Reference no: EM132236383
Maximize 12X1 + 10X2 + 8X3 Total Profit
Subject to X1 + X2 + X3 > 160 At least a total of 160 units of all three products needed
X1 + 3X2 + 2X3 ≤ 450 Resource 1
2X1 + X2 + 2X3 ≤ 300 Resource 2
2X1 + 2X2 + 3X3 ≤ 400 Resource 3
And X1, X2, X3 ≥ 0
Where X1, X2, and X3 represent the number of units of Product 1, Product 2, and Product 3 to be manufactured.
The QM for Windows output for this problem is given below.
Solution List:
Variable Status Value
X1 Basic 100
X2 Basic 100
X3 NONBasic 0
surplus 1 Basic 40
slack 2 Basic 50
slack 3 NONBasic 0
slack 4 NONBasic 0
Optimal Value (Z) 2200
Linear Programming Results:
X1 X2 X3 RHS Dual
Maximize 12 10 8
Constraint 1 1 1 1 >= 160 0
Constraint 2 1 3 2 <= 450 0
Constraint 3 2 1 2 <= 300 2
Constraint 4 2 2 3 <= 400 4
Solution 100 100 0 2200
Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
X1 100 0 12 10 20
X2 100 0 10 6 12
X3 0 8 8 -Infinity 16
Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 0 40 160 -Infinity 200
Constraint 2 0 50 450 400 Infinity
Constraint 3 2 0 300 275 400
Constraint 4 4 0 400 320 420
(a) What are the ranges of optimality for the profit of Product 1, Product 2, and Product 3?
(b) Find the dual prices of the four constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?
(c) If the profit contribution of Product 1 changes from $12 per unit to $15 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results. Do not solve the problem).
(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above. Do not solve the problem).