$[30$ points$]$ Consider the following model:

Maximize $Z=,3x_1+2x_2$

subject $to3x_1+x_{2}12$

$,x_1+x_{2}6$

$,5x_1+3x_{2}27$

$,x_{1}0,x_{2}0.$

$($a$)[5$ points$]$ Construct the dual problem for this primal problem.

$($b$)[10$ points$]$ Solve by the original simplex method $($in tabular form$).$ Identify the

complementary basic solution for the dual problem obtained at each iteration.

$($c$)[10$ points$]$ Solve the dual of this problem manually by the dual simplex method.

$($d$)[5$ points$]$ Compare the resulting sequence of basic solutions obtained in part $(c)$ with the

complementary basic solutions obtained in part $(b).$