$1$: Adding a distribution center $($DC$)$ to the problem introduces an intermediary step in the supply chain. Now, we must account for transportation costs not only from factories to customers but also from factories to the distribution center and then from the distribution center to the customers. We'll assume a single distribution center for simplicity.

Now we have four factories $($F$1,$ F$2,$ F$3,$ F$4),$ one distribution center $($DC$),$ and six customers $($C$1,$ C$2,$ C$3,$ C$4,$ C$5,$ C$6).$ The factories send goods to the DC$,$ which in turn distributes them to the customers. We need to decide whether to open or close each factory and minimize the total costs, including operating factories, transporting goods to the DC$,$ and then from the DC to the customers.

Additional Assumptions:

Fixed costs of operating the distribution center $($DC$)$: $$20,000$

Variable transportation costs per unit from factories to the DC:

DC