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Let us illustrate the use of an elementary GA based on a real‐coded version of Figure 1.8. In particular, let us attempt to find the value of x that maximizes the function

(1.6A-1)

For the purposes of solving this problem, we will use linear coding with x_mn,1 = x_mn,2 = 0 and x_{mx, 1} = x_{mx, 2} = 5. In order to form the mating pool, we will use two‐way tournament selection. We will also assume simple‐blend crossover with α = 0.5 and a probability of crossover of 0.5. We will use total mutation with the probability of a gene mutation of 0.1. In this example, we will consider three generations with a population size of 8. The code for this example can be obtained from Sudhoff [6].

Table 1.3 lists the data for the evolution. The first population’s parameters were determined by random initialization. Next, the parameter vector for each member of the population was decoded yielding xⁱ for every member of the population. This is then evaluated by calculating fⁱ = f(xⁱ), where the fitness is given by (1.6A-1). At this point, the maximum, median, and mean fitness values of the population are 0.2213, 0.04123, and 0.06940, respectively. The next step is the creation of the mating pool using tournament selection. Applying simple‐blend crossover, mutation, and associated gene repair operators yields the second population. Decoding to obtain the parameter vector for each element and then evaluating the fitness, it can be seen that maximum and mean fitness values of the population have increased to 0.4886 and 0.09160 while the median fitness is decreased to 0.03661. Repeating the process a third time, the maximum, median, and mean fitness increase to 0.7719, 0.08486, and 0.1774, respectively. After the third generation, taking the individual with the highest fitness yields x^* = [1.942 3.233]^T. This is starting to approach the correct solution, which is x = [2 3]^T. Note, however, that a different run will produce different results since a population size of 8 over three generations is inadequate to find a consistent solution.

Before concluding this example, it is interesting to repeat it with a more substantial population size and over a larger number of generations. Figure 1.13 illustrates an optimization run over eight generations with a population size of 25 individuals. In Figure 1.13(a), the best fitness in the population, median fitness of the population, and mean fitness of the population are shown versus generation. All three of these quantities can be seen to increase with generation number, though not monotonically. Figure 1.13(b) depicts a plot of the individuals in terms of their parameter vectors. Therein, an o, x, +, *, star, diamond, triangle, and pentagram depict members of the first through eighth generations, respectively. In addition, the darkness of the shading of each point is proportional to generation number. Thus, it can be seen that as the evolution progresses, the population moves toward the objective function maximizer. At the end of the run, the estimate of the solution is x^* = [2.133 2.755]^T, which yields f(x^*) = 0.797. With the given limited number of generations and small population size, the results vary from run to run. This variance could be eliminated with a larger population and more generations, but it would make the resulting figures (particularly Figure 1.13(b)) hard to trace.

Table 1.3 Example of Real‐Coded Genetic Algorithm Evolution

Generation 1
P[1]	0.32100.8296	0.82220.5707	0.57180.2860	0.69910.7963	0.44160.4462	0.46570.2790	0.67540.9037	0.90850.7472
x i	1.60514.1478	4.11092.8534	2.85911.4301	3.49573.9813	2.20792.2311	2.32831.3952	3.37694.5183	4.54263.7360
f i	0.1492	0.0295	0.0689	0.0148	0.2213	0.0530	0.0104	0.0081
M[1]	0.57180.2860	0.46570.2790	0.82220.5707	0.32100.8296	0.46570.2790	0.82220.5707	0.44160.4462	0.32100.8296
Generation 2
P[2]	0.57180.2860	0.13550.3321	0.89750.7424	0.65340.6579	0.46570.2790	0.52790.0321	0.36270.6971	0.84670.5786
x i	2.85911.4301	0.67741.6606	4.48743.7118	3.26683.2895	2.32831.3952	2.63940.1604	1.81333.4857	4.23362.8932
f i	0.0689	0.0313	0.0086	0.0419	0.0530	0.0155	0.4886	0.0249
M[2]	0.13550.3321	0.57180.2860	0.36270.6971	0.57180.2860	0.84670.5786	0.84670.5786	0.57180.2860	0.65340.6579
Generation 3
P[3]	0.13550.3321	0.57180.2860	0.54620.3365	0.38830.6467	0.84670.5786	0.84670.5786	0.62350.5216	0.60170.4223
x i	0.67741.6606	2.85911.4301	2.73081.6824	1.94173.2334	4.23362.8932	4.23362.8932	3.11742.6082	3.00852.1114
f i	0.0313	0.0689	0.1008	0.7719	0.0249	0.0249	0.1625	0.2335

Figure 1.13 Fitness and gene values for Example 1.6A.

While Example 1.6A helped to illustrate the operation of a real‐coded GA, practical GAs typically have several additional features to increase their efficacy. We will now briefly consider some of these additional operators.