SAMPLE MIDTERM (QMIS 205)

Part I: True/False

1. Given that x and y are decision variables, the equation 3xy = 9 is linear.

2. A feasible solution is one that satisfies at least one of the constraints of a linear programming

problem.

3. Decision variable is one of the parameters in a linear programming model.

4. Alternative (multiple) optimal solution occurs only if there is no optimum solution.

5. Unbounded feasible region always results in an unbounded solution.

6. A problem is a single criterion problem if there is only one objective function to be optimized.

7. Profit is the difference between volume multiplied by price and the sum of fixed and variable costs.

8. In LP the constraints usually appear as linear expressions that are ≤ or = or ≥ than a parameter.

9. In LP < or > are not allowed for the constraints.

10. Fixed cost increases as the number of produced products increases.

11. In LP the objective function always consists of either maximizing or minimizing some value.

12. An infeasible solution violates at least one of the constraints.

13. Linear programming problems can be solved graphically, if they have any number of decision

variables and only 2 constraints.

14. The feasible solution area contains infinite solutions to the linear program.

15. Sensitivity ranges can be computed only for the right-hand sides of constraints.

16. The sensitivity range for a constraint quantity value is the range over which the shadow price is valid.

17. A change in the value of an objective function coefficient will always change the value of the

optimal solution.

Part II: Multiple Choice

1. Which of the following could not be a constraint for a linear programming problem?

a. 1A + 2B ≤ 3.

b. 1A + 2B ≥ 3.

c. 1A + 2B.

d. 1A + 2B = 3.

e. 1A + 2B + 3C ≤ 3.

2. What is the optimal solution for the following problem?

Maximize P = 3x + 15 y

subject to 2x + 4y ≤ 12

5x + 2y ≤ 10

and x ≥ 0, y ≥ 0.

a. (x, y) = (0, 3).

b. (x, y) = (2, 0).

c. (x, y) = (0, 0).

d. (x, y) = (1, 5).

e. (x, y) = (2, -3).

3. Which one of the following is true about the optimal solution for a linear programming problem:

a. The optimal solution always uses all the limited resources available

b. The optimal solution does not necessarily use up all the limited resources available

c. The optimal solution always leaves at least one limited resource unused

d. none of the above.

e. a and c

4. All variables in the solution of a linear programming problem are either positive or zero because of the

existence of:

a. an objective function

b. structural constraints

c. limited resources

d. none of the above.

e. a and c

5. It is possible to solve a linear programming problem graphically if there are no more than two:

a. non-negativity constraints

b. decision variables

c. inequalities

d. constraints

e. objective functions

6. In linear programming, non-negativity implies that a variable cannot have:

a. a negative coefficient in the objective function

b. a negative coefficient in a constraint

c. a negative right hand side value

d. none of the above.

e. b and c

7. If for an optimal solution for a linear programming problem, the slack variable is equal to zero, then:

a. the solution is not feasible

b. the solution is not possible

c. the entire amount of resource identified with the constraint in which the slack variable appears

has been used up

d. all of the above

e. a and b

8. Which of the following are components of a linear programming model for decision making?

a. Decision variables.

b. An objective function.

c. Constraints.

d. All the data.

e. All of the above.

9. Which of the following is an inequality or equation that expresses a restriction in a mathematical model?

a. Decision variable.

b. Parameter.

c. Constraint.

d. Objective function.

e. None of the above.

Consider the Linear Programming problem given below to answer the questions in this section:

Max 8X + 7Y

s.t. 15X + 5Y < 75 Constraint (1)

10X + 6Y < 60 Constraint (2)

X + Y < 8 Constraint (3)

X, Y 0

1. A feasible solution to the above problem is:

a. X = 5, Y = 3

b. X = 3, Y = 5

c. X = 4, Y = 6

d. X = 3, Y = 6

e. None of the above

2. What is the value of the objective function at (X=3, Y=6)

a. 58

b. 66

c. 61

d. 74

e. None of the above

3. Which of the following points represents an infeasible solution

a. (0, 0)

b. (2, 3)

c. (3, -2)

d. (4, 3)

e. None of the above

4. If the optimal solution is (X=3 , Y=5), which constraints are fully utilized (i.e., which constraints

are binding)

a. 2 & 3

b. 1 & 2

c. 1 & 3

d. All constraints are fully utilized

e. None of the constraints are fully utilized

Answer the following questions using the problem given below:

An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time,

radio time, and newspaper ads. Information about each medium is shown below:

Medium Cost Per Ad # Reached Exposure

Quality

TV 500 10000 30

Radio 200 3000 40

Newspaper 400 5000 25

If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget

is $10000, develop the model that will maximize the number reached and achieve an exposure quality of at

least 1000.

If the decisions Variables are defined as following:

Let T = the number of TV ads

Let R = the number of radio ads

Let N = the number of newspaper ads

1. What is the objective function?

a. Max 10000T + 3000R + 5000N

b. Max 500T + 200R + 400N

c. Max 30T + 40R + 25N

d. Max 500T + 3000R + 25N

e. Max 1000T + 500R + 30N

2. What is the advertising budget constraint?

a. 500T + 200R + 400N ≥ 10000

b. 10000T + 3000R + 5000N 10000

c. 500T + 200R + 400N ≤ 10000

d. 500T + 200R + 400N =10000

e. 30T + 40R + 25N ≤ 10000

3. Which constraint represents the condition: "the number of TV ads cannot exceed the number of radio ads

by more than 4?"

a. T - R 4

b. T - R ≥ 4

c. T - R = 4

d. T + R ≥ 4

e. T + R 4

4. Which constraint represents the condition: "achieve an exposure quality of at least 1000.”

a. 30T + 40R + 25N 1000

b. 30T + 40R + 25N = 1000

c. 30T + 40R + 25N ≤ 1000

d. 30T + 40R - 25N 1000

e. 30T - 40R + 25N 1000

5. Which constraint represents the condition: "achieve an exposure quality of at most 3000.”

a. 30T + 40R + 25N 3000

b. 30T + 40R + 25N = 3000

c. 30T + 40R + 25N ≤ 3000

d. 30T + 40R - 25N 3000

e. 30T - 40R + 25N ≤ 3000

Refer to the following sensitivity report for a maximization problem and answer the questions below:

ADJUSTABLE CELLS . Final Objective Allowable Allowable

Cell Name Value Coefficient Increase Decrease .

$B$9 Number of Tables 30 5 1E + 30 3

$C$9 Number of Chairs 0 4 6 1E + 30 ----------------------------------------------------------------------------------------------------------------------------------------------------------.

CONSTRAINTS . Final Shadow Constraint Allowable Allowable

Cell Name Value Price R. H. S. Increase Decrease .

$D$6 Assembly 90 0 120 1E + 30 30

$D$7 Painting 60 2.5 60 20 60

___________________________________________________________________ .

1 What is the optimal value of the objective function?

a. 30.

b. 90.

c. 60.

d. 150.

e. 120.

2 If you were to decrease only one of the resources, which one of the following would be the best

action to take?

a. decrease Assembly by 30 units.

b. decrease Painting by 60 units

c. decrease Assembly by 31 units

d. decrease Painting by 61 units.

e. decrease Assembly by 29 units.

3 If you were to increase only one of the resources, which one of the following would be the best

action to take?

a. increase Assembly by 90 units.

b. increase Assembly by 120 units.

c. increase Painting by 2.5 units.

d. increase Painting by 20 units.

e. increase Assembly by 30 units.

4 What is the allowable range for the Painting constraint’s RHS value?

a. 60≤RHS≤120

b. 120≤RHS≤80

c. 0≤RHS≤80

d. 60≤RHS≤20

e. none of the above

5 Which of the following is true about the allowable range for the objective function value of the first

variable (Number Of Tables)

a. it has no upper limit

b. it has no lower limit

c. it can decrease by 5

d. it can increase by 3

e. none of the above

6 If the objective function coefficient for number of tables and number of chairs change to 4 and 6

respectively, then:

a. The optimum solution will not change.

b. The problem have to be resolved to find the optimum solution

c. The shadow prices are valid

d. There will be no optimum solution

e. None of the above

7 If the RHS value of Assembly decrease by 20 and the RHS value of Painting increase by 10 then:

a. The optimum solution will not change.

b. The problem have to be resolved to find the new optimum solution

c. The shadow prices are valid

d. The shadow prices may or may not be valid

e. None of the above

Spread Sheet Modeling

Questions 1 through 4 in this part refer to the following spreadsheet model for a cost minimization problem.

123456789

10

A B C D E F

Activity 1 Activity 2Unit Cost $15 $20

Benefit Totals Needed

A 1 2 10 >= 10B 2 3 16 >= 6C 1 1 6 >= 6

Activity 1 Activity 2 Total Cost

Solution 2 4 $110

Contribution per Unit

1 Where are the data cells located?

a. B2:C2, B5:C7, and F5:F7.

b. B2:C2.

c. B10:C10.

d. F10.

e. None of the above.

2 Where are the changing cells located?

a. B10:C10.

b. B2:C2, B5:C7, and F5:F7.

c. B2:C2.

d. F10.

e. None of the above.

3 Where is the target cell located?

a. B2:C2.

b. B2:C2, B5:C7, and F5:F7.

c. F10

d. B10:C10.

e. None of the above.

4 Where are the output cells located?

a. B2:C2.

b. B2:C2, B5:C7, and F5:F7.

c. B10:C10.

d. F10.

e. None of the above.

5 Where are the decision variables located?

a. B10:C10.

b. B2:C2, B5:C7, and F5:F7.

c. B2:C2.

d. F10.

e. None of the above.