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**Linear Programming Practice Problem **

**Fall 2015 **

Use the following information to answer the questions on the next page.

Max Z: 40x1+30x2 subject to:

Constraint A

Constraint B

Constraint C

Constraint D

Non-Neg

Non-Neg

10x1+5x2<=500

4x1+3x2<=300 x1+2x2<=200

3x1+5x2<=300 x1>=0 x2>=0

ROW SLACK OR SURPLUS DUAL PRICES

A) 0.000000 3.142857

B) 57.142857 0.000000

C) 85.714287 0.000000

D) 0.000000 2.857143

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

X1 40.000000 20.000000 21.999998

X2 30.000000 36.666664 10.000000

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

A

B

C

D

500.000000 181.818176 199.999985

300.000000 INFINITY 57.142857

200.000000 INFINITY 85.714287

300.000000 199.999985 150.000000

**QUESTIONS **

1.

Given the information, what is the optimal combination of X1 and X2 to produce to maximize Z? What is the corresponding Z amount you earn?

2.

Which constraints would you like more of to earn more profit/revenue today?

3.

Which constraints are non-binding?

4.

If the objective function coefficient for X1 was to increase from $40 to $50, how much additional revenue would be generated?

5.

If the objective function coefficient for X2 was to increase from $30 to $70, how much additional revenue would be generated?

6.

If the objective function for X2 was to decrease from $30 to $25, what would the new total Z amount be for the day?

7.

If you were able to purchase 100 more units of Constraint A at a cost of $4.14, would you? If so, what would be the additional Z earned?

8.

If you were able to purchase 50 more units of Constraint C at a cost of $5, would you? Explain.

9.

If you were able to purchase 150 more units of Constraint D at a cost of $2.00, would you? If so, what would be the additional Z earned?

10.

If Constraint B was to decrease today to 275 units, what is the impact to the problem?

**ANSWERS **

1.

Given the information, what is the optimal combination of X1 and X2 to produce to maximize Z? What is the corresponding Z amount you earn?

X1 = 28.57, x2 = 42.83

Z = $2428.57

2.

Which constraints would you like more of to earn more profit/revenue today? How do you know?

A and D. They have zero slack (all used up) and a positive dual price.

3.

Which constraints are non-binding?

B and C. They have a positive slacks and zero dual price.

4.

If the objective function coefficient for X1 was to increase from $40 to $50, how much additional revenue would be generated?

This is an increase of $10, which is in the allowable range. Thus, the optimal combo stays the same.

$10 x 28.57 = $285.70 extra dollars are generated.

5.

If the objective function coefficient for X2 was to increase from $30 to $70, how much additional revenue would be generated?

You cannot tell-- because the increase of $40 is outside the allowable range. The optimal combination would change. You must resolve the problem.

6.

If the objective function for X2 was to decrease from $30 to $25, what would the new total Z amount be for the day?

This is a decrease of $5, which is in the allowable range. Thus, the optimal combo stays the same. --

-$5 x 42.83 = -$214.15. The new total Z amount = $2428.57 - 214.15 = $2,214.42

7.

If you were able to purchase 100 more units of Constraint A at a cost of $4.14, would you? If so, what would be the additional Z earned?

No. The additional units of 100 is in the acceptable range, which means the dual price of $3.14 is in effect. This value is LESS than the cost-- making the purchase too expensive. We would not make the purchase.

8.

If you were able to purchase 50 more units of Constraint C at a cost of $5, would you? Explain.

No. We do not need any more C to produce today. We already have 85 units left over. We would not purchase C to produce more for today. Nor can we judge whether it would be wise to purchase the C at this cost for future use because we don't know if it's an acceptable price.

9.

If you were able to purchase 150 more units of Constraint D at a cost of $2.00, would you? If so, what would be the additional Z earned?

Yes. 150 additional units is in the range, thus, the dual price of $2.86 is in effect. The cost is $2.00, giving us a net profit of $.86 for each unit purchased. If we buy 150 units, we generate 150 x $.86 =

$129 extra Z. The new total profit would be $2428.57 + $129 = $2,557.57.

10.

If Constraint B was to decrease today to 275 units, what is the impact to the problem?

There would be no impact to the problem. The allowable decrease for Constraint B is 50 units. The decrease to 275 represents a decrease of only 25 units. The binding constraints would remain as is-

- as would the answer.