Consider the following linear programming problem $P$ with two resources and two activities:

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

subject $to{x}_1+2x_2{b}_1$

$,x_1+x_2{b}_2$

and

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

Let $x_3$ and $x_4$ denote the slack variable $of$ functional constraint $(1)$ and $(2),$ respectively.

When $c_1=2,c_2=3,b_1=30,$ and $b_2=20c_{2}2c_{2}4.$

Using this information and the results obtained $in$ part $(a),$ answer the following questions

$(do$ not $re-$optimize$)$:

$(i)If$ the unit profit $c_2$ increased $by0\mathrm{.}5(i.e.,$ from $3to3\mathrm{.}5),$ what $is$ the maximum

decrease $in\frac{c}{?}so$ that the optimal solution remains unchanged?

$(ii)If{b}_1$ becomes $35,$ what the optimal profit $Z^{*}$ would $be?$

$(c)$ Suppose that a new process $(constraint)on$ the two activities has been added: $3x_1+x_{2}20.$

Will the optimal solution remain unchanged? $If$ not, $re-$optimize $to$ find the new optimal

solution.