7. CAPÍTULO 3. El Carnaval de Negros y Blancos
7.2. Antecedentes Históricos
= n
i f
F E i
E
0
) 27 (
1 , (7.65)
whereEC and EF are total earth excavation and embankment (cu yd).
7.2.1.4 Estimation of earthwork cost
In highway design, one major way to minimize the cost of earthwork is to balance (i.e., equalize) the volumes of cuts and fills. However, when earth is excavated and hauled to form an embankment, the material may be compacted and the final volume may be less than its original quantity. This phenomenon is called “shrinkage”. The amount of shrinkage varies with the soil type and the depth of the fill. Let se denote the earth shrinkage factor. Then the net earthwork volume will be
F e C
N E s E
E = − . (7.66)
If the net earthwork EN is positive, then the extra earth must be shipped to a landfill. If EN is negative, then the deficiency must be supplemented from a borrow pit. Let Kl be the transportation cost for moving one cubic yard of earth to a landfill, and Kb represent the transportation cost for moving one cubic yard of earth from a borrow pit. Further denote KC and KF as cutting and filling cost per cubic yard. Then the total earthwork cost for an alignment alternative can be computed as
{
,0}
min{
,0}
max N b N
l F F C C
E K E K E K E K E
C = + + − , (7.67)
where CE is total earthwork cost ($), including excavation, embankment, and transshipment cost.
7.2.2 User costs
One of the key factors in estimating user costs is average running speed, through which we can compute vehicle operating costs and travel time costs. In section 4.6, we have already presented a formula to estimate average running speed based on various geometric design features and traffic conditions (see eqns (4.46) to (4.48)). In optimizing horizontal alignments, the average hilliness H used in eqns (4.46) to (4.48) is ignored due to lack of information about vertical profiles. In optimizing 3-dimensional alignments, however, H can be determined from the corresponding vertical alignments.
The calculation of H is given in eqn (4.39). To compute H , the vertical distance between adjacent sag and crest must be determined. Using the same notation, we find that
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(1) If gi>gi−1 and gi+1<gi (sag and then crest), then )
( ) ( i1 r i
r
i z V z V
h = + − , (7.68)
where hi = vertical distance between the ith sag and the following (i+1)th crest )
( i
r V
z and zr(Vi+1) are road elevations at control points Vi and Vi+1, and can be computed following the procedures in section 7.2.1.2.
(2) If gi<gi−1 and gi+1>gi (crest and then sag), then )
( )
( − +1
= r i r i
i z V z V
h . (7.69)
With eqns (7.68), (7.69), (4.38) to (4.40), and (4.46) to (4.48), we are able to estimate average running speed for a 3-dimensional alignment, which will be further used for computing vehicle operating costs and travel time costs. For detailed discussion and formulas, please refer to sections 4.6.1 and 4.6.2.
7.3 Final model and its properties
In this section, the final model formulations for optimizing non-backtracking 3-dimensional alignments are presented. We will first discuss the constraints that must be considered in this model and then covert them into penalty functions.
Finally the penalty costs will be incorporated into the objective function.
Recall that the decision variables in this model include the abscissa di and the ordinate zi of each intersection point Pi on its associated vertical cutting plane.
The domain constraints of di’s are defined in such a way that Pi’s are restricted within the study region, as shown in eqns (4.4) to (4.7). However, the ordinates zi’s do not have any domain constraints. Instead, zi’s are restricted by the gradient limitations required by highway designs. The maximum grade depends on the design speed and the surrounding topography, and is usually treated as a design control parameter. Let Gmax denote the maximum allowable gradient (%).
Then the gradient constraint can be expressed as n i
G
gi ≤ max,∀ =0,..., , (7.70) where gi = the road grade at tangent segment ViVi+1, obtained with eqn (7.7).
As shown in section 4.6.3, the minimum radius constraints are incorporated into the objective function by adding penalties to accident costs. In addition, the horizontal alignment generated by Algorithm 4.1 will satisfy the continuity constraint. For vertical alignment, the continuity constraint is also satisfied.
However, the minimum required sight distance may be violated. Taking all these into account, the problem can be formulated as
Intelligent Road Design 147
Model 3 – Model for optimizing non-backtracking 3-dimensional alignments
E where the horizontal alignment is produced with Algorithm 4.1 while the vertical alignment is generated with Algorithm 7.1
CT = total cost ($)
CN = location-dependent cost ($), computed with Algorithms 4.2 and 4.3
CL = length-dependent cost ($), given in eqn (4.35)
CU = user cost ($), given in eqn (4.71) with updated average running speed
Lm = minimum required length of vertical curve at Vi )
,
(di zi = the coordinates of the intersection points on the ith vertical cutting plane.
In the optimization process, penalty functions representing constraints (7.73) and (7.74) will be added to the objective function. Let Cg(i) denote the penalty cost for violating gradient constraint at the ith tangent. Then Cg(i) is given the following function form:
( )
i nwhere α = penalty cost when gradient constraint is violated 0 α and 1 α (2 α2>1) are user-specific coefficients.
The total penalty cost for violating gradient constraints, denoted by CG, is then obtained by the summation of individual penalty cost at each tangent. That is
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Similarly, the penalty cost for violating the minimum length of vertical curve is computed as
( )
i nwhere Cm(i) = penalty cost for violating the minimum length of vertical curve at Vi
β = fixed penalty cost coefficient for violating the minimum length 0
of vertical curve
β and 1 β (2 β2>1) are user-specified coefficients.
where CM = total penalty cost for violating minimum length of vertical curve.
Adding the above penalty costs into the objective function, we may rewrite Model 3 as: The penalty function approach allows the constraints to be violated slightly during the search. The magnitudes of penalty coefficients α and i β determine i the tradeoff between constraint violations and other costs. The selection of α i and β depends on the terrain and the functional categories of the highway. i Larger coefficients tend to force the final solutions to satisfy the constraints as closely as the users wish.
The objective function defined in the above model includes most of the important costs considered in highway design. The design constraints, including minimum radius, maximum gradient, and minimum length of vertical curve are all taken into account by penalizing the violations of constraints. Moreover, the horizontal alignment generated by Algorithm 4.1 and the vertical alignment produced by Algorithm 7.1 will hold the continuity condition (defined in eqn (3.2)) and the first continuously differentiable condition (defined in eqn (3.3)). In addition, the necessary conditions of alignments defined in eqns (3.13) and (3.14) are also satisfied. To complete the model, we now need an efficient search algorithm to solve it. In the next section, we will define an appropriate genetic encoding scheme and operators to perform the search.
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7.4 Genetic encoding and initial population
In Model 3, each intersection point is determined by two variables, the abscissa and ordinate on the associated vertical cutting plane. For an alignment with n intersection points, the encoded solution will consist of 2n genes. Therefore, the chromosome is defined as
]
It can be seen that the mappings between the genes in a chromosome and the coordinates of the intersection points are
i
The alleles of odd genes in a chromosome will be limited to the range of the corresponding abscissa. That is
n i
d
diL ≤λ2i−1≤ iU ,∀ =1,..., . (7.84) Eqn (7.84) is the domain constraint of Model 3 as shown in eqn (7.80). The constraint will be always satisfied throughout the solution algorithm by restricting the mutation range of the genes.
To maintain a large variety in the gene pool, the initial population of intersection points includes the following 3 categories:
(1) Intersection points lying on the straight line connecting the start and end points
In this case, the set of intersection points represents a straight alignment, which reduces length-dependent cost to a minimum. The chromosome is defined as
]
z in the above equation is simply determined by
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(2) Intersection points lying randomly on the vertical cuts with random elevations
In this case, the odd genes of the chromosomes (i.e., di’s) are randomly generated from continuous uniform distributions defined by the boundaries of the corresponding vertical cuts:
] ,
1 [
2i− =rc diL diU
λ , ∀i=1,...,n. (7.87)
The even genes (i.e., zi ’s) are randomly generated from the ranges determined by the gradient constraints. Note that given the odd genes, the Xand
Ycoordinates of the decoded intersection points can be determined by eqn (7.1), and thus the corresponding horizontal alignment can be obtained with Algorithm 4.1. Consequently, the distance along the horizontal alignment between any two successive control points Vi and Vi+1 can be computed with eqn (7.6). Recall that the ith even gene represents the elevation at Vi, namely λ2i =zi =zVi . Assume that the even genes are generated from the first one (i.e., λ ) to the last 2 one (i.e., λ ). Then the range of the 2n ith (i=1,....,n) even gene, denoted by the interval [ziL,ziU], is determined according to the following equations:
horizontal alignment, computed from eqn (7.6)
Gmax = maximum allowable gradient (%).
Hence, we get
Note that the first term on the right-hand side of eqn (7.88) is the lower bound of the elevation at Vi determined by the gradient constraint from Vi−1 to Vi,
Intelligent Road Design 151 while the second term is the lower bound dominated by the gradient constraint from Vn+1 to Vi. The final lower bound ziL is then taken as the one that is larger.
Similarly, the final upper bound ziU is determined by the allowable upper bounds from Vi−1 to Vi or from Vn+1 to Vi, whichever is smaller.
(3) Intersection points lying randomly on the vertical cuts with elevations possibly close to the existing ground elevations
This population type is similar to the previous category. The odd genes of the chromosomes are generated by eqn (7.87). The even genes, which represent the elevations of intersection points, are set as close as possible to the existing ground elevations at the corresponding control points Vi’s, but should be within the allowable range calculated by eqns (7.88) and (7.89). The ground elevation at the ith control point Vi, denoted by zg(i), is determined by the XY coordinates of Vi. In the format of study region, zg(i) is the elevation of the cell whose indexes are computed by eqns (7.41) and (7.42). If zg(i)<ziL, then λ is set to 2i
ziL. If zg(i)>ziU, then λ is set to 2i ziU. If ziL<zg(i)<ziU, then λ is taken as 2i )
(i zg .
According to the discussions in section 5.7, the population size is set to be proportional to the number of decision variables. In Model 3, if n intersection points are used to represent the alignment, the total number of genes in a chromosome will be2n. Thus, a population size np=10n is recommended.
7.5 Genetic operators
Eight different types of genetic operators are employed in solving Model 3. The first four are mutation-based operators, while the last four operators are crossover-based. The operators discussed in this section are similar to those developed for Models 1 and 2. However, in order to take the effects of 3-dimensional alignments into account, several modifications are made to facilitate the search. To fit the nature of the problem, all operators are intentionally designed to work on the decoded intersection points rather than a single encoded gene. We will now briefly discuss each operator in turn.
7.5.1 Uniform mutation
For a given chromosome Λ=[λ1,λ2,...,λ2n−1,λ2n], if the kth intersection point is selected for uniform mutation, where k=rd[ n1, ], then λ2k−1 will be replaced by
] ,
1 [
2′k− =rc dkL dkU
λ . (7.91)
Since the allowable ranges of elevations at intersection points depend on the horizontal alignment, the other encoded gene of the kth intersection point (i.e.,
2k)
λ , which represents the elevation, will not be changed until the new horizontal alignment is determined. As mentioned in section 5.5, a curve
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elimination procedure is required to prevent a horizontal alignment from getting trapped at a local optimum. Hence, λ must be changed after applying this 2k procedure. The procedure for Model 3 is shown below:
Algorithm 7.2 Curve elimination procedure for Model 3