Q$\mathrm{.}4.(28$ points$)$ Consider the following problem. $\backslash [\backslash$begin$\{$array$\}\{$lrrrl$\}\backslash$text $\{$ Max $\}$ & $4$ x$\_\{1\}$ & $+8$ x$\_\{2\}$ & $+6$ x$\_\{3\}$ & $\backslash \backslash \backslash$text $\{$ s$.$t$.\}$ & $-3$ x$\_\{1\}$ & $+$x$\_\{2\}$ & $+2$ x$\_\{3\}$ & $\backslash$leq $20\backslash \backslash$ & $-$x$\_\{1\}$ & $+$x$\_\{2\}$ & $+2$ x$\_\{3\}$ & $\backslash$leq $22\backslash \backslash$ & x$\_\{1\}$ & $+$x$\_\{2\}$ & & $\backslash$leq $18\backslash \backslash$ & x$\_\{1\},$ & x$\_\{2\},$ & x$\_\{3\},\backslash$geq $0$ & $\backslash$end$\{$array$\}\backslash ]($a$)(4)$ Write the dual of the problem $($b$)(8)$ Solve the problem using the Tableau Method. At each iteration of the algorithm, identify

B

$1$

$,$ and

w$=$c

B

$$

B

$1$

$.($c$)(8)$ At each iteration of the tableau method, state whether

w

is feasible to the dual or not. If not feasible, state which constraints of the dual problem does

w

violate and how much? Do you see any hint in the tableau of that iteration? $($d$)(8)$ At each iteration of the algorithm show that the complementary slackness condition holds.