Consider the following linear program.

Min $8x+12y$

$s.t.$

$1x+3y6$

$2x+2y8$

$6x+2y12$

$x,y0$

$($a$)$ Use the graphical solution procedure to find the optimal solution.

What is the value of the objective function at the optimal solution?

lat $(x,n)=(,)$

$($b$)$ Assume that the objective function coefficient for $x$ changes from $8$ to $6.$ Use the graphical solution procedure to find the new optimal solution. Does the optimal solution change?

Does the optimal solution change?

$($d$)$ The computer solution for the linear program in part $($a$)$ provides the following objective coefficient range information.

$\backslash$table$[[$Variable$,\backslash$table$[[$Objective$],[$Coefficient$]],\backslash$table$[[$Allowable$],[$Increase$]],\backslash$table$[[$Allowable$],[$Decrease$]]],[x,8\mathrm{.}00000,4\mathrm{.}00000,4\mathrm{.}00000],[,12\mathrm{.}00000,12\mathrm{.}00000,4\mathrm{.}00000]]$

How would this objective coefficient range information help you answer parts $($b$)$ and $($c$)$ prior to re$-$solving the problem?

$.$ Since the change in part $($c$)$ this range, we know the optimal solution $-$ Select$-$ change.