Consider the following $LP$

max,$5x_1+8x_2+6x_3,$

subject $to{x}_1+x_2+x_3,10$

$,-x_1+2x_2+4x_3,10$

$,4x_1+4x_2+2x_3,8$

$,x_1,x_2,x_3,0$

Answer the following questions "manually" without using CPLEX.

$(a)$ Apply Simplex method $to$ solve. What $is$ the optimal basis? What are the basic and

non$-$basic variables?

$(b)By$ how much can the objective coefficient for $x_1{x}_1?$

$(e)By$ how much can the objective coefficient for $x_2{x}_2?$

$(h)By$ how much can the $rhsof$ the $2^{nd}2x_1+4x_2+2x_{3}8$ will there $be$ a change $to$ the optimal

basis?

$(k)If$ the $3rd$ constraint changed $to4x_1+5x_2+2x_{3}8$ will there $be$ a change $to$ the optimal

basis?

$(l)$ Assume $we$ add $an$ activity corresponding $to$ the decision variable $x_4,$ with objective

coefficient $c_4=10,$ and constraint coefficients $2,4,$ and $1$ corresponding $to$ the $1^{st},2^{nd},$

and $3^{rd}$ constraints. Will there $be$ a change $to$ the optimal basis?