DSCI 3870: Management Science
Exam # 2, Version A
Date
Nov 7th,2013
Allotted Time 78 Min
Name
SID
Please read this carefully
The questions, which you attempt today, consist of True/False and multiplechoice questions worth 133 points total. Some of these questions are based on
descriptive cases. There are a total of 34 questions. Of these, 32 questions
cover the 133 points and any 2 questions can be considered as
“embedded” bonus questions which will be taken into account when your
test is graded.
Please answer all the questions (1 to 34) on the scantron sheet provided in the
order that they appear. After you are done, please turn in the scantron and this
question booklet.
Please note that you have to enter your name and Student ID Number (SID#) in
the above area and on the scantron. Failure to do so will result in a grade of zero
This is an open book exam. As mentioned in the syllabus, you are allowed to
use a programmable calculator and/or laptop computer. Besides the
textbook, you are allowed to use only those notes, which have been handed
out in class, and notes, which you have recorded during, class sessions. Be
sure to allocate you time wisely between the multiple choice and T/F questions.
Acronyms such as LP, LHS (left-hand side), RHS (right-hand side) etc. have been
used liberally. If you are unsure about an acronym, please ask Dr. Kulkarni/the
invigilator immediately.
Best of luck!!
1
1.
The amount that the objective function coefficient of a decision variable would
have to improve before that variable would have a positive value in the solution is
the
A. dual price.
B. surplus variable.
C. reduced cost.
D. upper limit.
2. The dual price for a constraint that compares funds used with funds available is
0.058. This likely means that
A. the cost of additional funds is 5.8%.
B. if additional funds can be obtained at a rate of at most 5.8%, they
should be.
C. if additional funds can be obtained at a rate of more than 5.8%, they
should be.
D. the objective function value is 5.8%
3. The range of feasibility measures
A. the right-hand-side values for which the objective function value will
not change.
B. the right-hand-side values for which the values of the decision
variables will not change.
C. the right-hand-side values for which the dual prices will not change.
D. each of the above is true.
4. If a decision variable is not positive in the optimal solution, its reduced cost is
A. what its objective function value would need to be before it could
become positive.
B. the amount its objective function value would need to improve before
it could become positive.
C. zero.
D. its dual price.
5. The objective function for portfolio selection problems usually is maximization of
expected return or
A. maximization of investment types
B. minimization of cost
C. minimization of risk
D. maximization of number of shares
2
6. In the game that we played in class, I discussed an alternate solution that was not
optimal (it was $10 less than the optimal in terms of profit) but it was more
practical. What was the key difference between this “practical” solution versus the
optimal solution?
A. It involved making all products
B. It involved using all the available inventory
C. It involved selling leftover inventory in the open market
D. It involved making some products and buying some.
7. In a linear programming problem, the binding constraints for an optimal solution
are 5X + 3Y < 30 and 2X + 5Y < 20.As long as the slope of the objective
function stays between _______ and _______ respectively, the current optimal
solution point will remain optimal.
A. 5/3, 2/5
B. 3/5, 5/2
C. 20 , 30
D. 3/2 , 1
Use the following graphical representation for a maximization problem to answer
the two questions which follow. The formulation is embedded in the picture shown
below :
3
Over what range can the objective function coefficient for x2 (let’s call it c2) vary,
without changing the current optimal solution?
A. 0 < c2 <
B. 1 < c2 < 5
C. 4 < c2 < 8
D. –  < c2 < 5
8.
9.
A.
B.
C.
D.
What is the absolute value of the shadow price corresponding to Constraint 3?
0
2
2.5
3
The next five questions are based on the following case:
A large jewelry store is placing an order with its supplier. Four types of jewelry can be
ordered: necklaces, bracelets, rings, and earrings. It is assumed that every piece ordered
will be sold, and their profits in dollars, respectively, are 100,120,150, and 125. There
are several conditions that the store needs to worry about. One of these is the amount of
display space. They decided to develop an LP model whose output determines how many
necklaces, bracelets, rings, and earrings the jewelry store should order to maximize profit.
Constraint 1 measures display space in units, Constraint 2 measures time to set up the
display in hours. Constraints 3 and 4 are marketing restrictions that dictate maximum
and minimum amounts of combinations of items that can be sold. See the formulation
and MS Excel reports and answer the questions that follow.
Let:
X1= the number of necklaces
X2= the number of bracelets
X3= the number of rings
X4= the number of earrings
MAX:100X1+120X2+150X3+125X4
S.T.
X1+2X2+2X3+2X4≤108 - (Constraint 1)
3X1+5X2+X4≤120
- (Constraint 2)
X1+X3≤25
- (Constraint 3)
X2+X3+X4≥50
- (Constraint 4)
4
Objective Cell (Max)
Final
Name Value
Cell
$H$5
profit
7475
Decision Variable Cells
Name
X1
X2
X3
X4
Final
Value
8
0
17
33
Reduced
Cost
0
-5
0
0
Objective
Coefficient
100
120
150
125
Allowable Allowable
Increase Decrease
1E+100
12.5
5
1E+100
12.5
25
25
5
Name
Constraint 1
Constraint 2
Constraint 3
Constraint 4
Final
Value
108
57
25
50
Shadow
Price
75
0
25
-25
Constraint
R.H. Side
108
120
25
50
Allowable Allowable
Increase Decrease
15.75
8
1E+100
63
33
17
4
8.5
Cell
$B$3
$C$3
$D$3
$E$3
Constraints
Cell
$F$7
$F$8
$F$9
$F$10
10. How many necklaces and rings should be stocked?
A. 8 necklaces and 17 rings
B. 8 necklaces and no rings
C. 17 rings and 33 necklaces
D. 8 rings and 17 necklaces
11. How much space will be left unused and how much time will be used?
A. 108 units of space will be left unused and 63 hours of time will be used.
B. No space will be left unused and 57 hours of time will be used.
C. 57 units of space will be left unused and 108 hours of time will be used
D. 63 units of space will be left unused and 0 hours of time will be used.
12. By how much will the marketing restriction captured by Constraint 4 be exceeded
by?
A. It will be exceeded by 50.
B. It will not be exceeded.
C. It will be exceeded by 25.
D. It will be fall short by 25.
5
13. To what value can the profit on necklaces drop to and the profit on rings increase
to, respectively, before the current optimal solution changes?
A. 87.5 and 162.5
B.  and 25
C. 12.5 and 25
D. 112.5 and 137.5
14. You are offered the chance to obtain more space. The offer is for 15 units of space
at a total cost of $1500. Should you accept the offer and why or why not?
A. Yes, because the marginal cost of every unit of space being offered is less
than the shadow price
B. No, because the marginal cost of every unit of space being offered is less
than the shadow price
C. Yes, because the marginal cost of every unit of space being offered is
more than the shadow price
D. No, because the marginal cost of every unit of space being offered is more
than the shadow price
Please answer the next question based on the information in the table below
which displays the range of optimality for Variable 1.
Variable
1
Lower
Limit
60
Current
Value
100
Upper
Limit
120
15. What will happen to the optimal solution if the objective function coefficient for
Variable 1 decreases by 20?
A. Nothing. The values of the decision variables, the dual prices, and the
objective function will all remain the same.
B. The value of the objective function will change, but the values of the
decision variables and the dual prices will remain the same.
C. The same decision variables will be positive, but their values, the
objective function value, and the dual prices will change.
D. The problem will need to be resolved to find the new optimal solution and
dual price.
16. In the 3-part video that we saw in class, one of the applications of Management
Science was:
A. Scheduling police officers in a major city
B. Optimal placement of advertisements on websites
C. Breast cancer diagnosis and prognosis
D. Electricity grid optimization
6
The next four questions are based on the following case:
Karga coffee roaster blends three types of coffee beans (Brazilian, Colombian and
Peruvian) and sell it at Health Coffee Drink retail chain. Each kind of bean has different
fat and calorie contents. Information regarding the two ingredients in the three coffee
beans and cost of the three coffee beans is shown below. One pound is sixteen ounces.
Brazilian
Colombian
Peruvian
Cost ($/pound)
5
6
7
Fat (in grams/pound)
75
60
85
Calories (per ounce)
15
20
18
Karga would like to create a blend that has an average fat content of at least 78
grams/pound and it should have at least 1600 calories overall. The available quantities are
1500lb of Brazilian, 1200lb of Colombian and 2000lb of Peruvian beans. Karga wants to
make 4000lb of the blend at the lowest possible cost.
Let
B = Pounds of Brazilian coffee beans to blend
C = Pounds of Colombian coffee beans to blend
P = Pounds of Peruvian coffee beans to blend
17. What is the appropriate objective function?
A. Max: 5B+6C+7P
B. Min: 5B+6C+7P
C. Max: B+C+P
D. Min: B+C+P
18. Which of the following captures the constraint on the fat requirement?
A. 75B+60C+85P≥78
B. 3B18C+7P≤0
C. 75B+60C+85P≤78
D. 3B18C+7P≥0
19. Which of the following captures the constraint on calories?
A. 15B+20C+18P≥1600
B. 15B+20C+18P≥100
C. 15B+20C+18P≤1600
D. 15B+20C+18P≤100
20. What does the constraint P≤2000 indicate?
A. The available quantity for Brazilian bean
B. The available quantity for Colombian bean
C. The available quantity for Peruvian bean
D. The available quantity for all bean
The next four questions refer to the following case:
7
The Denton County Sheriff’s Department schedules police officers for 8-hour shifts. The
beginning times for the shifts are 8:00 am, Noon, 4:00 pm, 8:00 pm, Midnight and 4:00
am. An officer beginning a shift at one of these times works for the next 8 hours. During
normal weekday operations, the number of officers needed varies depending on the time
of the day. The department staffing guidelines require the following minimum number of
officers on duty (see the table below). The Denton County Sheriff’s Department would
like to formulae a linear program and determine the number of police officers that should
be scheduled to begin the 8-hour shifts at each of the six times (8:00 am, noon, 4:00 pm ,
8:00 pm, midnight and 4:00 am) to minimize the total number of officers required. Let
X1= the number of officers beginning work at 8:00 am, X2= the number of officers
beginning work at noon, and so on.
Time of Day
8:00 am - Noon
Noon - 4:00 pm
4:00 pm - 8:00 pm
8:00 pm - Midnight
Midnight - 4:00 am
4:00 am - 8:00 am
Minimum Officers on Duty
10
6
5
6
4
7
21. An appropriate objective function would be:
A. Min: 10X1 + 6X2 + 5X3 + 6X4 + 4X5 + 7X6
B. Max: 10X1 + 6X2 + 5X3 + 6X4 + 4X5 + 7X6
C. Min: X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8
D. Min: X1 + X2 + X3 + X4 + X5 + X6
22. The constraint for 4:00 pm – 8:00 pm is given as:
A. X1 + X2 ≥ 5
B. X2 + X3 ≥ 5
C. X1 +X2 + X3 ≤ 5
D. X1 +X2 + X3 ≥ 5
23. The constraint X1 + X2 ≥ 6 is for:
A. Noon - 4:00 pm
B. 4:00 pm - 8:00 pm
C. 8:00 pm - Midnight
D. Midnight - 4:00 am
24. Scheduling exactly 38 officers (i.e. 10+6+….+7), to begin work at 8:00 am and
none to begin work at any of the remaining shift times is a feasible solution to the
problem.
A True
B. False
8
The next three questions are based on the following case:
Futurama Kitchen Appliances Ltd (FKA) produces trendy microwaves. Two of its
microwave models are the market leaders. FKA Genius is one of the models and is sold
for a price of $ 120 and FKA Miracle is sold for a price of $150. Producing FKA Genius
requires 3 standard heating coils and 4 IC boards and producing FKA Miracle requires 2
heavy duty heating coils and 3 IC board. There are presently 90 standard heating coils
and 100 heavy duty heating coils available. There are 198 IC boards available. Variable
definition for the linear programming problem, the feasible region and the sensitivity
analysis are provided below. You are required to answer the questions which follow
using this information, the GLP figure and the MS Excel analysis as needed.
Let, G = number of Genius models produced
M = number of Miracle models produced
M
50
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Hv. Dty. Ht. Coil:
0.0 G +
2.0 M = 100.0
Payoff: 120.0 G + 150.0 M = 8940.0
IC Board:
4.0 G +
3.0 M = 198.0
St. Ht. Coil:
0
1
2
3
4
5
6
7
8
9
3.0 G +
0.0 M = 90.0
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Optimal Decisions(G,M): (12.0, 50.0)
St. Ht. Coil:
3.0G +
Hv. Dty. Ht. Coil:
IC Board:
4.0G +
0.0M <= 90.0
0.0G +
2.0M <= 100.0
3.0M <= 198.0
Futurama.
1
2
3
4
5
6
7
8
A
Decision Variables
Quantity
Profit Contribution
Subject To
Stand Coil
Heavy Duty Coil
IC Boards
B
Genius
C
Miracle
120
150
Profit
8940
2
3
LHS
36
100
198
3
4
9
D
E
F
<=
<=
<=
RHS
90
100
198
G
Microsoft Excel 12.0 Sensitivity Report
Adjustable Cells
Final Reduced
Cell
Name
Value
Cost
$B$3 Quantity Genius
12
0
$C$3 Quantity Miracle
50
0
Objective
Coefficient
120
150
Allowable
Increase
80
1E+30
Allowable
Decrease
120
60
Constraint
R.H. Side
90
100
198
Allowable
Increase
1E+30
32
72
Allowable
Decrease
54
48
48
Constraints
Cell
$D$7
$D$8
$D$9
Name
Stand Coil LHS
Heavy Duty Coil LHS
IC Boards LHS
Final
Value
36
100
198
Shadow
Price
0
30
30
25. What is the allowable increase for the RHS value of the constraint “Heavy Duty
Coil”?
A. 32
B. 30
C. 48
D. 72
26. What is the allowable decrease for the RHS value of the constraint “IC board”?
A. 32
B. 30
C. 48
D. 72
27. What is the shadow price for the “IC board” constraint in the above sensitivity
analysis?
A. 32
B. 30
C. 48
D. 72
10
The next three questions are based on the following case:
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
TV
Radio
Newspaper
500
200
400
10000
3000
5000
Exposure
Quality
30
40
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 if at least 1000. Define variables as follows:
Let, T = the number of TV ads
R = the number of radio ads
N = the number of newspaper ads
28. What is the appropriate objective function?
A. Min.10000T + 3000R + 5000N
B. Min.30T + 40R + 25N
C. Max.10000T + 3000R + 5000N
D. Max.30T + 40R + 25N
29. What does the constraint 500T + 200R + 400N  10000 capture?
A. Exposure requirement
B. Reach requirement
C. Budget restriction
D. Objective function
30. The restriction on TV ads relative to radio ads is captured by the constraint:
A. T - R  4
B. T - R > 4
C. T – 4 R > 0
D. T – 4 R < 0
11
Please answer the next four questions based upon the case given below and the
attached MS Excel output:
Mamma Mia Coffee Inc., one of the largest coffee producing companies in Italy,
produces two types of coffees. Coffee 1 and Coffee 2 are produced by processing raw
coffee beans. Up to 103.33 lb of raw coffee beans may be purchased at a cost of $18/lb.
One pound of raw coffee beans can be used to produce either 1.5 lb of Coffee 1 or 0.55 lb
of Coffee 2. Using a pound of raw beans to produce 1.5 lbs of Coffee 1 requires 3.0
hours of labor and to produce 0.55 lb of Coffee 2 requires 2.5 hours of labor. A total of
280 hours of labor are available, and at most 160 lbs of Coffee 1 and Coffee 2 taken
together can be sold. Not more than 80 lbs of Coffee 2 and not more than 50 lbs of Coffee
1 can be sold. Coffee 1 sells for $45/lb, and Coffee 2 sells for $57/lb. (Yes, they are that
expensive; they are imported Baba Budan coffees from the famed Nilgiri Hills
plantations).
Let,
B = lbs of raw beans processed
C1 = lbs of raw beans used to produce Coffee 1
C2 = lbs of raw beans used to produce Coffee 2
To maximize profit, Mamma Mia Coffee Inc. should solve the following LP:
Max. (45×1.5)C1 + (57×0.55) C2 – 18 B
s.t.
B > C1 + C2
(Constraint 1)
3.0 C1 + 2.5 C2 < 280
(Constraint 2)
B < 103.33
(Constraint 3)
1.5 C1 + 0.55 C2 < 160 (Constraint 4)
0.55 C2 < 80
(Constraint 5)
1.5 C1 < 50
(Constraint 6)
C1, C2, B > 0
Using the Excel output shown on the next page, answer the following questions.
31. If Coffee 1 sold for $47/lb, then the new optimal profit would_________.
A. stay the same
B. increase by @ $33.33
C. increase by @ $100
D. increase by $67.5
12
32. Next month, the shipment of raw beans is going to be limited to 100 lbs. due to a
change in packaging standards. What is the new optimal profit?
A. @ $2571
B. @ $2598
C. @ $2581
D. @ $2540
33. Considering the current optimal production plan, what is the most that Mamma
Mia Inc. should be willing to pay for another hour of labor?
A. $18
B. $28
C. $22
D. Nothing
34. What would be the new optimal profit of Mamma Mia Inc., if they found out that
market conditions dictate at most 40 lbs. of Coffee 1 being sold henceforth?
A. $2560.4
B. $2574.5
C. $2343.5
D. $241
ANSWER REPORT:
Target Cell (Max)
Cell
$D$5
Name
Profit
Original
Value
0
Final Value
2584.5
Original
Value
0
0
0
Final Value
33.33
70
103.33
Adjustable Cells
Cell
$A$2
$B$2
$C$2
Name
C1
C2
B
Constraints
Cell
Name
$E$8
Constraint 1
$E$9
Constraint 2
$E$10 Constraint 3
$E$11 Constraint 4
$E$12 Constraint 5
Cell Value
0.00
275
103.33
88.50
38.50
Formula
$E$8>=$G$8
$E$9<=$G$9
$E$10<=$G$10
$E$11<=$G$11
$E$12<=$G$12
13
Status
Binding
Not Binding
Binding
Not Binding
Not Binding
Slack
0.00
5
0
71.50
41.50
$E$13
Constraint 6
50.00
$E$13<=$G$13
Binding
0
SENSITIVITY REPORT:
Adjustable Cells
Cell
$A$2
$B$2
$C$2
Name
C1
C2
B
Final
Value
33.333
69.997
103.330
Reduced
Cost
0.000
0.000
0.000
Objective
Coefficient
67.5
31.35
-18
Allowable
Increase
1E+30
36.15
1E+30
Allowable
Decrease
36.15
13.35
13.35
Final
Value
0.00
275
103.33
88.50
38.50
50.00
Shadow
Price
-31.35
0.00
13.35
0.00
0.00
24.10
Constraint
R.H. Side
0
280
103.33
160
80
50
Allowable
Increase
70
1E+30
2
1E+30
1E+30
15.025
Allowable
Decrease
2
5
70
71.5
41.5
50
Constraints
Cell
$E$8
$E$9
$E$10
$E$11
$E$12
$E$13
Name
Constraint 1
Constraint 2
Constraint 3
Constraint 4
Constraint 5
Constraint 6
14