Assume the function is defined as $f(x,y)=x^2+y^2-4xy$, and $1\leq x \leq 4,1\leq y \leq 4$. The Genetic Algorithm is selected to maximize the function. If the first population for pairs of $(x,y)$ is defined as $S=\{A=(1,2), B=(2,1), C=(2,2), D=(2,3), E=(3,1) \}$.
a) Calculate the fitness of each of individuals (A,B,C,D,E) in population if: $\text{fitness function}=f(x,y)$
b) Calculate the probability of each individual and sort them in descending order. Which individual has the maximum fitness (probability)? $p_{i}=\frac{f_{i}}{\sum_{j=1}^{N} f_{j}}$
c) Draw the roulette wheel and calculate the boundaries for each individual
d) If we use two individuals and their arithmetic mean for crossover each time, and for mutation, we add 0.1 to x and subtract 0.1 from y for each individual created after crossover, what will be the next population with five members?
For part (d), use the following random numbers in order whenever you need them in the selection process:
$\text{random numbers} = \{0.780,0.220,0.776,0.507,0.822,0.765,0.288,0.881,0.895,0.421\}$
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