$1.$ Consider the minimum cost flow problem shown below, where the

b

i

$$

values $($net flows generated$)$ are given by the nodes, the

C

ij

$$

values $($costs per unit flow$)$ are given by the arcs, and the

U

ij

$$

values $($arc capacities$)$ are

U

AD

$$

$=20,$U

BC

$$

$=8$

$,$ and other

U

ij

$$

$=\backslash$infty

$.($a$)$ Formulate the linear programming model of this problem. $(15$ Points$)($b$)$ Consider the optimal solution of the problem shown below: $\backslash$begin$\{$tabular$\}\{|$c$|$c$|\}\backslash$hline Variable & Value $\backslash \backslash \backslash$hline XAB & $0\backslash \backslash \backslash$hline XAC & $20\backslash \backslash \backslash$hline XAD & $20\backslash \backslash \backslash$hline XBC & $8\backslash \backslash \backslash$hline XBD & $12\backslash \backslash \backslash$hline XCE & $18\backslash \backslash \backslash$hline XDE & $32\backslash \backslash \backslash$hline $\backslash$end$\{$tabular$\}($b$1)$ Determine the objective value of the minimum cost flow problem. $(5$ Points$)($b$2)$ Show the optimal solution on the network. $(10$ Points$)$