Consider the linear program:

Maximize $z=x_1+3x_2,$

subject to:

$x_1+x_{2}8,($ resource $1)$

$-x_1+x_{2}4,($ resource $2)$

$x_{1}6(resource3),$

$x_{1}0$

$x_{2}0$

The optimal tableau determined by the simplex method $($after adding slack variables$)$:

$($a$)$ Using the optimal tableau, determine

$($i$)$ the optimal solution

$($ii$)$ the shadow prices on the three constraints

$($iii$)$ the range on the objective coefficient of each variable, holding the coefficient of the other

variable at its current value, for which the solution to part $($i$)$ remains optimal

$($b$)$ Answer the same questions in part $($a$)$ by using excel solver $($attach the answer and sensitivity

report$).$ Verify that your answers in part $($a$)$ are correct. From the excel file, also find the

range on the availability of each resource, holding the availability of the other resources at their

current values, for which the shadow prices in part $($ii$)$ remain optimal.