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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Consider the following linear program:

MAX	3x₁ + 4x₂ ($ Profit)

s.t.	x₁ + 3x₂ £ 12
	2x₁ + x₂ £ 8
	x₁ £ 3
	x₁, x₂³ 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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