The linear programming problem whose output follows is used to determine how many bottles of Hell$-$bound red nail polish $(x_1),$ Blood red nail polish $(x_2),$ Bile green nail polish $),$

and Pepto$-$Bismol pink nail polish $)$ a beauty salon should stock. The objective function measures profit; it's assumed that every piece stocked will be sold. Constraint $1$ measures

display space in units. Constraint $2$ measures time to set up the display in minutes. $($Bile green nail polish does not require any time to prepare its display$).$ Constraints $3$ and $4$ are

marketing restrictions. Constraint $3$ indicates that the maximum demand for Hell$-$bound red and Bile green polish is $25$ bottles. Constraint $4$ specifies that the minimum demand for

Blood red, Bile green and Pepto$-$Bismol pink nail polish bottles combined is at least $50$ bottles.

MAX $100x_1+120x_2+150x_3+125x_4$

such that: $x_1+2x_2+2x_3+2x_{4}110$

$,3x_1+5x_2+x_{4}120$

$,x_1+x_{3}25$

$,x_2+x_3+x_{4}50$

$,x_1,x_2,x_3,x_{4}0$

Optimal Solution:

Objective Function Value $=$$$7,625\mathrm{.}00$

Objective Cell $($Max$)$

Variable Cells Constraints

Variable Cells

Constraints How much space will be left unused?

How many minutes of idle time remaining for setting up the display?

To what value can the per bottle profit on Hell$-$bound red nail polish drop before the solution $($product mix$)$ would change?

By how much can the per bottle profit on Bile green nail polish increase before the product mix would change?

What is the lowest value for the amount of time available to setup the display before the product mix would change?

You are offered the chance to obtain more space. The offer is for $13$ units and the total price is $$980.$ What should you do$?$ Why?