Describe how to construct an incremental network in the Ford$-$Fulkerson algorithm

in order to find the maximal flow through a network flow model with minimal overall cost.

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

S$1$ S$2$ F$134.$

F$223$

$($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?