Models for optimizing horizontal alignment

The models for optimizing horizontal alignment are more complex and require substantially more data than those for optimizing vertical alignment (OECD, 1973). The main reasons are that the design of horizontal alignment involves more political, socioeconomic, and environmental issues, and that the costs associated with a horizontal alignment are significantly affected by the vertical alignment. However, the cost reductions due to the optimization of horizontal alignment are substantially higher because most of major costs such as land cost, construction cost, social and environmental cost are very sensitive to the configuration of the horizontal alignment.

The progress in developing models for optimizing horizontal alignment is slow and their number is relatively small. In the literature, three basic approaches are noted: calculus of variations, network optimization, and dynamic programming. These are discussed in turn.

2.4.1 Calculus of variations

The calculus of variations has been developed to quantify the optimization process for a certain class of optimization problems. Many applications have led to the theory of optimal control. The basic problem in calculus of variations is to seek a curve connecting two end points in space which minimizes the integral of a function (Wan, 1995). To some extent, the optimization of highway alignment is similar to the typical problem in calculus of variations. Howard, Bramnick, and Shaw (1968) borrowed the idea to derive the Optimum Curvature Principle (OCP) for highway horizontal alignment. A similar application of the calculus of

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variations in transportation can be seen in Thomson and Sykes (1988), where a maritime route is optimized through a dynamic ice field.

The basic approach of the OCP is to hypothesize the existence of a criterion function defined in the region concerned between two cities or points where it is contemplated to build a highway. In other words, it is assumed that there exists a continuous cost surface above the 2 dimensional region of interest. The OCP then generates a route that winds along the low cost valleys and skirts the high cost mountain to connect two end points.

The OCP states that the curvature of an optimally located highway at each point is equal to the logarithmic directional derivative (percentage rate of change) of the criterion function (local cost function) perpendicular to the route.

During optimization, several routes are initialized from the start point in different directions. The sub-optimal routes are obtained by numerical integration of the OCP for each starting angle, and by selecting the paths that terminate at the end point. The optimal route is the best of this discrete set.

Shaw and Howard (1981) proposed two numerical integration methods for applying the OCP. One is to integrate the criterion function along a sequence of circular arcs joined together (arc of circle algorithm). The other uses a Taylor series expansion of degree 3 to approximate the path (intrinsic equation procedure). An example was utilized to test the performance of the two methods.

The result shows that the intrinsic equation algorithm is superior to the arc of circle algorithm. Shaw and Howard (1982) further applied the OCP to find the horizontal alignment of an expressway in flat south Florida, where construction cost was optimized. They also conducted an error analysis to establish error bounds when the true answer is unknown. The numerical example shows that a saving of nearly 5% in the cost over the straight line was gained by an increase in length of only about 0.5%.

Although Howard, Bramnick, and Shaw (1968) stated that the determination of a local cost function is not a weakness of the OCP, that determination still seems to be a major problem in applying the OCP. In practice, the authors suggested that Bicubic Spline interpolation over several discrete points be used to derive the continuous smooth surface of the local cost function. However, the local cost function may not be continuous since at least the cost of right-of-way is usually not continuous between different zones or land use patterns. Thus additional processing is required. Moreover, the local cost function in the OCP implies that the highway cost at any individual point is determined, which is somewhat unrealistic. As discussed in section 2.2, the highway costs depend on various factors, including length, location, area, volume, and VMT. It seems that there are some approximations and assumptions behind the determination of the local cost function in the OCP.

Regardless of the above disadvantages, the optimum route derived by the OCP is continuous and a global optimum is guaranteed. These can be regarded as the main advantages of this approach.

Intelligent Road Design 19 2.4.2 Network optimization

The basic idea of this approach is to formulate the optimization of horizontal alignment as a network problem, in which the alignment is represented by the arcs connecting from the start point to the end point. Then some well-developed network optimization techniques such as a shortest path algorithm are used to solve the problem.

According to OECD (1973), early studies at MIT and Miami University belong to this category. A similar concept was used by Turner (1971) to develop the Generalized Computer Aid Route Selection (GCARS) system. The GCARS system is initialized by establishing cost models (a surface covering the region of interest) for right-of-way, pavement, and earthwork (maintenance and operating costs are not considered and left to engineers for judgments). Total cost is then summed by a linear combination of different cost components with relative weightings. A grid network is finally formed from the cost model matrix by joining all nodes and assigning the cost of traversing it to each link.

Due to some obvious drawbacks in the early version, the GCARS system was further improved by allowing movement on the diagonals of the network matrix, modifying the shape and size of network grids, and incorporating the environmental impact into the model (Turner, 1978).

Athanassoulis and Calogero (1973) also employed network optimization techniques to approach the optimal route problem. Unlike Turner’s model, where link costs are calculated by averaging the costs of two end nodes, Athanassoulis defined cost lines (like river, bridge) and cost areas (such as lake, wetland) as a basis for calculating link costs. Then the cost between any pair of nodes was accurately calculated by the summation of the length in each cost area multiplying the associated unit cost. Athanassoulis’s model is the only one found in this search, which explicitly and precisely estimates link costs.

For finding the optimal route location, instead of utilizing the shortest path algorithm, Athanassoulis formulated it as a modified transportation problem, where one unit of goods is shipped from the start point to the end point. (In fact, this is the so-called transshipment problem or generalized network flow problem.) All nodes in the resulting network are considered as possible sources as well as possible destinations so that the optimal route can move in any direction to join any two nodes without passing through the other nodes. The same network will take more time to be solved by Athanassoulis’s approach than by Turner’s and more storage space is required due to the dimension of the embedded cost matrix.

It must be mentioned that neither Turner nor Athanassoulis considered the vertical profile in their models, although Turner’s GCARS system did consider earthwork cost. Later, Parker (1977) developed a two-stage approach to optimize a route corridor subject to a gradient constraint. Although that approach was claimed to simultaneously optimize both horizontal and vertical alignments, it is more reasonable to categorize it as a 2-dimensional alignment optimizer because the vertical alignment was predetermined before optimizing the horizontal route.

In Parker’s approach, the region of interest is first divided into sub-squared zones where the mean elevation of each zone is taken to be the ground elevation at the

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zone centroid. A smooth surface is then constructed through the zone centroid elevations such that any horizontal alignment from the origin to the destination will intersect the surface with a feasible vertical alignment with respect to gradient constraints. The smoothed surface is derived by a binomial expansion of a predetermined order in both dimensions of horizontal plan. Then a linear programming regression model is run to minimize the residuals (deviations between projected surface and ground surface) with respect to the coefficients in the binomial function of the projected surface, while subject to gradient constraints and some other system equations. To some extent, the surface smoothing procedure is slightly similar to Turner’s work, where the projected surface is derived with a linear regression model without considering gradient constraints.

In the second stage of Parker’s approach, the centroids of the sub zones form a grid network. Then a multiple-path program is used to derive a set of best routes which minimize the absolute residuals (which can be regarded as a measure of earthwork cost). By its nature, the second stage is also similar to Turner’s approach, but Turner considers more cost components in the cost function.

Another study by Trietsch (1987a) presented a family of methods for the preliminary design of highway alignment. The methods were proposed for optimizing 3-dimensional alignment, but the models are 2-dimensional for the same reasons mentioned above. Trietsch’s paper includes 16 models with different structures and cost estimation procedures. Some of them fall into the network optimization category, while the others belong to the class of dynamic programming models (discussed in the next subsection). Trietsch (1987b) further applied his results to a comprehensive design of highway networks, which is capable of satisfying all bilateral (two-way) transportation demands and minimizing total cost.

Four basic search grids were proposed by Trietsch: rectangular, square, ellipse, and honeycomb. The rectangular grid is the basic search grid mostly used in a dynamic programming approach, but can also be formulated as a network problem. The apparent drawback of the rectangular grid in a dynamic programming model is that it is incapable of handling backtracking alignments, which are sometimes necessary in mountainous terrain. The other three network structures are proposed for possible backward bends with different detailed level of moving angles. The most sophisticated one, honeycomb grid, allows twelve different moving directions at each node. Since early studies in alignment optimization do not take the curvature constraints into account, Trietsch presented a modified shortest path algorithm to force the search direction at each node within a limited range of angles.

The optimal alignment derived by the network approach is a piecewise linear trajectory, which is very rough for highway alignment. In fact it is a corridor rather than an alignment. In Turner’s model, the node size is about 1 km × 1 km, whereas in Trietsch’s honeycomb approach, the diameter of the hexagons is around 500 meters. If a more accurate alignment is desired, the number of nodes must be increased and the size of the node must be decreased so that the network

Intelligent Road Design 21 can cover the whole region of interest. However, the resulting network problem quickly becomes too large to solve.

Other implied shortcomings of this approach are the calculation and storage requirements for link costs. To apply shortest path algorithm, all link costs must be determined before performing the search. If the resulting network is large, the calculation efforts and the computer storage space (usually for a matrix) are considerable.

2.4.3 Dynamic programming

Dynamic programming is developed for solving the optimization problems of complex and large-scale systems. The principal assumption of dynamic programming is that a problem can be divided into a number of sub-problems (stages) and that the contributions to the objective function value from each sub-problem are independent and additive.

Dynamic programming was widely used in optimizing highway alignments, especially for vertical alignment (see section 2.5). The stages in a dynamic programming model for optimizing horizontal alignment are usually evenly spaced lines perpendicular to the axis connecting the start and end points of the alignment. At each stage, the states are the nodes or grids on the perpendicular line. During the search, the objective function is usually evaluated from the last stage back to the first stage. To ensure the resulting alignment satisfies the curvature constraints, only a limited number of nodes in the next stage are permitted to connect to the node at the current stage. Studies using dynamic programming for optimizing horizontal alignment and 3-dimensional alignment (which can also be applied to optimize horizontal alignment) include Trietsch (1987a), France (see OECD, 1973), Hogan (1973), and Nicholson (1976).

By nature, a dynamic programming model for optimizing horizontal alignment can also be formulated as a shortest path problem, but the reverse is not always true. When compared with the shortest path algorithm, the advantage of using dynamic programming is its efficiency and lower storage requirements, because it assumes that the decision cost (for alignment, it is the link cost) at each state only depends on the remaining stages but is independent of previous stages. (This idea is known as the principle of optimality.) Therefore, the associated link costs can be deleted from memory after finishing the search of a stage. However, an obvious drawback of dynamic programming is that it is difficult to handle alignments with backward bends due to its basic assumption.

As with the network optimization approach, the optimal alignment derived by dynamic programming is not precise unless the number of stages and the number of states at each stage are sufficiently large. However, these are undesirable elements, especially when the distance between successive stages is very small, because it becomes almost impossible to fit smooth curves even though we control the exit direction (angle) at each state. Therefore, dynamic programming models usually utilize a coarse grid and then possibly a refined grid in a second search.

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