$8.($Sensitivity Analysis$)$ Use the following output to answer the questions.

MIN $4$X$1+5$X$2+6$X$3$

S$.$T$.$

$1)$ X$1+$X$2+$X$3<85$

$2)3$X$1+4$X$2+2$X$3>280$

$3)2$X$1+4$X$2+4$X$3>320$

Objective Function Value $=400\mathrm{.}000$

Variable Value Reduced Costs

X$10\mathrm{.}0001\mathrm{.}500$

X$280\mathrm{.}0000\mathrm{.}000$

X$30\mathrm{.}0001\mathrm{.}000$

Constraint Slack$/$Surplus Dual Prices

$-------------------------------------------$

$15\mathrm{.}0000\mathrm{.}000$

$240\mathrm{.}0000\mathrm{.}000$

$30\mathrm{.}000-1\mathrm{.}250$

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit

$--------------------------------------------------------$

X$12\mathrm{.}5004\mathrm{.}000$ No Upper Limit

X$20\mathrm{.}0005\mathrm{.}0006\mathrm{.}000$

X$35\mathrm{.}0006\mathrm{.}000$ No Upper Limit

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit

$------------------------------------------------------------$

$180\mathrm{.}00085\mathrm{.}000$ No Upper Limit

$2$ No Lower Limit $280\mathrm{.}000320\mathrm{.}000$

$3280\mathrm{.}000320\mathrm{.}000340\mathrm{.}000$

a$.$ What is the optimal solution, and what is the value of the profit contribution?

b$.$ Which constraints are binding?

d$.$ Compute and interpret the ranges of optimality