$6.$ Consider the following min$-$cost flow problem given in the figure below with a directed network over $5$ nodes. Note that for each arc $\backslash (($i$,$ j$)\backslash ),$ we specify $\backslash (\backslash$left$($c$\_\{$i j$\},$ u$\_\{$i j$\}\backslash$right$)\backslash )$ where $\backslash ($ c$\_\{$i j$\}\backslash )$ is the cost of using unit capacity of flow on that arc, and $\backslash ($ u$\_\{$i j$\}\backslash )$ is the upper bound on the flow on that arc.

a$.$ Explain why the problem is infeasible. You can demonstrate a cut in an appropriate network that shows that the demands cannot be satisfied. Verify using a solver that the problem is infeasible.

b$.$ Suppose we can add capacities to the arcs by paying cost equal to the cost of unit flow on each arc. For instance, we can increase the capacity of arc $\backslash ((1,2)\backslash )$ by paying a cost of $4$ for each unit increase in capacity. Formulate an optimization problem that adds capacities of the arcs at minimum possible cost to make the problem feasible. Note we do not consider the cost of the flow; only the cost we pay to add additional capacities to the arcs. Solve using your favorite solver.