Smart$-$Tech is utilising flow shop manufacturing system. The company has threemachines $($M$1,$ M$2$ and M$3)$ and $8$ jobs in their manufacturing operations, they would like to find out their optimal job sequence. The job processing time of each job for M$1,$ M$2$ and M$3$ machines are given in Table Q$3.$

Table Q$3$

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$c$\}$

Job number $\backslash \backslash$

$\backslash ((\backslash$mathrm$\{$Ji$\})\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

Processing time $\backslash \backslash$

in Machine$1\backslash \backslash$

$($in minute$)$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

Processing time $\backslash \backslash$

in Machine$2\backslash \backslash$

$($in minute$)$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

Processing time $\backslash \backslash$

in Machine$3\backslash \backslash$

$($in minute$)$

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

$\backslash$hline $1$ & $4$ & $5$ & $8\backslash \backslash$

$\backslash$hline $2$ & $9$ & $6$ & $10\backslash \backslash$

$\backslash$hline $3$ & $8$ & $2$ & $6\backslash \backslash$

$\backslash$hline $4$ & $6$ & $3$ & $7\backslash \backslash$

$\backslash$hline $5$ & $5$ & $4$ & $11\backslash \backslash$

$\backslash$hline

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

$($a$)$ According to the above processing time, explain if the extended Johnson algorithm can be applied.

$(6$ marks$)$

$($b$)$ What is the optimal job sequence?

$(6$ marks$)$

$($c$)$ Find the jobs' total completion time, jobs' total waiting time, and machines' total idle time. Plot the make$-$span results on the Gant chart and illustrate the waiting time and idle time on it$.$

$(13$ marks$)$