An overview of the cost function

As mentioned in chapter 2, the costs associated with highway transportation include planning and administrative costs, construction costs, maintenance costs, user costs, and environmental costs. Since planning and administrative costs are insensitive to alignment alternatives, they will be excluded from the proposed model. Note that most existing models only consider in their objective function construction costs, or even just one of its components such as earthwork cost.

The absence of a comprehensive cost function in existing models seems due to

62 Intelligent Road Design

the lack of studies on the relations between these cost items and the design variables and also, possibly to the mathematical difficulties of minimizing intricate cost functions with previously available optimization methods. In this book, we attempt to establish a relatively comprehensive objective function (i.e., to satisfy condition (1) of a good model, as described in section 2.7). The solution algorithm is then developed based on the nature of the final model Thus, we endeavor to solve the actual problem, rather than fit the problem into an existing solution tool.

The way we formulate the objective function is to categorize all cost items into location-dependent, length-dependent, area-dependent, volume-dependent and VMT-dependent costs. With this approach, all cost items can be easily related to the design variables. This is very important because the objective function must be able to reflect the features of the alignment, which is described by the set of decision variables. In the previous section, we proposed Algorithm 4.1 to generate the alignment based on a given set of decision variables. (In the proposed model, these are the coordinates of the set of intersection points.) With the resulting alignment, we must quantify all dominating and sensitive costs in terms of decision variables. Assuming that the projected traffic demand is given and the number of traffic lanes is fixed, then area-dependent cost (e.g., pavement cost) and VMT-dependent cost (e.g., environmental cost) can be further converted into length-dependent cost. Without the detailed vertical profile of the alignment, we are not able to precisely calculate the volume-dependent cost (e.g., earthwork cost) because we are optimizing the horizontal alignment here.

However, to some extent, earthwork cost is also location-dependent. For example, an alignment alternative passing through a mountainous cell will incur higher earthwork cost than one passing through a flat zone. Although the relation is loose, it does exist and we will provide some mechanisms to clarify it. The only major cost that cannot be easily classified into a single category is user cost.

As discussed in chapter 2, user costs include vehicle operating, accident, and travel time cost, which are affected by different features of the alignment.

Therefore, we need an additional section to discuss them. Note that the estimations of user costs are highly dependent on future traffic, which cannot be precisely forecast. For design, Annual Average Daily Traffic (AADT) is usually employed. The prediction of traffic demand is beyond the scope of this book. We assume that AADT is obtained from other studies and is considered to be given information for our proposed model. The precision of AADT is not considered an issue here.

Since each of the aforementioned cost items consists of complex computations, we will discuss them separately in different sections. In the next one, the estimation of location-dependent costs is presented. Section 4.5 will explain the estimation of length-dependent costs. The estimation of user costs is then discussed in section 4.6.

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4.4 Location-dependent costs

Each cell in the study region represents a zone of different construction cost.

Building a highway through different cells may incur different costs. Let C( ji, ) denote the cell bounded by x=x_O+iD_x, x=x_O+(i+1)D_x, y= y_O+ jD_y, and is an index used to identify inaccessible cells, environmentally sensitive areas, or historical preservation areas, where the alignment is not allowed to pass. If somehow a cell is inaccessible or sensitive, then K_I( ji, ) will be assigned to a very large positive number. Since our objective is to achieve the minimal cost, a high cost cell will be avoided during the optimum search.

(x_O , y_O) x=x_O+D_x

Figure 4.9: Cell definition for the study region.

64 Intelligent Road Design )

, ( ji

K_I is very useful in formulating the problem. For example, if the region of interest is not rectangular, it can be modeled as a collection of cells in a rectangular grid, where inaccessible regions are represented by cells with high availability cost via K_I( ji, ). Thus, any study region could be similarly transformed into a format acceptable to the proposed model. Figure 4.10 shows an example of such mapping.

The unit earthwork cost K_E( ji, ) cannot be precisely calculated without the actual vertical alignment. However, one can imagine that if the alignment passes through a cell whose elevation is much different from the elevation at start and end points, then the resulting earthwork cost will be higher. Based on this observation, Parker (1977) has developed a linear programming regression model to construct the projected road surface and estimate the earthwork volume for each cell. (For more detailed discussion, refer to section 2.4.2.) Another approach to estimate earthwork cost without presenting a vertical profile is proposed by Sthapit and Mori (1994). In their model, the earthwork cost is estimated by linear regression analysis, given the ground elevations along the horizontal alignment. The independent variables are the average slope of the ground profile (called the slope factor) and the average difference between the actual change in ground elevations and the maximum allowable change in ground elevations (called the hill factor). The coefficient of determination (R²) shows that their model has significant statistical performance.

Both methods for estimating earthwork cost should be valid for the proposed model; However, Parker’s approach is preferred because it also uses a data format which is consistent with the proposed model.

Here we assume that K_N( ji, ) only includes land acquisition cost, soil stabilization cost, availability or environmental impact, and earthwork cost. In fact, more cost items can be added as long as the cost information is available and all components are properly weighted according to their relative importance.

feasible region

Legend: High cost cell

Figure 4.10: The transformation of a non-rectangular shape into a rectangle.

Intelligent Road Design 65 To compute location-dependent cost for an alignment alternative, we must first locate the cells through which the alignment passes and then calculate the costs incurred in each cell. To simplify, we assume that the highway width is fixed. Then the associated location-dependent cost C_N for a highway alignment is

W = the width of the highway, which is assumed to be fixed along the alignment.

In eqn (4.14), L( ji, ) is a function of the set of decision variables d_i’s.

However, the function form is not explicit. In fact, we are unable to express )

, ( ji

L explicitly in terms of d_i ’s. L( ji, ) is computable only after d_i’s are specified. Once d_i’s are given, a human could easily calculate L( ji, ) by looking at the map to see what cells the alignment passes through, and then calculate the location-dependent costs incurred in each cell. Finally, C_N can be obtained by applying eqn (4.14). Unfortunately, computers do not possess this visual intuition, and therefore, must be programmed more precisely to identify the cells that the alignment passes through, and then calculate the associated location-dependent cost.

As discussed in the previous section, the alignment generated by Algorithm 4.1 contains tangent sections and circular curves. For notational convenience, we further denote T₀ =P₀ =S and C_n+1=P_n+1=E as the start and end points of the alignment. Then we observe that T_i and C_i+1 are linked by a straight-line section for all i=0,...,n, whereas C_i and T_i are connected by a circular curve with radius R_i for all i=1,...,n (see Figure 4.11 for an example). In some extreme cases, where the tangent section between two intersection points is completely eliminated by two circular curves, the point of tangency pertaining to one intersection point will coincide with the point of curvature pertaining to the next intersection point. For example in Figure 4.11, T₄ and C₅ are the same

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Figure 4.11: An example of points of tangency and curvature.

Since the logical and mathematical requirements for determining the length of the alignment in each cell are different for tangent sections and circular curves, they will be discussed separately in the following subsections:

4.4.1 Location-dependent costs of tangent sections

Since each of the cells is a convex set, any linear combination of two points in a cell will fall into the same cell. It is noted that each tangent section will intersect horizontal cell grids (parallel to X coordinate) and/or vertical cell grids (parallel to Y coordinate), or none, depending on the slope and length of the section.

Therefore, if we can find all of these intersection points and sort them in either the X or Y coordinate, then the mid-point of any two consecutive points will give a clue about the cell where the line segment occurs.

Intelligent Road Design 67 grids. The intersection points are then sorted by the magnitude of their X coordinates. If m_i approaches ±∞(i.e., x_Pi+1 =x_Pi), then Link(i) only intersects horizontal grids. Therefore, the intersection points are sorted by the magnitude of their Y coordinates. If x_Pi+1 ≠x_Pi and y_Pi+1 ≠ y_Pi (the most general case), we should find out all intersection points of Link(i) at both vertical and horizontal grids, and then sort them according to their X coordinates.

Let Q₁⁽ⁱ⁾, Q₂⁽ⁱ⁾,…….,Q be the resulting points after sorting, including two _l⁽ⁱ_i⁾ end points of Link(i)(see Figure 4.12 for a general example), and assume that the coordinates of Q⁽ⁱ_j⁾ are represented by (x_Q⁽ⁱ⁾_j,y_Q⁽ⁱ⁾_j). Then the line segment between two consecutive points Q⁽ⁱ_j⁾ and Q⁽_jⁱ₊⁾₁ will exist in exactly one cell with the following indexes:

1^st index:

where



)

Figure 4.12: Sorted intersection points of a tangent section.

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The indexes calculated with eqns (4.18) and (4.19) indicate the cell where the j^th segment of Link(i) is located. To simplify notation, let the unit location-dependent cost of the cell be further denoted by _LK_N⁽ⁱ_⋅⁾_j. Then the location-dependent costs of the alignment along all tangent sections can be calculated as



where C^T_N = the location-dependent costs of the alignment along all tangent sections

With the above approach, we would be able to calculate the location-dependent costs for the tangent sections of a given alignment. The algorithm for calculating C^T_N is summarized as follows:

Algorithm 4.2 Computation of location-dependent costs for tangent sections