Consider the following LP:

MIN $Z=5x1+22+103$

subject $to$

$x1-2x2-x310$

$3x2+x310$

$x1+22+2320$

and $x1,x2,x30$

$($a$)$ Using the two$-$phase method, find the optimal solution to the primal problem above.

$($b$)$ Write directly the dual of the primal problem, without using the method of transformation.

$($c$)$ Determine the optimal values of the dual variables from the optimal tableau of the primal problem. Note that the answer obtained by solving the dual by the simplex algorithm will not be evaluated.

$($d$)$ Suppose that you do not know the optimal solution to the dual. Using the complementary slackness and optimal dual solution obtained in part $($a$),$ find the optimal solution to the dual.

$($e$)$ Using the dual$-$simplex method, find the optimal solution to the primal problem.

$($f$)$ What are the allowable decrease and allowable increase in the RHS value of the first constraint of the primal problem to keep the current basis unchanged?

$($g$)$ What is the cost of one unit increase in the RHS value of the first constraint of the primal problem?

$($h$)$ Suppose that we want to decrease the objective function value $($cost$)$ of the first decision variable. In what values can you decrease this cost so that the optimal solution of the primal problem remains optimal?