Solution and goodness test

Since the terrain is not extremely difficult, we may expect that the optimal alignment is non-backtracking. Therefore, Model 3 is employed to run the test example. The parameters for running the proposed genetic algorithm are summarized in Table 9.2.

172 Intelligent Road Design

Table 9.2: Genetic parameters for test example 1.

Parameter Value Number of intersection points n 10

Population size n_p 100

Maximum number of generations n_T 1500

Number of operators in each generation 4 for each type of genetic operator Coefficient of selective pressure q 0.1

Parameter for non-uniform mutation ξ 6

Recall that the solution algorithm is probabilistic and the result is dependent on the random seed. For this reason, we run the program 10 times. The results are given in Table 9.3. The sample mean of these ten runs is 102.873 million dollars and the standard deviation is 0.767 million, about 0.7% of the mean value. A small coefficient of variation (i.e., standard deviation ÷ the mean) indicates that the solutions found with different random seeds are consistent and the algorithm converges to solutions of similar quality. Table 9.3 also shows that the 6^th run ends up with the lowest total cost (101.764 million). The graphical presentations of the horizontal and vertical alignments for the 6th run are displayed in Figure 9.4 and Figure 9.5.

Figure 9.4 shows that the optimal horizontal alignment skirts the two small hills to minimize earthwork cost and avoids high cost cells to minimize location-dependent cost. The optimal vertical alignment in Figure 9.5 also shows that the alignment is very close to the ground elevation in order to minimize earthwork excavation and embankment. The results are consistent with our expectations.

Table 9.3: Program results for test example 1.

Run Objective Value Found (Units: $1,000,000)

1 103.338 2 104.481 3 102.940 4 102.719 5 103.263 6 101.764 7 102.720 8 103.112 9 102.396 10 101.992 Mean 102.873

Standard Deviation 0.767

Intelligent Road Design 173

Figure 9.4: The optimized horizontal alignment for test example 1.

Vertical alignment Elevation

66 234

0

Location of vertical control points

1 2 3 4 5 6 7 8 9 10 11

Figure 9.5: The optimized vertical alignment for test example 1.

In order to visualize the evolution of the program, we plot the objective value versus the generation number for the 6^th run in Figure 9.6. The figure shows that the objective values in the first few generations are extremely high. However, the value drops sharply down to 120 million at the 10^th generation, and to 107 million at the 43^rd generation. After that, the improvement in the objective value becomes very slow. The final objective value of $101.764 million is reached at the 1498^th generation.

174 Intelligent Road Design

0 50 100 150 200 250

0 500 1000 1500

Number of generations

Objective value (1E+6)

Objective value

43

Figure 9.6: Objective value through successive generations for test example 1.

Although the solution found by the proposed algorithm seems to be reasonable, we wish to assess how good the solution is. Since we do not know the exact optimal solution to the problem (note that no existing methods can guarantee finding the global optimum), it is very hard to prove the goodness of the solution found by the proposed algorithm. Therefore, we design an experiment to statistically test the goodness of the algorithm. The experiment is initialed by randomly generating solutions to the problem. For each of them we then evaluate its objective value. This procedure is a sampling process. To maximize the generality and satisfy the statistical requirements, the sample must be created in such a way that the solutions are representative and independent of each other. This is similar to the way we generate the second type of initial population for the genetic algorithm, as presented in section 7.4.

The next step in the experiment is to fit a distribution to the objective values for the random sample. The fitness of the distribution can be checked with the Chi-Square or K-S tests (Neter, et al, 1982). Since the sample is randomly generated, the fitted distribution should be able to reflect the actual distribution of the objective value for the real population. Based on this distribution, we can compare the solution found by the proposed algorithm and calculate the cumulative probability of the solution in the distribution. A lower cumulative probability means that most solutions end up with a higher objective value than the one found by the proposed algorithm. The lower the probability, the better the solution.

Intelligent Road Design 175 Following the experiment, we first create a random sample of 40,000 observations. The objective value of the best solution in this sample is 164 million. The worst solution yields an objective value of 2630 million. The sample mean is 602 million and the standard deviation is 234 million. Next, we must fit an appropriate distribution to the objective value. After trying different distributions, it is found that the best fitting one is the Gamma distribution shown below.

TC= 164+Gamma(129, 3.4). (9.1)

The above distribution shows that the objective value has an offset of 164 million. In other words, the minimum value in the distribution of objective value is 164, which is much higher than the solution (101.764) found by the proposed algorithm. Figure 9.7 shows the relative positions of the solution found at the 4^th and 1498^th generations in the distribution diagram. It is found that after 4 generations, the objective value (150) is lower than the lower bound (164) of the fitted Gamma distribution. Of course, the final solution (101.764) dominates all possible values in the distribution.

The above analysis indicates that the solution found by the proposed algorithm is remarkably good when compared with other possible solutions to the problem. However, the result also raises an interesting question: why can the solution found by the proposed algorithm easily dominate other solutions, even at the first generation. There are two main reasons: (1) the proposed initial population members in the genetic algorithm are generated in such a way that they carry very useful information about the optimal solution, and (2) the genetic operators are designed in such a way that the population is very likely to evolve toward the optimal solution. As discussed earlier, in this book several modifications are made to the classical GAs and special logic is added to the

101 (1498 generations) 150 (4 generations)

164 2630

Figure 9.7: The fitted distribution of the objective value for test example 1.

176 Intelligent Road Design

evolution procedure, based on our understanding of the problem’s nature. This enables the search algorithm to find a relatively good solution.