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# Ranges of optimality

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

Maximize 2X1 + 5X2 + 4X3 Total Profit
Subject to X1 + X2 + X3 > 150 At least a total of 150 units of all three products needed
X1 + 3X2 + 2X3 â?¤ 300 Resource 1
2X1 + X2 + 2X3 â?¤ 250 Resource 2
2X1 + 2X2 + 3X3 â?¤ 300 Resource 3
And X1, X2, X3 â?¥ 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 2 5 4
Constraint 1 1 1 1 >= 150 -.5
Constraint 2 1 3 2 <= 300 1.5
Constraint 3 2 1 2 <= 250 0
Constraint 4 2 2 3 <= 300 .5
Solution-> 75 75 0 Optimal Z-> 525

Linear Programming Results:
Variable Status Value
X1 Basic 75
X2 Basic 75
X3 Basic 0
surplus 1 NONBasic 0
slack 2 NONBasic 0
slack 3 Basic 25
slack 4 NONBasic 0
Optimal Value (Z) 525

Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
X1 75 0 2 -Infinity 2.2
X2 75 0 5 2 6
X3 0 0 4 3.75 Infinity

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 -.5 0 150 120 150
Constraint 2 1.5 0 300 250 450
Constraint 3 0 25 250 225 Infinity
Constraint 4 .5 0 300 300 350

2. (a) Determine the optimal solution and optimal value and interpret their meanings.
(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

3. (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 2 changes from \$5 per unit to \$5.50 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).
(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 again).

#### Solution Preview

Hi,

Please find below detailed assistance to your problem.

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2. (a) Determine the optimal solution and .optimal value and interpret their meanings.
-> Optimal solution refers to values of decision variables and optimal value is maximized or minimized value of the objective function. Optimal solution is the values of decision variables for which value of the objective function is maximized or minimized. Optimal solution is to produce x1 = 75 units and x2 = 75 units. x3 should not be produced. Optimal value of the solution is profit = 525

(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.
-> Slack is defined as unused quanityt of resource. Slack is for less than or equal to inequalities. Surplus is defined as overshooting the minimum requirement. In the optimal solution
Constraint 1: Surplus of 0, which means ...

#### Solution Summary

Solution shows optimal value and interpretation of their meanings.

\$2.19