rightofway cost for highway optimization. Transp Res Rec J Transp Res Board 1719:241–249CrossRefGoogle Scholar
 26.De Smith MJ (2006) Determination of gradient and curvature constrained optimal paths. Comput Aided Civ Infrastruct Eng 21(1):24–38CrossRefGoogle Scholar
 27.Cheng JF, Lee Y (2006) Model for threedimensional highway alignment. J Transp Eng 132(12):913–920CrossRefGoogle Scholar
 28.Lai X, Schonfeld P (2016) Concurrent optimization of rail transit alignments and station locations. Urban Rail Transit 2(1):1–15CrossRefGoogle Scholar
 29.Li W, Pu H, Schonfeld P, Zhang H, Zheng X (2016) Methodology for optimizing constrained 3dimensional railway alignments in mountainous terrain. Transp Res Part C Emerg Technol 68:549–565CrossRefGoogle Scholar
 30.Li W, Pu H, Schonfeld P, Yang J, Zhang H, Wang L, Xiong J (2017) Mountain railway alignment optimization with bidirectional distance transform and genetic algorithm. Comput Aided Civ Infrastruct Eng 32(8):691–709CrossRefGoogle Scholar
 31.Pu H, Zhang H, Li W, Xiong J, Hu J, Wang J (2019) Concurrent optimization of mountain railway alignment and station locations using a distance transform algorithm. Comput Ind Eng 127:1297–1314CrossRefGoogle Scholar
 32.Kim E (2001) Modeling intersections and other structures in highway alignment optimization. University of Maryland, College ParkGoogle Scholar
 33.Tat CW, Tao F (2003) Using GIS and genetic algorithm in highway alignment optimization. Paper presented at the Intelligent transportation systems, 2003. proceedings. IEEEGoogle Scholar
 34.Kim E, Jha MK, Lovell DJ, Schonfeld P (2004) Intersection modeling for highway alignment optimization. Comput Aided Civ Infrastruct Eng 19(2):119–129CrossRefGoogle Scholar
 35.Kim E, Jha MK, Schonfeld P (2004) Intersection construction cost functions for alignment optimization. J Transp Eng 130(2):194–203CrossRefGoogle Scholar
 36.Kim E, Jha MK, Schonfeld P, Kim HS (2007) Highway alignment optimization incorporating bridges and tunnels. J Transp Eng 133(2):71–81CrossRefGoogle Scholar
 37.Kim E, Jha MK, Son B (2005) Improving the computational efficiency of highway alignment optimization models through a stepwise genetic algorithms approach. Transp Res Part B Methodol 39(4):339–360CrossRefGoogle Scholar
 38.Kang M, Yang N, Schonfeld P, Jha M (2010) Bilevel highway route optimization. Transp Res Rec J Transp Res Board 2197:107–117CrossRefGoogle Scholar
 39.Kang MW (2008) An alignment optimization model for a simple highway network. University of Maryland, College ParkGoogle Scholar
 40.Kang MW, Schonfeld P, Jong JC (2007) Highway alignment optimization through feasible gates. J Adv Transp 41(2):115–144CrossRefGoogle Scholar
 41.Kang MW, Schonfeld P, Yang N (2009) Prescreening and repairing in a genetic algorithm for highway alignment optimization. Comput Aided Civ Infrastruct Eng 24(2):109–119CrossRefGoogle Scholar
 42.Wright P, Dixon K (2004) Highway engineeringm, 7th Edn. Wiley, HobokenGoogle Scholar
 43.Jha MK, Jha MK, Schonfeld P, Jong JC (2006) Intelligent road design, vol 19. WIT Press, Ashurst LodgeGoogle Scholar
 44.O’Connor C (1971) Design of bridge superstructures. Wiley, New YorkGoogle Scholar
 45.Li X, Engelbrecht AP (2007) Particle swarm optimization: an introduction and its recent developments. Paper presented at the Proceedings of the 9th annual conference companion on genetic and evolutionary computationGoogle Scholar
 46.Kang MW, Jha MK, Schonfeld P (2012) Applicability of highway alignment optimization models. Transp Res Part C Emerg Technol 21(1):257–286CrossRefGoogle Scholar
 47.Kazemi SF, Shafahi Y (2013) An integrated model of parallel processing and PSO algorithm for solving optimum highway alignment problem. Paper presented at the ECMSGoogle Scholar
 48.Lindamood, Strong, McLeod (2009) Railway track design: practical guide to railway engineering, chapter 6. American Railway Engineering and Maintenance of Way Association, MarylandGoogle Scholar
 49.Kennedy J, Eberhart R (1995) Particle swarm optimization. Proc IEEE Int Conf Neural Netw 4:1942–1948CrossRefGoogle Scholar
 50.Tu S, Guo X, Tu S (2008) Optimizing highway alignments based on improved particle swarm optimization and ArcGIS. In: The first international symposium on transportation and development–innovative best practices (TDIBP 2008). American Society of Civil Engineers, China Academy of Transportation SciencesGoogle Scholar
 51.Shi Y, Eberhart R (1998) A modified particle swarm optimizer. In: 1998 IEEE international conference on evolutionary computation proceedings. IEEE world congress on computational intelligence (Cat. No. 98TH8360). IEEE, pp 69–73Google Scholar
 52.Eberhart R, Simpson P, Dobbins R (1996) Computational intelligence PC tools. Academic Press Professional, IncGoogle Scholar
 53.Hickerson T (1964) Route location and design, 5th edn. McGrawHill, New YorkGoogle Scholar