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Homework answers / question archive / 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 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?
See the attached file for complete solution. The text here may not be copied exactly as some of the symbols / tables may not print. Thanks
What is the lowest value for the amount of time available to setup the display before the solution (product mix) would change?
The constraint for amount of time available is constraint 2. What we want is the amount of slack/surplus available. So look at the Constraint Slack / Surplus Dual Prices table and look for constraint 2. We get
Constraint Slack / Surplus Dual Prices
2 63 0
The slack available for constraint 2 is 63. Since the current time available is 120, the minimum time we can go up to without changing the solution is (120-63)=57
Answer is 57. Hope this clears the point.
By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?
The variable for Green nail polish is X3. What we want is the objective coefficient range, so look at the objective coefficient range table in the output for X3. We get
Variable Lower Limit Current Value Upper Limit
X3 125 150 162
Since we are looking for increase, look at upper limit. It is 162. While the current value is 150. Thus, we can increase the per bottle profit by upto 12 per bottle for the green basil nail polish before the solution would change.
