# interpret the output of a linear programming model - LINDO

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?

#### Solution Preview

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 ...

#### Solution Summary

This problem shows how to interpret the LINGO / LINDO output of a linear programming problem