A TL carrier operates within a region divided into three zones: Zone A$,$ Zone B$,$ and Zone C$.$ Each zone can import a certain good$$ from other zones. For instance, Zone A can import fuel from Zone B or Zone C$.$ Each zone can supply up to the capacity of the arcs. For instance, Zone A can supply up to $80($AB:$50+$ AC:$30)$ shipments. The objective is to minimize the total transportation cost between zones.

$$Demand for each zone:

$$ Zone A: $40$

$$ Zone B: $40$

$$ Zone C: $30$

$$ Distances $($in miles$)$ between zone hubs $($assuming Cities $1,4,$ and $7$ are selected as hubs$)$:

$$ A to B: $300$ miles

$$ A to C: $450$ miles

$$ B to C: $200$ miles

$$ Capacity $($number of shipments per day$)$ between zones:

$$ A $-$ B: $50$ shipments

$$ A $-$ C: $30$ shipments

$$ B $-$ C: $50$ shipments

$$ Cost per Mile: $$2$ per mile