Quantitative Methods

1. 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), basic 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 bright red polish is 25 bottles, while constraint 4 specifies that the minimum demand combined for bright red, green, and pink nail polish bottles is at least 50 bottles.
MAX 100x1 + 120x2 + 150x3 + 125x4
Subject to: 1. x1 + 2x2 + 2x3 + 2x4 <= 108
2. 3x1 + 5x2 + x4 <= 120
3. x1 + x2 <= 25
4. x2 + x3 + x4 >= 50
x1, x2 , x3, x4 >= 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
What is the lowest value for the amount of time available to setup the display before the solution (product mix) would change?

2. 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 combined for bright red, green, and pink nail polish bottles is at least 50 bottles.
MAX 100x1 + 120x2 + 150x3 + 125x4
Subject to: 1. x1 + 2x2 + 2x3 + 2x4 <= 108
2. 3x1 + 5x2 + x4 <= 120
3. x1 + x2 <= 25
4. x2 + x3 + x4 >= 50
x1, x2 , x3, x4 >= 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
By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?

3. The production manager for the Whoppy soft drink company is considering the production of two kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup, is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case, and profits for diet soft drink are $2.00 per case. What is the optimal daily profit?