Your colleague is working on the following LP problem:

Maximize $z=-x_1,+x_2+2x_3$

subject $to{x}_1+2x_2,-x_{3}2$

$2x_1+3x_2+x_{3}6$

$-2x_1+4x_2+2x_{3}8$

$x_1,x_2,x_{3}0.$

He tells you that he has found an optimal solution with a dictionary where $x_2,x_5,x_6$ are

non$-$basic and $x_1,x_3,x_4$ are basic.

$($a$)$ Consider the initial dictionary and the dictionary that is thought to be optimal $($by

your colleague$).$ For each of these dictionaries, give the corresponding

$B,N,$vec$(c_B{)}^T,$vec$(c{)}_N^T,$vec$(x{)}_B,$vec$(x{)}_N,$vec$(b),$ and remember to write in the basis headings $($that is$,$ label

the columns of matrices with the corresponding variables$).$

$($b$)$ What is the basic solution of your colleague's $($thought$-$to$-$be$-$optimal$)$ dictionary?

Is this a feasible solution? You have to show your work.

$($c$)$ Is your colleague's $($thought$-$to$-$be$-$optimal$)$ dictionary really optimal? You have to

justify your answer.

$($d$)$ What is the optimal objective value?

$($e$)$ Your colleague then decides to change the third constraint to

$-2x_1+4x_2+2x_{3}8+$

Over what range of $$ is your colleague's dictionary feasible and optimal. What is

the corresponding solution and optimal value?

$($f$)$ Your $($increasingly indecisive$)$ colleague then decides to leave the third constraint

as it was originally, but instead changes the objective function to

$z=(-1)x_1+x_2+2x_3$

Over what range of $$ is the dictionary feasible and optimal. What is the corre$-$

sponding solution and optimal value?