Excel's Solver tool has been used to generate the sensitivity analysis report:

Variable Cells

Model Variables Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease

X$1$ Deluxe $15\mathrm{.}000\mathrm{.}0010\mathrm{.}0012\mathrm{.}502\mathrm{.}50$

X$2$ Professional $17\mathrm{.}500\mathrm{.}0015\mathrm{.}005\mathrm{.}008\mathrm{.}33$

Constraints Name Final Value Shadow Price Constraint

R$.$H$.$ Side Allowable Increase Allowable Decrease

$1$ Aluminum $100\mathrm{.}003\mathrm{.}13100\mathrm{.}0060\mathrm{.}0046\mathrm{.}67$

$2$ Steel $80\mathrm{.}001\mathrm{.}2580\mathrm{.}0070\mathrm{.}0030\mathrm{.}00$

Based on the above solution, answer the following questions:

$1.$ Determine the objective coefficient ranges.

$2.$ Suppose the profit on deluxe frames is increased to $$20.$ Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $$20?$

$3.$ If the unit profit on deluxe frames were $$6$ instead of $$10,$ would the optimal solution change?

$4.$ Given that aluminum is a sunk cost, what is the maximum amount the company should pay for $50$ extra pounds of aluminum?