$($b$)$ A factory $\backslash ($ F$\_\{1\}\backslash )$ can produce up to $5$ units of a product$/$week at a cost of $3$ per unit. A second factory $\backslash ($ F$\_\{2\}\backslash )$ can produce up to $3$ units of the product$/$week at a cost of $4$ per unit. They both supply two shops $\backslash ($ S$\_\{1\}\backslash )$ and $\backslash ($ S$\_\{2\}\backslash ).$ Shops $\backslash ($ S$\_\{1\}\backslash )$ and $\backslash ($ S$\_\{2\}\backslash )$ can both sell $3$ units per week. The cost per unit of transporting the products from the factories to the shops is given by

$($i$)[8$ Marks$]$ Construct a flow network for the factory$-$shop system described above that clearly shows the capacities and costs.

$($ii$)[10$ Marks$]$ Use the Ford$-$Fulkerson algorithm to find a flow through the network found in $($i$)$ that minimizes the overall cost.

What is the minimal capacity cut?