Four factories supply stuff to four shops. The production capacities of the factories are $7,12,8$ and 9 units per week, and the requirements of the shops are 8 units per week each. If the costs of transporting a unit of stuff from factory $i$ to shop $j$ is the $(i, j)$ th element in the matrix

$$\left(\begin{array}{cccc} 6 & 10 & 3 & 5 \\ 4 & 8 & 6 & 12 \\ 3 & 4 & 9 & 2 \\ 5 & 7 & 2 & 6 \end{array}\right)$$

find a minimal-cost allocation of the outputs of the factories to the shops.

Suppose that the cost of producing one unit of stuff varies across the factories, being $3,2,4,5$ respectively. Explain how you would modify the original problem to minimise the total cost of production and of transportation, and find an optimal solution for the modified problem.