# QUANTITATIVE ANALYSIS for BUSINESS OPERATIONS

Indicate whether the statement is true or false.
1. ​An extreme point of the feasible region can include negative values of coordinates.

a.
True

b.
False

2. In stationary time series there is no significant upward or downward trend in the data over time.

a.
True

b.
False

3. Mathematical programming is an approach that involves determining how to allocate the resources in such a way as to maximize profits or minimize costs.

a.
True

b.
False

4. Examining the effect of changes ​in the RHS values of constraints is part of the answer report.

a.
True

b.
False

5. The TREND( ) function can be used to calculate the estimated values for linear regression models.

a.
True

b.
False

6. Shadow prices represent the marginal values of the resources in an LP problem,

a.
True

b.
False

7. Seasonality is a regular, repeating pattern in the data that takes longer than 1 year to complete.

a.
True

b.
False

8. Objective cell, variable cells and constraint cells are terms used in Excel solver to describe the purpose of the cells. ​

a.
True

b.
False

9. Solver can be used to estimate model parameters when the time series is stationary and additive seasonal effects are present.

a.
True

b.
False

10. The weighted moving average technique is a special case of the moving average technique.

a.
True

b.
False

11. If a shadow price is positive for a maximization problem, a unit increase in the RHS value of the associated constraint results in a decrease in the optimal objective function value.

a.
True

b.
False

12. ​Good decisions always result in good outcomes.

a.
True

b.
False

13. ​Humans usually do not make errors in estimation due to anchoring and framing effects.

a.
True

b.
False

14. ​Excel and other spreadsheets contain an add-on called solver.

a.
True

b.
False

Indicate the answer choice that best completes the statement or answers the question.
Exhibit 3.1 The following questions are based on this problem and accompanying Excel windows. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate \$30 of profit and desks generate \$40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let
X1 = Number of Beds to produce

X2 = Number of Desks to produce
The LP model for the problem is
MAX:
30 X1 + 40 X2
Subject to:
6 X1 + 4 X2 ≤ 36 (carpentry)

4 X1 + 8 X2 ≤ 40 (varnishing)

X2 ≤ 8 (demand for desks)

X1, X2 ≥ 0

A
B
C
D
E
1

Jones Furniture

2

3

Beds
Desks

4
Number to make:

Total Profit:
5
Unit profit:
30
40

6

7
Constraints:

Used
Available
8
Carpentry
6
4

36
9
Varnishing
4
8

40
10
Desk demand

1

8

15. Refer to Exhibit 3.1. What formula should be entered in cell D8 in the accompanying Excel spreadsheet to compute the amount of carpentry used?

a.
=B4*B5+C4*C5

b.
=SUMPRODUCT(B8:C8,\$B\$4:\$C\$4)

c.
=SUM(B5:C5)

d.
=SUM(E8:E10)

16. Why would someone wish to use a spreadsheet model?

a.
To implement a computer model.

b.
Because spreadsheets are convenient.

c.
To analyze decision alternatives.

d.
All of these.

17. In a model Y=f(x1, x2), Y is called:

a.
a dependent variable.

b.
an independent variable.

c.
a confounded variable.

d.
a convoluted variable.

18. In a mathematical formulation of an optimization problem, the objective function is written as z=2×1+3×2. Then:

a.
x1 is a decision variable

b.
x2 is a parameter

c.
z needs to be maximized

d.
2 is a first decision variable level

20. What needs to be done to the two constraints in order to convert the problem to a standard form?
MAX:
8 X1 + 4 X2
Subject to:
5 X1 + 5 X2 ≤ 20

6 X1 + 2 X2 ≤ 18

X1, X2 ≥ 0

a.
a slack variable needs to be added to each constraint to convert them to equalities.

b.
nothing.

c.
they need to be combined to a single constraint.

d.
they need to be subtracted side-by-side

Exhibit 3.3 The following questions are based on this problem and accompanying Excel windows. Jack’s distillery blends scotches for local bars and saloons. One of his customers has requested a special blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products, call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50 1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B. The customer also specified that the blend have an alcohol content of at least 85%. Product A contains 95% alcohol while product B contains 78%. The blend is sold for \$12.50 per bottle. Product A costs \$7 per liter and product B costs \$3 per liter. The company wants to determine the blend that will meet the customer’s requirements and maximize profit.
Let
X1 = Number of liters of product A in total blend delivered

X2 = Number of liters of product B in total blend delivered

MIN:
7 X1 + 3 X2
Subject to:
X1 + X2 = 1.5 * 50 (Total liters of mix)

X1 ≥ 0.48 * 1.5 * 50 (X1 minimum)

X2 ≥ 0.5 * 50 (X2 minimum)

.0.95 X1 + 0.78 X2 ≥ 0.85 * 1.5 * 50 (85% alcohol minimum)

X1, X2 ≥ 0

A
B
C
D
E
1

Jacks’ Distillery

2

3

A
B

4
Liters to use

Total Cost:
5
Unit cost:
10.5
4.5

6

7
Constraints:

Supplied
Requirement
8
Total Liters
1
1

75
9
A required
1

36
10
B required

1

25
11
85% alcohol
0.95
0.78

63.75
2

3

A
B

4
Liters to use

Total Cost:
5
Unit cost:
10.5
4.5

6

7
Constraints:

Supplied
Requirement
8
Total Liters
1
1

75
9
A required
1

36
10
B required

1

25
11
85% alcohol
0.95
0.78

63.75

21. Refer to Exhibit 3.3. What formula should be entered in cell D11 in the accompanying Excel spreadsheet to compute the total liters of alcohol supplied?

a.
=B4*B5+C4*C5

b.
=SUMPRODUCT(B11:C11,\$B\$4:\$C\$4)

c.
=SUM(B5:C5)

d.
=SUM(E8:E10)

22. Binding constraints have

a.
zero slack.

b.
negative slack.

c.
positive slack.

d.
surplus resources.

24. A linear formulation means that:

a.
the objective function and all constraints must be linear

b.
only the objective function must be linear

c.
at least one constraint must be linear

d.
no more than 50% of the constraints must be linear

26. Which step of the problem-solving process is considered the most important?

a.
Identify problem.

b.
Analyze model.

c.
Test results.

d.
Implement solution.

28. The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch
MAX:
150 X1 + 250 X2
Subject to:
2 X1 + 5 X2 ≤ 200

3 X1 + 7 X2 ≤ 175

X1, X2 ≥ 0
How much profit is earned per each unit of product 2 produced?

a.
150

b.
175

c.
200

d.
250

29. The symbols X1, Z1, Dog are all examples of

a.
decision variables.

b.
constraints.

c.
objectives.

d.
parameters.

30. Which of the following categories of modeling techniques involves determining the value of a dependent variable based on specific values of independent variables?

a.
Biased models.

b.
Descriptive models.

c.
Predictive models.

d.
Prescriptive models.

31. When performing sensitivity analysis, which of the following assumptions must apply?

a.
All other coefficients remain constant.

b.
Only right hand side changes really mean anything.

c.
The X1 variable change is the most important.

d.
The non-negativity assumption can be relaxed

32. The first step in formulating a linear programming problem is

a.
Identify any upper or lower bounds on the decision variables.

b.
State the constraints as linear combinations of the decision variables.

c.
Understand the problem.

d.
Identify the decision variables.

e.
State the objective function as a linear combination of the decision variables.

33. Mathematical programming is referred to as

a.
optimization.

b.
satisficing.

c.
approximation.

d.
simulation.

Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table.

Maximum

Assembly
Inspection
Computer
Profit per
demand for
Wiring Hours
Hours
Hours
Model
Model (\$)
product
Required
Required
Required
Plain
30
80
.4
.5
.2
Fancy
40
90
.5
.4
.3

Hours Available
50
50
22

Let
X1 = Number of Plain computers to produce

X2 = Number of Fancy computers to produce

MAX:
30 X1 + 40 X2
Subject to:
.4 X1 + .5 X2 ≤ 50 (wiring hours)

.5 X1 + .4 X2 ≤ 50 (assembly hours)

.2 X1 + .2 X2 ≤ 22 (inspection hours)

X1 ≤ 80 (Plain computers demand)

X2 ≤ 90 (Fancy computers demand)

X1, X2 ≥ 0

A
B
C
D
E
1

Byte Computer Company

2

3

Plain
Fancy

4
Number to make:

Total Profit:
5
Unit profit:
30
40

6

7
Constraints:

Used
Available
8
Wiring
0.4
0.5

50
9
Assembly
0.5
0.4

50
10
Inspection
0.2
0.3

22
11
Plain Demand
1

80
12
Fancy Demand

1

90

34. Refer to Exhibit 3.2. Which of the following statements will represent the constraint for just assembly hours?

a.
B4:C4 ≤ B5:C5

b.
D9 ≤ E9

c.
D8:D10 ≤ E8:E10

d.
E8:E10 ≤ D8:D10

36. A solvable problem must have:

a.
​a feasible region that is not an empty set.

b.
​the best solution.

c.
no more than two constraints.

d.
​no more than two decision variables.

37. The coefficients in an LP model (cj, aij, bj) represent

a.
random variables.

b.
numeric constants.

c.
random constants.

d.
numeric variables.

38. If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to the objective function value?

a.
increase of 100

b.
increase of 110

c.
decrease of 100

d.
increases but by unknown amount

42. A financial planner wants to design a portfolio of investments for a client. The client has \$400,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 30% of the money in any one investment, at least one half should be invested in long-term bonds which mature in six or more years, and no more than 40% of the total money should be invested in B or C since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.
Investment
Return
Years to Maturity
Rating
A
6.45%
6
1-Excellent
B
8.50%
5
3-Good
C
9.00%
8
4-Fair
D
7.75%
4
2-Very Good

Formulate the LP for this problem.

44. A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
Time period
Required # of Nurses
Wage (\$/hr)
12 am − 4 am
20
15
4 am − 8 am
30
16
8 am − 12 pm
40
13
12 pm − 4 pm
50
13
4 pm − 8 pm
40
14
8 pm − 12 am
30
15
What values would you enter in the Analytic Solver Platform (ASP) task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem? Objective Cell: Variables Cells: Constraints Cells:
Let
Xi = number of nurses working in time period i; i = 1,6

MIN:
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6
Subject to:
1X1 + 1X2 ≥ 30

1X2 + 1X3 ≥ 40

1X3 + 1X4 ≥ 50

1X4 + 1X5 ≥ 40

1X5 + 1X6 ≥ 30

1X1 + 1X6 ≥ 20

Xi ≥ 0

A
B
C
D
E
F
G
H
I
1

Nurse
Hiring

2

3

Hours for each shift

4

Mid
4am
8am
Noon
4pm
8pm
Nurses
Wages per
5
Shift
4am
8am
Noon
4pm
8pm
Mid
Scheduled
Nurse
6
1
1
1
0
0
0
0

\$15
7
2
0
1
1
0
0
0

\$16
8
3
0
0
1
1
0
0

\$13
9
4
0
0
0
1
1
0

\$13
10
5
0
0
0
0
1
1

\$14
11
6
1
0
0
0
0
1

\$15
12
Available:

Total Wages:
13
Required:
20
30
40
50
40
30

46. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert’s grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert’s task somewhat by adding the following stipulations:

homework can account for up to 25% of the grade, but must be at least 5% of the grade;

the project can account for up to 25% of the grade, but must be at least 5% of the grade;

the mid-term and final must each account for between 10% and 40% of the grade but cannot account for more than 70% of the grade when the percentages are combined; and

the project and final exam grades may not collectively constitute more than 50% of the grade.
Formulate an LP model for Robert to maximize his numerical grade.

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