$($a$)$ Considering a manufacturing firm with fluctuating demand, how would you design an inventory management strategy that balances the costs of holding excess inventory against the risks of stockouts, ensuring optimal production planning and control efficiency? $[4$ Marks$]$

$($b$)$ There are three jobs $($namely $P,Q$ and $R),$ each of which must go through machines $A,B$ and $C$ in the order ABC. Processing Time $($in hours$)$ are given below:

$\backslash$table$[[$Jobs$,$P$,$Q$,$R$],[$Machine A$,5,7,6],[$Machine B$,2,1,4],[$Machine C$,3,7,5]]$

Using Assignment Method, find the sequence that minimum the total elapsed time Required to complete job