Question $4(45$ marks$)$:

Consider the following network.

Some nodes are demand nodes and the others are supply nodes. $($For example, node $1$ needs $3$ items so it is a demand point. However, node $4$ supplies $5$ items so it is a supply node$)$

Demand nodes satisfy their demand by using their input flow. $($For example, if the input flow to node $1$ is $5,$ the output flow of node $1$ will be $5$ minus its demand$)$

Supply nodes provide their supply in their output flow. $($For example, if the input flow to node $4$ is $2,$ the output flow of node $4$ will be $2$ plus its supply$)$

There is a shipment cost from each node to another node. $($For example, the shipping cost of $2$ items from node $4$ to node $2$ is $2**3=6.)$

Each arc has a capacity. $($For example, the capacity of the arc from $4$ to $2$ is $3$ items. This implies that at most $3$ items can be sent from node $4$ to node $2.)$

All demands must be satisfied, and the objective is to minimize the total shipping cost.

Formulate the problem as a Linear Program. $($NO NEED TO SOLVE IT$)$

All parameters can be found in the following tables.

$\backslash$table$[[$Node$,1,2,3,4,5,6],[$Demand$,3,1,2,-,3,-],[$Supply$,-,-,-,5,-,4]]$

$\backslash$table$[[$Arc$,\backslash$table$[[$Cost$],[($per item$)]],$Capacity$],[13,1,3],[15,2,3],[21,3,6],[32,1,3],[34,2,3],[42,3,3],[46,1,6],[53,2,3],[56,2,3],[63,1,6]]$