Question $1$:

Consider the transportation network below. In this network, the numbers in parentheses on the arcs indicate the transportation time $($in days$)$ and the unit cost of the flow on the corresponding arc, respectively. We would like to prepare a $3-$day low$-$cost distribution plan.

The table below indicates the nodes where the product is supplied and demanded and the quantities for each day.

$\backslash$table$[[$Day$,$Supply,Demand$],[1,\backslash$table$[[*5$ units at node $1],[*1$ unit at node $2],[*3$ units at node $3]],],[2,*1$ unit at node $1,*1$ unit at node $4],[3,,\backslash$table$[[*5$ units at node $4],[*4$ units at node $5]]]]$

Consider the flow given in the table below.

$\backslash$table$[[$Day $1,$Day $2],[*3$ units from node $1$ to node $3,*1$ unit from node $1$ to node $4],[-2$ units from node $1$ to node $4,*4$ units from node $3$ to node $4],[-1$ unit from node $2$ to node $3,*4$ units from node$],[-3$ units from node $3$ to node $4,]]$

Is this flow optimal?

Why is it optimal or why is it not optimal?

$1$

If the flow is not optimal, find a better flow by applying an iteration of the network simplex algorithm.

$($Hint: Create a new time$-$space network by duplicating each node for each day. Thus, the nodes will be such that node $1\mathrm{.}1,$ node $1\mathrm{.}2,$ node $1\mathrm{.}3,$ and so on$.$ For instance, node $1\mathrm{.}1$ represents node $1$ on day $1$; node $1\mathrm{.}2$ indicates node $1$ on day $2.)$