Problem $1.(35$ points$).$ Each week a small logistics company has to move products from the manufacturing centers in cities A$,$ B$,$ and C$,$ to the ports, located in cities D$,$ E$,$ F$,$ and G$.$ The weekly production is $3$ tons, $5$ tons, and $6$ tons, in cities A$,$ B$,$ and C$,$ respectively; whereas the weekly required amounts are $3$ tons, $2$ tons, $4$ tons, $5$ tons, in cities D$,$ E$,$ F$,$ and G$,$ respectively. Due to truck availability constraints, only $2$ tons can be served weekly from city C to city F$,$ and $2$ tons can be sent weekly from city C to city G$.$ The distances $($in miles$)$ between the cities are summarized next.

$($Look in the picture for the table$)$

a$.(15$ points$).$ Model this problem as a minimum cost flow problem. Identify the nodes, arcs, supplies$/$demands$,$ costs, and capacities of the network. $($You might draw the network and label the nodes$/$arcs with the required information.$)$

b$.(20$ points$).$ Provide an LP formulation to minimize the weekly total distance traveled and that meets all demand and capacity constraints.