Question: Answer Questions 2 and 3 based on the following LP problem.

Maximize 6X1 + 5X2 + 4X3 Total Profit

Subject to X1 + X2 + X3 > 220 At least a total of 220 units of all three products needed

X1 + 3X2 + 2X3 ≤ 500 Resource 1

2X1 + X2 + 2X3 ≤ 350 Resource 2

3X1 + 2X2 + 3X3 ≤ 600 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.

Original problem with Answers:

X1 X2 X3 RHS Dual

Maximize 6 5 4

Constraint 1 1 1 1 >= 220 0

Constraint 2 1 3 2 <= 500 0.8

Constraint 3 2 1 2 <= 350 2.6

Constraint 4 3 2 3 <= 600 0

Solution-> 110 130 0 Optimal Z-> 1310

Linear Programming Results:

Variable Status Value

X1 Basic 110

X2 Basic 130

X3 NONBasic 0

surplus 1 Basic 20

slack 2 NONBasic 0

slack 3 NONBasic 0

slack 4 Basic 10

Optimal Value (Z) 1310

Ranging Results:

Variable Value Reduced Cost Original Val Lower Bound Upper Bound

X1 110 0 6 2.5 10

X2 130 0 5 3 18

X3 0 2.8 4 -Infinity 6.8

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound

Constraint 1 0 20 220 -Infinity 240

Constraint 2 .8 0 500 400 550

Constraint 3 2.6 0 350 300 357.1429

Constraint 4 0 10 600 590 Infinity

1. What are the ranges of optimality for the profit of Product 1, Product 2, and Product 3?
2. 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?
3. If the profit contribution of Product 2 changes from $5 per unit to $8 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 again).
4. 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 again).

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