Assume that you are required to minimize the Ackley function benchmark below for $n=4$

using $16-$bit binary string representation. The bound for the variables is $-2,2.$

$f(x)=-20*exp(-0\mathrm{.}2\sqrt[\phantom{2}]{\frac{1}{n}*{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i^2})-$exp$(\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}cos(2x_i))+20+e$

Write the objective values for the following strings:

a$)1011110010110010$

b$)0011110010110010$

c$)1111110110010011$

d$)0000000000000001$

Note: You may opt to upload program or spreadsheet as part of your online submission if

you do not wish to present the detail workings in your handwritten submission.

An evolutionary algorithm uses permutation string coding for a $12-$city closed$-$loop

travelling salesman problem. The visitation sequence is coded as permutation string

denoted as $($:$a_1,a_2,$dots,$a_{12}$:$).$ Assume that the distance between any two nodes i and $j$

denoted as $d_{ij}$ is specified by the equation below, where the symbol $"\%"$ denotes the

modulo operation.

$d_{ij}=((a_i+a_j)\%5)+1$ for $ij.$

For the pair of chosen parents shown below in Figure $1,$ show the offspring produced for

the three types of recombination operations:

a$.$ Partially matched crossover $($PMX$)$ where the shaded cells indicate the sub$-$string

chosen for the PMX$.$

b$.$ Cycle recombination

c$.$ Edge recombination crossover, constructing the two offspring starting from node

$1$ and node $7.$

Figure $1$