True/False. Indicate whether the following statements are True or False and why? (you will not get full credit without an explanation.)

1. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables.

2. Only binding constraints form the shape (boundaries) of the feasible region.

3. The constraint 5x1 - 2x2 < 0 passes through the point (20, 50).

4. The point (3, 2) is feasible for the constraint 2x1 + 6x2 £ 30.

5. An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the problem.

6. The reduced cost for a positive decision variable is 0.

7. For any constraint, either its slack/surplus value must be zero or its dual price must be zero.

8. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective.

9. For a minimization problem, a positive dual price indicates the value of the objective function will increase.

10. The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.

Part II. Short Problems.

1. Solve the following system of simultaneous equations.

6X + 2Y = 50

2X + 4Y = 20

2. Use this graph to answer the questions.

Max 20X + 10Y

s.t. 12X + 15Y < 180

15X + 10Y < 150

3X - 8Y < 0

X , Y > 0

a. Which area (I, II, III, IV, or V) forms the feasible region?

b. Which point (A, B, C, D, or E) is optimal?

c. Which constraints are binding?

d. Which slack variables are zero?

3. Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below.

Fliptop Model Tiptop Model Available

Plastic 3 4 36

Ink Assembly 5 4 40

Molding Time 5 2 30

The profit for either model is $1000 per lot.

a. What is the linear programming model for this problem? (show the

objective function and constraints).

4. Use the following Management Scientist output to answer the questions.

LINEAR PROGRAMMING PROBLEM

MAX 31X1+35X2+32X3

S.T.

1) 3X1+5X2+2X3>90

2) 6X1+7X2+8X3<150

3) 5X1+3X2+3X3<120

OPTIMAL SOLUTION

Objective Function Value = 763.333

Variable Value Reduced Cost

X1 13.333 0.000

X2 10.000 0.000

X3 0.000 10.889

Constraint Slack/Surplus Dual Price

1 0.000 -0.778

2 0.000 5.556

3 23.333 0.000

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit

X1 30.000 31.000 No Upper Limit

X2 No Lower Limit 35.000 36.167

X3 No Lower Limit 32.000 42.889

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit

1 77.647 90.000 107.143

2 126.000 150.000 163.125

3 96.667 120.000 No Upper Limit

a. Give the solution to the problem.

b. Which constraints are binding?

c. What would happen if the coefficient of x1 increased by 3?

d. What would happen if the right-hand side of constraint 1 increased by 10?

1. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables.

2. Only binding constraints form the shape (boundaries) of the feasible region.

3. The constraint 5x1 - 2x2 < 0 passes through the point (20, 50).

4. The point (3, 2) is feasible for the constraint 2x1 + 6x2 £ 30.

5. An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the problem.

6. The reduced cost for a positive decision variable is 0.

7. For any constraint, either its slack/surplus value must be zero or its dual price must be zero.

8. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective.

9. For a minimization problem, a positive dual price indicates the value of the objective function will increase.

10. The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.

Part II. Short Problems.

1. Solve the following system of simultaneous equations.

6X + 2Y = 50

2X + 4Y = 20

2. Use this graph to answer the questions.

Max 20X + 10Y

s.t. 12X + 15Y < 180

15X + 10Y < 150

3X - 8Y < 0

X , Y > 0

a. Which area (I, II, III, IV, or V) forms the feasible region?

b. Which point (A, B, C, D, or E) is optimal?

c. Which constraints are binding?

d. Which slack variables are zero?

3. Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below.

Fliptop Model Tiptop Model Available

Plastic 3 4 36

Ink Assembly 5 4 40

Molding Time 5 2 30

The profit for either model is $1000 per lot.

a. What is the linear programming model for this problem? (show the

objective function and constraints).

4. Use the following Management Scientist output to answer the questions.

LINEAR PROGRAMMING PROBLEM

MAX 31X1+35X2+32X3

S.T.

1) 3X1+5X2+2X3>90

2) 6X1+7X2+8X3<150

3) 5X1+3X2+3X3<120

OPTIMAL SOLUTION

Objective Function Value = 763.333

Variable Value Reduced Cost

X1 13.333 0.000

X2 10.000 0.000

X3 0.000 10.889

Constraint Slack/Surplus Dual Price

1 0.000 -0.778

2 0.000 5.556

3 23.333 0.000

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit

X1 30.000 31.000 No Upper Limit

X2 No Lower Limit 35.000 36.167

X3 No Lower Limit 32.000 42.889

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit

1 77.647 90.000 107.143

2 126.000 150.000 163.125

3 96.667 120.000 No Upper Limit

a. Give the solution to the problem.

b. Which constraints are binding?

c. What would happen if the coefficient of x1 increased by 3?

d. What would happen if the right-hand side of constraint 1 increased by 10?

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