The following questions below apply to the linear program

Minimize $z=-101x_1+87x_2+23x_3$

subject $to6x_1-13x_2-3x_{3}11$

$6x_1+11x_2+2x_{3}45$

$x_1+5x_2+x_{3}12$

$x_1,x_2,x_{3}0$

with optimal basic solution

All of the following questions are independent.

a$)$ What is the solution of the linear program cbtained by decreasing the right$-$hand side of the

second constraint by $15?$

b$)$ By how much can the right$-$hand side of the second constraint increase and decrease without

changing the optimal basis?

c$)$ What is the solution of the linear program cbtained by increasing the coefficient of $x_1$ in the

objective by $25?$

d$)$ By how much can the objective coefficient of $x_3$ increase and decrease without changing the

optimal basis?

e$)$ Would the current basis remain optimal if a new variable $x4$ were added to the model with

objective coefficient $c_4=46$ and constraint coefficients $A_4=(12,-14,15{)}^T?$

f$)$ Determine the solution of the linear program obtained by adding the constraint

$5x_1+7x_2+9x_{3}50.$

g$)$ Determine the solution of the linear program obtained by adding the constraint

$12x_1-15x_2+7x_{3}10.$

h$)$ Determine the solution of the linear program obtained by adding the constraint

$x_1+x_2+x_3=30.$