Question: The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x₁), bright red nail polish (x₂), basil green nail polish(x₃), and basic pink nail polish(x₄) a beauty salon should stock. The objective function measures profit; it is 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. Note that 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 fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand for bright red, green and pink nail polish bottles combined is at least 50 bottles.

MAX 100x₁ + 120x₂ + 150x₃ + 125x₄

Subject to 1. x₁ + 2x₂ + 2x₃ + 2x₄ \(\le \) 108

2. 3x₁ + 5x₂ + x_{4 \(\le \)} 120

3. x₁ + x₃ \(\le \) 25

4. x₂ + x₃ + x_{4 \(\ge \)} 50

x₁, x₂ , x₃, x_{4 \(\ge \)} 0

Optimal Solution:

Objective Function Value = 7475.000

Variable Value Reduced Costs

X1 8 0

X2 0 5

X3 17 0

X4 33 0

Constraint Slack / Surplus Dual Prices

1 0 75

2 63 0

3 0 25

4 0 -25

Objective Coefficient Ranges

Variable Lower Limit Current Value Upper Limit

X1 87.5 100 none

X2 none 120 125

X3 125 150 162

X4 120 125 150

Right Hand Side Ranges

Constraint Lower Limit Current Value Upper Limit

1 100 108 123.75

2 57 120 none

3 8 25 58

4 41.5 50 54

a) To what value can the per bottle profit on fire red nail polish drop before the solution (product mix) would change?

b) By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?