$4.(25$ points$)$ Consider the minimum cost flow problem shown below, where the $\backslash ($ b$\_\{$i$\}\backslash )$ values $(+$ : supplies, $-$: demands$)$ are given by the nodes, the $\backslash ($ c$\_\{$i j$\}\backslash )$ values $($costs per unit flow$)$ are given by the arcs, and the $\backslash ($ u$\_\{$i j$\}\backslash )$ values $($arc capacities$)$ are given between nodes C and D $.$ Do the following work manually.

A$)$ Consider the following Basic feasible $($BF$)$ solution. Starting from this BF solution, apply the network simplex method. Identify the entering non$-$basic arc, the leaving basic arc, the next BF solution, and the total cost at each iteration until you reach the optimal BF solution. Show all your work. $(15$ points$)$