Q$.4.$ Consider the simple optimization problem using Genetic Algorithm: $[15]$ Maximize,

f$($x$)=64-$ x$^2$

$1<=$ x $<=64$

x is an integer

$($i$)$ How many binary digits are present in each of the string for the above problem?

$($ii$)$ What is the fitness function you would assume in the above problem?

$($iii$)$ Assume that $5$ random strings are created corresponding to x values, $63,4,1,10,36.$

Write the binary strings corresponding to the values of x and write down the fitness values.

$($iv$)$ Find out the expected number of copies of the best string in the above population in the mating pool under roulette$-$wheel selection, based on the following random numbers $0\mathrm{.}5469,0\mathrm{.}9575,0\mathrm{.}9649,0\mathrm{.}1576,0\mathrm{.}9706$

$($v$)$ Write the schema corresponding to x $>=33$

$($vi$)$ If single point crossover with probability $0\mathrm{.}9$ and a bit$-$wise mutation with a probability of $0\mathrm{.}01$ are used, how many copies of the above schema are expected in generation five?