Problem $2$:

Consider the following minimization problem:

MinTC$=,4$x$\_(1)+2$x$\_(2)$

s$.$t$.12$x$\_(1)+4$x$\_(2)>=600$

$,2$x$\_(1)+8$x$\_(2)>=800$

$,$x$\_(1),$x$\_(2)>=0$

$($a$)$ Graph the feasible region. Label your constraints.

$($b$)$ Are there any redundant constraints, and if so$,$ which on$($e$)/($s$)?$

$($c$)$ Label the extreme points of the feasible region. At each extreme point you are to state whether

each x$\_($j$)$ and s$\_($i$)$ are zero or non$-$zero.

$($d$)$ State the optimal values of x$\_(1),$x$\_(2)$ and TC$.$ Show the objective function on your graph of part $($a$).$

$($e$)$ Suppose the price of x$\_(2)($P$\_(2))$ increases to $4.$ What is the new optimal solution?

$($f$)$ Suppose the price of x$\_(2)($P$\_(2))$ increases to $18.$ What is the new optimal solution? State the values

of the x$\_($j$)$ and s$\_($i$).$