Traditional methods for alignment optimization
2.6 Models for simultaneously optimizing horizontal and vertical alignments vertical alignments
Although much progress has been made in developing models for optimizing vertical alignment, the development of models that simultaneously optimize horizontal and vertical alignments is not yet successful because there are more factors involved and more complexities in the geometric specification of a 3-dimensional alignment. There are two basic approaches found in the literature:
dynamic programming and numerical search. Although some of network optimization models such as those developed by Parker (1977) and Trietsch (1987a) claim the capability of simultaneously selecting both horizontal and vertical alignments, they employ a 2-stage optimization approach (i.e., vertical alignment is determined prior to horizontal alignment), and are essentially 2-dimensional optimizers. These models have already been discussed in section 2.1.
2.6.1 Dynamic programming
The basic structure of dynamic programming models for optimizing 3-dimensional alignment is to set the stages as equally spaced planes between the start and end points, which are, from a top view, perpendicular to the line segment connecting two end points of the alignment. At each stage, the search grids (i.e., states) are located on a 2-dimensional plan. Linking straight-line segments from the origin to the destination through the grids at each stage will generate a 3-dimensional alignment.
Hogan (1973) presented a dynamic programming model, OPTLOC, which is used by U.S. Forest Service for optimizing road alignment and profile. Generally a coarse search grid is used initially. Then successive iterations, accomplished by
Intelligent Road Design 29 refinement of the search grid, can be made to choose a route at any desired tolerance level.
Nicholson (1976) also employed similar approach to optimize route location.
At the first stage, the model searches a relative coarse grid of points for a preliminary alignment (or corridor). Then a discrete variational calculus is adopted to refine the alignment so that the resulting alignment can deviate from the grid points.
The application of dynamic programming in optimizing 3-dimensional alignment has several defects. First, it has difficulty in handling backtracking alignments. Secondly, the resulting solution is very rough for both horizontal and vertical alignments, and there are difficulties in dealing with both horizontal and vertical curvatures. Finally, the storage requirements may hinder this approach from searching a finer grid initially.
2.6.2 Numerical search
The only model found in the literature that simultaneously optimizes a “smooth”
3-dimensional alignment was developed by Chew, Goh, and Fwa (1989). This model is the extension of their continuous model for optimizing vertical alignment (Goh, Chew, and Fwa, 1988).
The problem is initially formulated as a calculus of variations problem. As mentioned in the last section, the necessary conditions are very difficult to solve.
The authors thus utilized a set of cubic spline functions to interpolate the alignment, then transformed the constraints into one-dimensional constraints by the method of constraint transcription used in the optimal control theory. Finally the model becomes a constrained nonlinear program structure with the coefficient vectors of spline functions as its decision variables.
It is noted that the objective function involves integrals, which are not easy to compute. Thus a numerical integration is suggested by the authors to facilitate the computation during search. The solution algorithm employed in their paper is the quasi Newton descent algorithm. Variable scaling is also considered to improve the convergence performance.
Like other models for optimizing vertical alignment by numerical search, the solution found in Chew’s model only guarantees a local optimum. In practice, different initial solutions with human judgments are used for running the model.
Furthermore, the variable scaling procedure requires interactions between the program and users during optimization. Therefore, the model is not fully automatic. Another potential defect embedded in this model is that it seems difficult to incorporate discontinuous location dependent costs (e.g., land acquisition cost) into the objective function because the algorithm requires a differentiable objective function.
Regardless of the above disadvantages and difficulties of formulation, this model possesses some good features. At least it successfully optimizes a smooth 3-dimensional alignment, which is not possible with the dynamic programming
30 Intelligent Road Design
approach or with any other known models besides those presented in the following chapters of this book.
2.7 Summary
In the previous three sections, several existing models for optimizing highway alignments are investigated. The potential advantages and disadvantages in each approach are summarized in Table 2.5 to Table 2.7 in a somewhat general way.
For example, the numerical search approach for optimizing vertical profile by Hayman’s model (1970) only produces a set of linear piecewise segments rather than a smooth alignment. However, this can be ameliorated if Goh, Chew, and Faw’s (1988) model is employed. Thus, the advantages and disadvantages listed for each approach are general to each method rather than a specific model.
Furthermore, the note “possibly finds the global optimum” appearing in the tables means that a particular approach can find the global optimum under its assumptions, which are, however, unrealistic for optimizing highway alignments.
Apparently, none of the approaches shown in Table 2.5 to Table 2.7 dominates the others, and there is always some trade off between them. The problem turns out to be: what approach (or model) is most promising for optimizing highway alignments. To answer this question, the features of a good model may be outlined. These are readily merged from Table 2.5 to Table 2.7.
Necessary conditions of a good model for optimizing highway alignments (1) Consider all dominating and sensitive costs
(2) Formulate all important constraints (3) Yield a realistic alignment
(4) Be able to handle alignment with backward bends (5) Simultaneously optimize 3-dimensional alignments (6) Find globally or near globally optimal solutions (7) Have an efficient solution algorithm
(8) Have low storage requirements (9) Have a continuous search space
(10) Automatically avoid inaccessible regions
(11) Be compatible with GIS (Geographical Information System) databases The first condition usually depends on the availability of data, but a good model should be able to deal with various cost items, including continuous and discrete cost functions. The second to the fifth conditions will be considered in more detailed and mathematical forms in the next chapter. The seventh and eighth conditions are less critical than in previous times. With technology developments, the speed and memory of modern computers has significantly increased. The last condition, “be compatible with GIS databases”, is the trend in transportation planning and analysis because most spatial data will be stored in GIS databases.
Intelligent Road Design 31 One may find that the aforementioned conditions are somewhat conflicting.
Emphasizing some aspects may neglect the others. It is very difficult and challenging to trade off these conditions. The history of research in this field shows that recent development has become very slow, or even stagnant. In fact, most of the models were developed in late 60’s and early 70’s, when many highways were built all over the world. After that only a few papers were published. The most promising one (Chew et al, 1989) was developed almost two decades later. Why? This seems due not only to the shortage of new highway projects, but also to the lack of theoretical breakthroughs. This book attempts to develop models that yield better tradeoffs among the above conditions. However, we are not going to solve the entire problem since each of these conditions deserves a major research effort.
32 Intelligent Road Design
Disadvantages
- Cannot deal with discontinuous cost items (requires well-behaved objective function) - Complex modeling and computation efforts - The resulting alignment is not smooth - Discrete solution set rather than continuous
search space
- Large memory requirements
- The resulting alignment is not smooth - Discrete solution set rather than continuous
search space
- Difficulty in handling backward bends Advantages
- Yields smooth alignment
- Possibly finds the global optimum - Has continuous search space - Simple and easy to use
- Well-developed algorithms for solving the problem exist
- Possibly finds the global optimum - Simple and easy to use
- Well-developed algorithms for solving the problem exist
- Possibly finds the global optimum Method
Calculus of variations
Network optimization
Dynamic programming
Table 2.5: Potential advantages and disadvantages of existing approaches for optimizing horizontal alignment.
Intelligent Road Design 33 Disadvantages
- Inefficient
- Discrete solution set rather than continuous search space
- The resulting alignment is not smooth - Discrete solution set rather than continuous
search space
- Only limited cost items and constraints can be formulated (must be linear)
- Gradient and curvature constraints are formulated for a limited number of points
- Multiple local optima exist
- Complex modeling and computation efforts
Advantages - Can yield a realistic alignment - Possibly finds the global optimum
- Can consider most of the important constraints - Simple and easy to use
- Well-developed algorithms for solving the problem exist - Possibly finds the global optimum
- Simple and easy to use
- Well-developed algorithms for solving the problem exist - Possibly finds the global optimum
- Can yield smooth alignment - Has continuous search space - Can yield a realistic alignment
- Can consider most of the important constraints and various costs
- Has continuous search space Method
Enumeration
Dynamic programming
Linear programming
Numerical search
Table 2.6: Potential advantages and disadvantages of existing approaches for optimizing vertical alignment.
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Intelligent Road Design
Disadvantages - The resulting alignment is not smooth - Discrete solution set rather than continuous
search space
- Large memory requirements
- Difficulty in handling backward bends - Multiple local optima exist
- Complex modeling and computation efforts - Difficulty in modeling discontinuous cost items Advantages
- Simple and easy to use
- Well-developed algorithms for solving the problem exist
- Possibly finds the global optimum
- Can yield a realistic alignment
- Can consider most of the important constraints and various types of costs
- Has continuous search space Method
Dynamic programming
Numerical search
Table 2.7: Potential advantages and disadvantages of existing approaches for simultaneously optimizing horizontal and vertical alignments.