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 $1$