
Genetic Algorithm
Find the value of $\displaystyle x $ that maximizes $\displaystyle \sin^{4}(x) $, $\displaystyle 0 \leq x \leq \pi $ to an accuracy of at least one part in a million. Use a population size of fifty and a mutation rate of $\displaystyle 1/(\text{twice the length of string}) $.
So randomly select a population of 50 binary string of length 8. Decode them into base 10. Look at their fitness levels (e.g. $\displaystyle \sin^{4}(x) $). Now exclude $\displaystyle 25 $ of the strings with the lowest fitness levels. Use crossover between random pairs of strings to get 25 "child strings." Now use a mutation rate of $\displaystyle 1/16 $ on this new population of strings? Because you dont want a population of strings with end digit 0. This will cause domination.
Is this generally correct? How would you decide the length of the strings?

The maximum value of $\displaystyle \sin x $ is $\displaystyle 1 $. So the maximum value of $\displaystyle \sin^{4} x $ is $\displaystyle 1 $. So maybe use a string length of $\displaystyle 3 $? Because $\displaystyle 111 = 7 $ in base 10.