Consider the following linear program: MAX 3x1 + 4x2 ($ Profit) s.t. x1 + 3x2 £ 12 2x1 + x 2 £ 8 x1 £ 3 x1, x2 ³ 0 The Management Scientist provided the following solution output:OPTIMAL SOLUTIONObjective Function Value = 20.000 Variable Value Reduced Cost X1 2.400 0.000 X2 3.200 0.000 Constraint Slack/Surplus Dual Price 1 0.000 1.000 2 0.000 1.000 3 0.600 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 1.333 3.000 8.000 X2 1.500 4.000 9.000 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 9.000 12.000 24.000 2 4.000 8.000 9.000 3 2.400 3.000 No Upper Limit a. What is the optimal solution including the optimal value of the objective function? b. Suppose the profit on x1 is increased to $7. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $7? c. If the unit profit on x2 was $10 instead of $4, would the optimal solution change? d. If simultaneously the profit on x1 was raised to $5.5 and the profit on x2 was reduced to $3, would the current solution still remain optimal?
Consider the following linear program: MAX 3x1 + 4x2 ($ Profit) s.t. x1 + 3x2 £ 12 2x1 + x 2 £ 8 x1 £ 3 x1, x2 ³ 0 The Management Scientist provided the following solution output:OPTIMAL SOLUTIONObjective Function Value = 20.000 Variable Value Reduced Cost X1 2.400 0.000 X2 3.200 0.000 Constraint Slack/Surplus Dual Price 1 0.000 1.000 2 0.000 1.000 3 0.600 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 1.333 3.000 8.000 X2 1.500 4.000 9.000 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 9.000 12.000 24.000 2 4.000 8.000 9.000 3 2.400 3.000 No Upper Limit a. What is the optimal solution including the optimal value of the objective function? b. Suppose the profit on x1 is increased to $7. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $7? c. If the unit profit on x2 was $10 instead of $4, would the optimal solution change? d. If simultaneously the profit on x1 was raised to $5.5 and the profit on x2 was reduced to $3, would the current solution still remain optimal?
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Question
Consider the following linear program:
| MAX | 3x1 + 4x2 ($ Profit) |
| s.t. | x1 + 3x2 £ 12 |
| 2x1 + x 2 £ 8 | |
| x1 £ 3 | |
| x1, x2 ³ 0 | |
The Management Scientist provided the following solution output:
OPTIMAL SOLUTION
Objective Function Value = 20.000
|
Variable
|
Value
|
Reduced Cost
|
|
X1
|
2.400
|
0.000
|
|
X2
|
3.200
|
0.000
|
|
Constraint
|
Slack/Surplus
|
Dual Price
|
|
1
|
0.000
|
1.000
|
|
2
|
0.000
|
1.000
|
|
3
|
0.600
|
0.000
|
OBJECTIVE COEFFICIENT RANGES
|
Variable
|
Lower Limit
|
Current Value
|
Upper Limit
|
|
X1
|
1.333
|
3.000
|
8.000
|
|
X2
|
1.500
|
4.000
|
9.000
|
RIGHT HAND SIDE RANGES
|
Constraint
|
Lower Limit
|
Current Value
|
Upper Limit
|
|
1
|
9.000
|
12.000
|
24.000
|
|
2
|
4.000
|
8.000
|
9.000
|
|
3
|
2.400
|
3.000
|
No Upper Limit
|
| a. | What is the optimal solution including the optimal value of the objective function? |
| b. | Suppose the profit on x1 is increased to $7. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $7? |
| c. | If the unit profit on x2 was $10 instead of $4, would the optimal solution change? |
| d. | If simultaneously the profit on x1 was raised to $5.5 and the profit on x2 was reduced to $3, would the current solution still remain optimal? |
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