$4-38.$ Using the artificial constraint procedure introduced in Problem $4-37,$ solve the following

problems by the dual simplex method. In each case, indicate whether the resulting solution is feasible, infeasible, or unbounded.

$($a$)$ Maximize $z=2x_3$

subject to

$-x_1+2x_2-2x_{3}4$

$-x_1+x_2+x_{3}2$

$2x_1-x_2+4x_{3}5$

$x_1,x_2,x_{3}0$

$($b$)$ Maximize $z=x_1-3x_2$

subject to

$x_1-x_{2}20$

$x_1+x_{2}40$

$2x_1-2x_{2}30$

$x_1,x_{2}0$

$*($c$)$ Minimize $z=-x_1+x_2$

subject to

$x_1-4x_{2}5$

$x_1-3x_{2}1$

$2x_1-5x_{2}1$

$x_1,x_{2}0$

$204$ Chapter $4$ Duality and Post$-$Optimal Analysis

$($d$)$ Maximize $z=2x_3$

subject to

$-x_1+3x_2-7x_{3}50$

$-x_1+x_2-x_{3}10$

$3x_1+x_2-10x_{3}80$

$x_1,x_2,x_{3}0$