Apply genetic algorithm $($GA$)$ to maximize the following fitness function:

$f(x)=12x^5-975x^4+28,000x^3-345,000x^2+1,800,000x$

subject to

$0x31.$

For simplification, Table $1$ is readily supplied with the chromosomes of initial population. Table

$2$ is meant to perform the genetic operators of selection and reproduction $($crossover and

mutation$)$ for iteration $1.$

Table $1$: Initialization

Table $2$: Selection and reproduction of iteration $1$

Fill up Tables $1$ and $2$ accordingly. Show steps to solutions whenever necessary.

Procedure for the selection and reproduction of this GA is as follows:

$1.$ Selection

a$)$ Identify the five chromosomes of initial population having the highest degree

of fitness to become parents;

b$)$ Sort them in descending order;

c$)$ Assign the uniform intervals of random number $[0,1].$ Since there are five most$-$fitting chromosomes, the intervals are $[0\mathrm{.}0000,0\mathrm{.}1999],[0\mathrm{.}2000,0\mathrm{.}3999]$

$[0\mathrm{.}4000,0\mathrm{.}5999][0\mathrm{.}6000,0\mathrm{.}7999],$ and $[0\mathrm{.}8000,0\mathrm{.}9999]$;

d$)$ Assume the random number, R $=0\mathrm{.}9665$ to reject one of these chromosomes;

e$)$ Two chromosomes among the five least fit of initial population are to be

determined;

f$)$ Sort the least$-$fitting chromosomes in descending order;

g$)$ Assign the uniform intervals of random number $[0,1].$ Since there are five least$-$fitting chromosomes, the intervals are $[0\mathrm{.}0000,0\mathrm{.}1999],[0\mathrm{.}2000,0\mathrm{.}3999],$

$[0\mathrm{.}4000,0\mathrm{.}5999],[0\mathrm{.}6000,0\mathrm{.}7999],$ and $[0\mathrm{.}8000,0\mathrm{.}9999]$;

h$)$ Assume the random number, R$=0\mathrm{.}5634$ to accept one of these chromosomes;

i$)$ Assign again the uniform intervals of random number $[0,1].$ Since there are four

remaining chromosomes, the intervals are $[0\mathrm{.}0000,0\mathrm{.}2499],[0\mathrm{.}2500,0\mathrm{.}4999],$

$[0\mathrm{.}5000,0\mathrm{.}7499],$ and $[0\mathrm{.}7500,0\mathrm{.}9999]$;

j$)$ Assume the random number, R$=0\mathrm{.}1270$ to accept one of these chromosomes;

k$)$ List the selected four most$-$fitting chromosomes in descending order followed

by the selected two least$-$fitting chromosomes in first$-$come$-$first$-$serve order for

pairing;

l$)$ For the first chromosome in the list, its partner is to be determined;

m$)$ Assign the uniform intervals of random number $[0,1].$ Since there are five

remaining chromosomes in the list, the intervals are$0\mathrm{.}0000,0\mathrm{.}1999,$

$[0\mathrm{.}2000,0\mathrm{.}3999],[0\mathrm{.}4000,0\mathrm{.}5999],[0\mathrm{.}6000,0\mathrm{.}7999],$ and $[0\mathrm{.}8000,0\mathrm{.}9999]$;

n$)$ Assume the random number, R$=0\mathrm{.}8204$ to accept one of these chromosomes;

o$)$ For the second chromosome in the list, its partner is to be determined;

p$)$ Assign again the uniform intervals of random number $[0,1].$ Since there are three

remaining chromosomes in the list, the intervals are $[0\mathrm{.}0000,0\mathrm{.}3333],$

$[0\mathrm{.}3334,0\mathrm{.}6666],$ and $[0\mathrm{.}6667,0\mathrm{.}9999]$;

q$)$ Assume the random number, R$=0\mathrm{.}0198$ to accept one of these chromosomes;

r$)$ Match the last couple.