Part $1)$

Three refineries $(1,2,3$ in the Figure below$)$ send gasoline to two distribution terminals $(7,8$ in the Figure below$).$

The pipeline network is served by $3$ pumping stations $(4,5,6$ in the Figure below$).$

The capacity of each pipe segment is shown on the arcs in the Figure below.

Determine:

$($a$)$ Production in each refinery that matches maximum capacity of the network

$($b$)$ The daily demand in each terminal that matches the maximum capacity of the network.

$($c$)$ The daily capacity of each pump that matches the maximum capacity of the network.

Part $2)$

Suppose the maximum daily capacity of pump $6$ in the network is limited to $50.$ Reformulate the problem and determine the maximum capacity of the network.