using genetic algorithms find the python code to calculate all the real$-$valued intersection points of the following three functions:

f$1($x$)=$ x $+1,$ f$2($x$)=$ x$^2$ and f$3($x$)=$ x$^5+1.$

Current Code:

# Genetic algorithm

# $-$ various problem types

# $-$ various chromosome types

import random

import math

# Create a chromosome

def create$\_$chromosome$($length$,$ gene$\_$type$=$'real', gene$\_$range$=(-2,2),**$kwargs$)$:

if gene$\_$type $==$ 'binary':

return $[$random$.$randint$(0,1)$ for $\_$ in range$($length$)]$

elif gene$\_$type $==$ 'integer':

return $[$random$.$randint$($gene$\_$range$[0],$ gene$\_$range$[1])$ for $\_$ in range$($length$)]$

elif gene$\_$type $==$ 'real':

return $[$random$.$uniform$($gene$\_$range$[0],$ gene$\_$range$[1])$ for $\_$ in range$($length$)]$

# Create the initial population

def create$\_$population$($pop$\_$size, chrom$\_$length, gene$\_$type$=$'real', gene$\_$range$=(-2,2))$:

return $[$create$\_$chromosome$($chrom$\_$length, gene$\_$type, gene$\_$range$)$ for $\_$ in range$($pop$\_$size$)]$

# Fitness function for max bits problem: maximise the number of $1\text{'}$s in a binary string

def fitness$\_$max$\_$bits$($chromosome$)$:

optimal$\_$value $=$ len$($chromosome$)$ # The best case is when all bits are $1$

return optimal$\_$value $-$ sum$($chromosome$)$ # Distance from the optimal value, $0$ being the best

def fitness$\_$intersection$($chromosome$)$:

x $=$ chromosome$[0]$

f$1=$ x $+1$

f$2=$ x$**2$

f$3=$ x$**5+1$

fitness$\_$value $=1/(1+$ abs$($f$1-$ f$2)+$ abs$($f$1-$ f$3)+$ abs$($f$2-$f$3))$

return abs$($fitness$\_$value$-1)$

# Selection function $-$ choose $2$ at random and select the one with fitness closest to $1$

def selection$($population$,$ fitness$\_$fn$=$fitness$\_$closest$\_$to$\_$one$)$:

selected $=$ random.sample$($population$,2)$ # Randomly select two individuals

return min$($selected$,$ key$=$fitness$\_$fn$)$ # Return the one with the fitness value closest to $1$

# Crossover function

def crossover$($parent$1,$ parent$2,$ gene$\_$type$=$'real', crossover$\_$rate$=0\mathrm{.}7)$:

if random.random$()<$ crossover$\_$rate:

if gene$\_$type $==$ 'binary':

# Single$-$point crossover for binary chromosomes

crossover$\_$point $=$ random.randint$(1,$ len$($parent$1)-1)$

return parent$1[$:crossover$\_$point$]+$ parent$2[$crossover$\_$point:$]$

elif gene$\_$type $==$ 'integer':

# Single$-$point crossover for integer chromosomes to avoid generating floats

crossover$\_$point $=$ random.randint$(1,$ len$($parent$1)-1)$

return parent$1[$:crossover$\_$point$]+$ parent$2[$crossover$\_$point:$]$

elif gene$\_$type $==$ 'real':

# Blending for real$-$valued chromosomes

return $[($p$1+$ p$2)/2$ for p$1,$ p$2$ in zip$($parent$1,$ parent$2)]$

return parent$1$ if random.random$()<0\mathrm{.}5$ else parent$2$

# Mutate a chromosome

def mutate$($chromosome$,$ mutation$\_$rate$=0\mathrm{.}01,$ gene$\_$type$=$'real', gene$\_$range$=(-2,2))$:

if gene$\_$type $==$ 'real':

return $[$gene if random.random$()>$ mutation$\_$rate else gene $+$ random.uniform$(-0\mathrm{.}1,0\mathrm{.}1)$ for gene in chromosome$]$

if gene$\_$type $==$ 'binary':

return $[$gene if random.random$()>$ mutation$\_$rate else $1-$ gene for gene in chromosome$]$

elif gene$\_$type $==$ 'integer':

return $[$gene if random.random$()>$ mutation$\_$rate else random.randint$($gene$\_$range$[0],$ gene$\_$range$[1])$ for gene in chromosome$]$

return chromosome

# Create the next generation

def create$\_$next$\_$generation$($population$,$ fitness$\_$fn$,$ crossover$\_$rate$=0\mathrm{.}7,$ mutation$\_$rate$=0\mathrm{.}01,$ gene$\_$type$=$'real', gene$\_$range$=(-2,2))$:

next$\_$generation $=[]$

for $\_$ in range$($len$($population$))$:

parent$1=$ selection$($population$,$ fitness$\_$fn$)$

parent$2=$ selection$($population$,$ fitness$\_$fn$)$

offspring $=$ crossover$($parent$1,$ parent$2,$ gene$\_$type, crossover$\_$rate$)$

mutated$\_$offspring $=$ mutate$($offspring$,$ mutation$\_$rate, gene$\_$type, gene$\_$range$)$

next$\_$generation.append$($mutated$\_$offspring$)$

return next$\_$generation

# Output the initial population

def print$\_$initial$\_$population$($population$,$ fitness$\_$fn$,$ gene$\_$type$)$:

print$("$Initial Population:"$)$

for i$,$ individual in enumerate$($population$)$:

if gene$\_$type $==$ 'real':

# Print real values with four decimal places

chromosome$\_$str $=\text{'},\text{'}.$join$($f$"\{$gene:$\mathrm{.}4$f$\}"$ for gene in individual$)$

elif gene$\_$type $==$ 'integer':

# Print integer values

chromosome$\_$str $=\text{'},\text{'}.$join$($f$"\{$gene$\}"$ for gene in individual$)$

elif gene$\_$type $==$ 'binary':

# Print binary values $(0$ or $1)$

chromosome$\_$str $=\text{'}\text{'}.$join$($str$($gene$)$ for gene in individual$)$

else:

chromosome$\_$str $=\text{'},\text{'}.$join$($str$($gene$)$ for gene in individual$)$

print$($f$"$Individual $\{$i$+1\}$: $[\{$chromosome$\_$str$\}],$ Fitness $=\{$fitness$\_$fn$($individual$)$:$\mathrm{.}4$f$\}")$

print$("---")$

# Perform the genetic algorithm

# Perform the genetic algorithm

def genetic$\_$algorithm$($pop$\_$size, chrom$\_$length, fitness$\_$fn$,$ generations, gene$\_$type$=$'real', gene$\_$range$=(-2,2),$ crossover$\_$rate$=0\mathrm{.}7,$ mutation$\_$rate$=0\mathrm{.}01)$:

population $=$ create$\_$population$($pop$\_$size, chrom$\_$length, gene$\_$type, gene$\_$range$)$

#