Maximize

$Z=c_1{x}_1+c_2{x}_2,$

subject $to$

$2x_1-x_2{b}_1$

$x_1-x_2{b}_2$

$x_{1}0,x_{2}0.$

Let $x3$ and $x4$ denote the slack variables for the respective functional constraints. When $c1$

$=3,c2=-2,b1=30,$ and $b2=10,$ the simplex method yields the following final simplex

tableau.

$(a)$ Use graphical analysis $to$ determine the allowable range for $c1$ and $c2.$

$(b)$ Use algebraic analysis $to$ derive and verify your answers $in$ part $(a).$

$(c)$ Use graphical analysis $to$ determine the allowable range for $b1$ and $b2.$

$(d)$ Use algebraic analysis $to$ derive and verify your answers $in$ part $(c)$

Caveat: allowable range $is$ a range $of$ parameter values within which the Optimal Basis &

Shadow Prices Remain Unchanged.