Consider the single$-$machine total tardiness minimization problem below.

$($a$)$ Apply the neighborhood search with adjacent pairwise interchange $($API$).$

$($b$)$ Apply the neighborhood search with pairwise interchange $($PI$).$

$($c$)$ Compare the results in parts $($a$)$ and $($b$).$

$($d$)$ Let the total tardiness value of the EDD schedule be an upper bound on the total tardiness for the reduced problem. To find the optimal schedule of jobs, apply the branch$-$and$-$bound algorithm. $\backslash$begin$\{$tabular$\}\{|$l$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline Job & $1$ & $2$ & $3$ & $4$ & $5\backslash \backslash$

$\backslash$hline Processing Time & $2$ & $4$ & $6$ & $8$ & $10\backslash \backslash$

$\backslash$hline Due Date & $6$ & $12$ & $30$ & $19$ & $12\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$