Recall that the decision variables in the proposed model are the coordinates di’s along vertical cutting lines. The objective function is the summation of various cost items. Hence the final model can be formulated as follows:
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Model 1 – Model for optimizing non-backtracking horizontal alignments
U L N d T
d
d C C C C
n
+ + =
Minimize
,..., , 2
1
, (4.73)
subject to diL ≤di ≤diU, for all i=1,...,n, (4.74) where the alignment is generated by Algorithm 4.1
CT = total cost ($)
CN = location-dependent cost ($), computed by Algorithms 4.2 and 4.3 CL = length-dependent cost ($), given in eqn (4.35)
CU = user cost ($), given in eqn (4.72)
diL and diU are lower and upper bound of the ith decision variable.
The above model form differs from that formulated in Model 0. Model 0 is a rigorous definition of the problem for optimizing highway alignments, but is not a solvable form. On the other hand, Model 1 is a computable form because we have already shown the relations between different cost items and the decision variables di’s. For any given set of decision variables, we will be able to compute the objective value. Regardless of gradient and vertical curvature constraints, and of the ability to deal with a backtracking alignment, we want to show the equivalence of Model 1 to Model 0 because Model 0 provides a more rigorous mathematical definition of alignment optimization problems.
The objective function defined in eqn (3.10) includes various cost items. The area-dependent cost CA and volume-dependent cost CV appearing in eqn (3.10) have been incorporated into length-dependent cost CL and location-dependent cost CN in eqn (4.73) respectively. Regardless of spiral transition curves, the alignment generated by Algorithms 4.1 will definitely satisfy the boundary conditions (eqns (3.11) and (3.12)), and alignment necessary conditions (eqns (3.13) and (3.14)). Moreover, both the continuity condition (defined in eqn (3.2)) and the first continuously differentiable condition (defined in eqn (3.3)) are also satisfied. The horizontal curvature constraint defined in eqn (3.15) is combined into user cost by introducing a big penalty into accident cost, and thus will be satisfied if the solution algorithm successfully locates the optimal alignment.
Finally, the inaccessibility constraint (eqn (3.18)) is incorporated into the location-dependent cost and the alignment is bounded by eqn (4.74). Therefore, it will also be satisfied.
Note that the objective function CT in eqn (4.73) is the summation of different cost components and thus, possible tradeoffs among various cost items may be made. For example, the increase in accident cost for a sharper curve may be compensated by the savings in location-dependent cost. The optimized alignment should have the lowest total cost by making the best tradeoffs among different cost components.
Intelligent Road Design 93 Next, we want to examine whether the proposed model satisfies the necessary conditions of a good model for optimizing highway alignment as described in section 2.7. The results are summarized in Table 4.5.
Table 4.5: Checklist of necessary conditions for a good alignment optimization.
model
No. Description of Conditions Check Box
(1) Consider all dominating and sensitive costs yes (2) Formulate all important constraints yes
(3) Yield a realistic alignment yes
(4) Be able to handle alignments with backward bends no (5) Simultaneously optimize 3-dimensional alignments no
(6) Find globally or near globally optimal solution not investigated yet (7) Have an efficient solution algorithm not investigated yet (8) Have low storage requirements possibly
(9) Have a continuous search space yes (10) Automatically avoid inaccessible regions yes
(11) Be compatible with GIS yes
The above table shows that conditions (4) and (5) are not satisfied in the proposed model. We are not yet able to investigate conditions (6) and (7) at this point because the solution algorithm has not been presented. For condition (8), the proposed model employs an efficient mechanism compatible with GIS data format for storing the information of the study region at a desired precision level.
However, the memory requirement also depends on the solution algorithm.
The main task remaining in developing the proposed model is to design an efficient solution algorithm. Note that the search space is an n-dimensional hyperspace rather than a 2-dimensioanl plane because the number of decision variables is n even though we are only optimizing a 2-dimensional alignment.
By carefully investigating the detailed structure of the model, we find that the objective function is an implicit function of the decision variables set. The computation of each cost component requires the coordinates of points of curvature, points of tangency, and points of intersection, as well as the intersection angle and radius at each circular curve. However, the information is not available until the decision variable set is given and Algorithms 4.1 has been applied to generate the alignment. Consequently, the problem turns out to be an implicit, constrained optimization model with a non-differentiable objective function, which is probably very noisy and has many local optima.
Due to the properties of the model, no gradient-based search algorithms are applicable to the problem. As a result, only direct search methods can be used.
This seems to be an unpromising approach to the problem because most existing direct search methods are unable to locate the global optimal solution unless a very exhaustive search is employed. Then how shall we solve the problem? Is
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there any method other than the conventional search algorithms with the potential to find the global optimal solution or at least a relatively good local optimum? Fortunately, with the development of artificial intelligence (AI) methods, it becomes possible to efficiently search for a relatively good solution without any gradient information. The proposed approach is known as Genetic Algorithms. In Appendix A, a brief introduction to the method is provided. A comparison of this method to other optimization techniques is also presented.