Example $2.$

The following network is given. Assume that the numbers on the edges represent the cost of traversing certain edges. Additionally, there is a limitation of $12$ units on the edges BC$,$ CF$,$ AE$,$ and DG$.$ Note that the edge CF is bidirectional, while the other edges are one$-$way.

a$)$ Set up a linear optimization model that transports $20$ units from point A to point G$.$

b$)$ Explain, using equations and text, how balance at point C is ensured.