$($a$)$ Suppose b $=(9,3)$T is changed to b$=(2,7)$T $.$ Find the optimal solution. $($You must use dual simplex method$)$;

$($b$)$ Suppose c $=(2,1,2,0)$T is changed to c$=(1,2,1,3)$T $.$ Use the last simplex table to find the optimal solution;

$($c$)$ Suppose a$1=(1,5)$T and a$3=(4,1)$T are changed to $$a$1=(3,1)$T$,$a$3=(1,4)$T$.$ Use the last simplex table to find the optimal solution.

$($d$)$ Suppose that a new constraint $2$x$1$ x$2+2$x$4<=2$ is added. Solve the new problem by dual simplex method.