Highway alignment optimization aims to connect two terminal points at minimum possible .... An initial population of random individuals is generated such that:.
A Genetic Algorithm Approach to Highway Alignment Development Botan A. AL-Hadad, Michael J. Mawdesley & Rod Stace University of Nottingham, UK
This paper published in the proceedings of the International Conference on Computing in Civil and Building Engineering (icccbe 2010): Nottingham, UK, 30 June – 2 July 2010.
Abstract In this paper, a novel optimization technique based on genetic algorithms (GA) is suggested to optimize highway horizontal alignment. The technique is based on station points to configure horizontal alignment rather than considering the existing conventional principles of design which uses horizontal intersection points (HIP), tangents, and curve fittings. This is expected to reduce the number of highway alignment design elements required and consequently reduce the constraints imposed on alignment planning and design. The introduction of this technique is motivated by the development of computers, computer software, and the highway field instruments (e.g. total station). The resulting solution found is promising and suggests further formulations should be investigated. Keywords: Highway Alignment, Genetic Algorithms, Optimization.
1
Introduction
Many parameters are involved in the process of highway alignment development. Access provision and economic generation for the communities where the highway is built, are two important parameters of the process. Construction and maintenance cost are other factors that need to be considered. Furthermore, fuel and vehicle wear and tear have critical impact on the overall cost/value over the life time of the highway. Developing alignments should also satisfy relevant national and/or international design standards. Moreover, the environmental impact of the highway must also be taken into account. All these parameters must be considered together. However an alignment that makes one of the parameters optimal will rarely, if ever, make all optimal. The problem of developing and selecting an optimal alignment is therefore very complex but very important. Highway alignment optimization aims to connect two terminal points at minimum possible cost subject to the design constraints. The most common cost components that can be considered are construction, maintenance, location, earthwork, environmental, social, and user costs. The weight of each component is affected by the user and/or designer preferences and/or the purpose that the alignment is built for. In conventional highway design projects, highway engineers and planners select several candidates as alternative solutions and evaluate their suitability for the region’s environment until coming up with the most suitable one (Wright and Ashford, 1998). This process starts by fixing several horizontal intersection points (HIP). The number of the HIPs may depend on the length of the highway and the topography. The number of HIPs increases as the complexity of the environment 1
increases. The successive HIPs are then connected by lines thus forming a horizontal piecewise linear trajectory. These lines are called tangents and the deflections in direction between successive tangents at HIPs are called Deflection angles. Later on and in a separate procedure horizontal circular or spiral curves are fitted at each HIP location to form the proposed horizontal alignment. This alignment then undergoes an evaluation process to know whether it suits the environment or not. The evaluation may take the form of cost consideration, damage to environment, and socio-economic issues subject to several design constraints and operational requirements that imposed on the alignment. This process, through numerous iterations, is repeated and should continue until finding the most suitable one. The selection of the final alternative is accomplished by focusing on the detailed design elements. HIPs, deflection angles, tangents, and radius of the curves are among the design elements of highway horizontal alignment. Most of these design elements are constrained by standard limits described by such documents as the Design Manual for Roads and Bridges (DMRB, 1992-2008) and AASHTO design standards (AASHTO, 1994). Researchers have tried to speed up the process of highway alignment planning and design and to find better solutions. Calculus of variations by Shaw and Howard (1982), numerical analysis by Chew et al (1989), and genetic algorithms by Jong (1998) and Tat and Tao (2003) are some of the techniques that have been used. All these have their advantages and drawbacks and all are based on the conventional design principles of highway alignment design which consider horizontal intersection points (HIP), tangents, and curve fittings. Since its introduction, despite the extreme development in computers and highway surveying field instruments technologies (e.g. total station), highway engineers and planners are still using the same design approach. None of the studies has exploited the technology development to explore the possibility of changing some ideas imposed on highway alignment planning and design. A question arises here, do we still need to keep the same planning and design approach or we need to change to reflect technology development? That is the question that this study seeks to answer. This study introduces a novel technique for alignment optimization. It suggests optimizing horizontal alignment of a highway through station points. Station points along the centre line of alignment are used to define the alignment configuration. This research study is inspired by the fact that any generated alignment by whatever method will finally consists of a series of station points and it will be implemented on the ground depending on those station points. Figure 1 shows the difference in alignment generation and configuration between the traditional and proposed method. HIP 2
Tangent
R1
PC 2
R1
PT 2
H Curve
?1
Tangent
Tangent
PC 1 H Curve
HIP 1
PT 1 Deflection Angle 1
Station Points
Deflection Angle 2
R2
?2
R2 Station Points
(a)
(b)
Figure 1: a) Alignment configuration with traditional method. b) Alignment configuration using station points.
The alignment generation through station points is intended to reduce the number of design elements and constraints imposed on alignments during the planning and design phases. The proposed method to perform the search is based on GAs. A GA is an evolutionary adoptive search technique inspired by the theory of natural selection and survival of the fittest (Beasley et al, 1993). Holland invented the technique in the early 1970s aiming to import mechanisms of natural evolution into computer systems. This technique has been modified in order to suit different problems and applications in different environments (Davis 1991; Mitchell 1996). A variety of studies have proven that GA is an efficient tool for planning and optimization problems. Mathews et al (1999) 2
applied GA to land use planning, Mawdesley et al (2002) used GA for construction site layout in project planning, Ford (2007) used GA for housing location planning, Jong (1998), Fwa et al (2002), and Tat and Tao (2003) used GA for alignment optimization problems.
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The Overall aims
The aim of this work is to generate an optimum horizontal highway alignment that joins two terminal points in a relatively complex environment. The optimal alignment should trade-off among the different cost components assigned to the problem. In this paper these costs are related to the location of the alignment, its length, and user costs.
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The Model Formulation
The study area is defined and divided into rectangular grid cells usually produced from a GIS model of the area under consideration. The size of the grid cells falls within the user preferences and depends on the desired accuracy. The smaller the grid sizes, the more precise the representation and the better will be the solution. Each grid cell has a unit cost value to represent land use costs. A two dimensional cost matrix is used to handle the unit cost values of the land parcels and to feed the model for alternative location evaluation. Figure 2 shows an illustration for a simple study area: cell width (w) X.max
X.min Y.max
Y.max Proposed highway alignment
High Unit Cost Area (Sensitive Area)
End
cell hight (h)
Start
X.min
High Unit Cost Area (Sensitive Area)
Y.min
X.max Y.min
Figure 2: Typical grid format of the study region.
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The Model Cost
4.1
The Components
The goodness of any alignment, in this model, is evaluated in terms of cost. The lower the cost, the better will be the solution. In general many costs could be included. However, the cost components that are considered in the cost formulation in this paper are as follows: 1. Length dependent cost (CLength): It is one of the most influencing factors in highway alignment optimization problems as it directly affects the construction, maintenance, and user costs. Highway alignment construction cost is a function of its length. To calculate the construction cost, the length of the alignment is multiplied by the unit construction cost as: CLength = L x Unit Construction Cost
(1)
Where L is the total length of the alignment. The alignment length is a function of the x and y coordinate of the station points (decision variables).
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n 1
L=
i 0
( X i 1 X i ) 2 (Yi 1 Yi ) 2 , for all i = 0, 1, 2 … (n-1)
(1-1)
Where n is the total number of station points. 2. Location dependent cost (CLocation): This represents the costs of land acquisitions and special requirements for construction at the locations where the alignment passes through. p
CLocation =
l xUCC k 1
(2)
k
Where: CLocation is the total alignment location cost; lk is the length of the alignment located in a grid cell (k) with a specific cost value; UCC is the unit cell cost of that cell; and p is the total number of cells that the alignment passes through. Thus, ) where L is the length of the alignment. 3. User costs (CUser): This cost represents the cost incurred by the users to travel along the road. Here it is calculated by multiplying the annual traffic volume (TV) by the design life of the road (T) to give the total traffic using the road. This in turn is multiplied by the unit cost of the vehicle travelling unit length (UTC) multiplied by the length of the alignment (L). Thus:
CUser = TV x T x UTC x L 4.2
(3)
The Fitness Function
In this paper, the three costs above are combined linearly to form the total cost and the aim of the process is therefore: Minimize; CTotal = a1.CTraffic + a2.CLocation + a3.CConstruction
(4)
Where a1, a2, and a3 are weighting factors of the individual cost components. This is called the fitness function. Other combinations could also be used.
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Chromosome Representation
Using the ideas of station points introduced above, the method defines alignment through generating station points along the centre line of the alignment. The x and y coordinates of the station points are considered as the decision variables of the alignment. The chromosome is as shown in Figure 3. It contains the two dimensional coordinates of each station point. The station points appear in the order in which they occur along the length of the alignment. Index (i) Individual ( i )
0
1
2
3
4
X0
X1
X2
X3
Y0
Y1
Y2
Y3
5
i
n
X4
Xi
Xn
Y4
Yi
Yn
Figure 3: Chromosome representation.
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Initial Population Generation
An initial population of random individuals is generated such that: All station points are within the study area. Xmin ≤ Xi ≤ Xmax, and Ymin ≤ Yi ≤ Ymax, where i = 1, 2, … , n-1. The first and last points (0 and n) are fixed as the required terminal points. The station points are sorted in order of their X values (Xi ≤ Xi+1). 4
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Reproduction
7.1
Selection
In this study parents are selected according to their fitness (based on ranking). The selected parents undergo crossover and mutation to produce offspring. The created offspring are then evaluated by the fitness function. The fittest individuals are merged into the population to breed in the next generation while the bad solutions die off (Davis, 1991).
7.2
Crossover
In this paper single point crossover is used to swap a segment of genes between two individuals to form two offspring.
7.3
Mutation
In this paper simple uniform mutation is modified to help in exploring the search space. This modified mutation is based on that described by Michalewicz (1996). It is adapted to associate a number of station points with a single mutated one. The trend of this mutation is to reduce the messiness of the alignment configuration and to produce a smoother alignment. This operator, as with simple uniform mutation, selects a gene position (k) randomly and assigns a new random value for both its X and Y coordinates. Then the operator generates two more positions (p 1 and p2) provided that p1 < k < p2. Then, all the genes that locate between p1 and k, and p2 and k on the other side are reallocated and put on a straight line connecting the newly generated gene at (k) with the selected genes at p1 and p2.
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Examples
This formulation has been tested using various scenarios. A typical example of a result is shown in Figure 5. The result indicates that the tested formulation has some merits which can be summarised as follows: 1) It enables the alignment to jump over the high cost and sensitive environmental features, 2) Produces as short length as possible, and 3) straightens the alignment as much as possible. However, as is clear in the results, it still keeps some sharp bends throughout the length of the alignment.
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More Modifier Techniques
A further algorithm has been tested. It is based on the assumption that any two successive station points should be joined by as straight a line as possible. In some circumstances this algorithm produces apparently smoother alignments. This is illustrated in Figure 4. The test result of this algorithm, as indicated in Figure 6, shows that the sharp bends were reduced significantly and the resulted alignment is much smoother than the alignment shown in Figure 5 without this modifier technique.
10
Conclusions
A two dimensional highway alignment optimization model based on genetic algorithm has been developed. A new technique has been introduced to define the alignment configuration. Station points along the centre line of the alignment have been used to generate alignments of different
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configurations. Specific genetic algorithm operators and modifier techniques were tested and the results indicate that the method holds promises. However, the results show that further development is required to generate more realistic alignments. More specific GA operators and further modification techniques will be investigated. station points
(a)
station points
(b)
Figure 4: A piece of alignment a) before applying modification, b) after modification.
Figure 5: An alignment with modified mutation.
Figure 6: An alignment with more modifier technique.
References American Association of State Highway and Transportation Officials (AASHTO), 1994. USA. BEASLEY, D., BULL, D.R., and MARTIN, R.R., 1993. An Overview of Genetic Algorithms: Part 1 Fundamentals, University of Computing, 15(2), 58-69. CHEW, E.P., GOH, C.J., and FWA, T.F., 1989. Simultaneous Optimization of Horizontal and Vertical Alignments of Highways, Transportation research, Part B, 23(5), 315-329. DAVIS, L., 1991. Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, USA. Design Manual for Roads and Bridges, (1992-2008). The Highways Agency, UK. FORD, M., 2007. A Genetic Algorithm Based Decision Support System for the Sustainable Location of development Allocations, Ph.D. Thesis, University of Nottingham, UK. JHA, M.K. and SCHONFELD, P., 2000. Integrating Genetic Algorithms and Geographic Information System to Optimize Highway Alignments, Transportation Research Record, 1719, 233-240. JONG, J. C., 1998. Optimizing Highway Alignments with Genetic Algorithms, Ph.D. dissertation, University of Maryland, College Park, USA. MATHEWS, K.B., CRAW, S., MACKENZIE, I., ELDER, S. AND SIBBALD, A.R., 1999. Applying Genetic Algorithms to Land Use Planning. In: Proceedings of the 18th Workshop of the UK Planning and Scheduling Spatial Interest Group, 1999, University of Salford, UK. MAWDESLEY, M.J., AL-JIBOURI, S.H. & YANG, H., 2002. Genetic Algorithms for Construction Site Layout in Project Planning, Journal of Construction Engineering and Management, 128 (5), 418-426. MITCHELL, M., 1996. An Introduction to Genetic Algorithms, Massachusetts Institute of Technology (The MIT Press), USA. SHAW, J.F.B. and HOWARD, B.E., 1982. Expressway Route Optimization by OCP, Journal of Transportation Engineering, ASCE, 108(3), 227-243 TAT, C.W. & TAO, F., 2003. Using GIS and Genetic Algorithms in Highway Alignment Optimization, Intelligent Transportation System, 2, 1563-1567. Wright, P.H. & Ashford, N.J., 1998. Transportation Engineering: Planning and Design, John Wiley and Sons, USA.
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