DSCI 3870: Management Science
Exam # 3
Version A
Date: 05/10/2012
Time: 100 minutes (1:30 – 3.10 pm.)
Name:
Student ID #:
Please read this carefully
This exam consists of 33 (T/F and multiple-choice) questions.
Please answer on the scantron sheet provided. I will not be responsible for lost sheets that are
turned in unstapled.
Please note that you have to enter your name and Student ID number in the above area.
Failure to do so will result in a grade of zero on the part of the exam in which the relevant
details have not been entered.
On the exam the following acronyms may have been used:
LP - Linear Program/Programming
IP – Integer Program/Programming
ILP – Integer Linear Program/Programming (used synonymously with IP)
NLP – Non-linear Program/ Programming
MILP – Mixed Integer Linear Program/ Programming
QP – Quadratic Program/Programming
An “(s)” appended to these acronyms denotes the plural.
This is an open book exam. Be sure to allocate your time wisely. All the best!
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Exam 3
For questions 1-3, consider the following: Burnside Marketing Research conducted a study
for Barker Foods on some designs for a new dry cereal. Three attributes were found to be
most influential in determining which cereal had the best taste: ratio of wheat to corn in the
cereal flake, type of sweetener (sugar, honey, or artificial), and the presence or absence of
flavor bits. Seven children participated in taste tests and provided the following part- worths
for the attributes:
Wheat/ Corn
Low
High
15
35
30
20
40
25
35
30
25
40
20
25
30
15
Child
1
2
3
4
5
6
7
Sugar
30
40
20
25
40
20
25
Sweetener
Honey Artificial
20
25
35
35
40
10
20
30
30
35
35
30
40
40
Flavor Bits
Present Absent
15
9
8
11
7
14
15
18
18
14
9
8
20
11
Assume that the overall utility (sum of part-worths) of the current favorite cereal is 70 for
each child. Your job is to design a product that will maximize the share of choices for the
seven children in the sample.
1. Suppose that the optimal solution indicates a cereal with high wheat/corn ratio, artificial
sweetener and no flavor bits. Then, Child # 7 will prefer this optimal cereal.
a. True
b. False*
c. Cannot be determined unless optimal attributes chosen are also given.
2. Suppose Child #1 prefers a particular solution. Then, which of the following children will
also DEFINITELY prefer this solution?
a.
b.
c.
d.
e.
Child #2
Child #3
Child #4
Child #5*
Child #6
3. Which child would definitely not switch from his/her current favorite cereal?
a.
b.
c.
d.
e.
All of the children may switch, given the appropriate options.
Child #2
Child #4
Child #5
Child #6*
2
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Exam 3
4. Consider the network below. Consider the LP for finding the shortest-route path from
node 1 to node 7. Bidirectional arrows () indicate that travel is possible both ways. Let Xij
= 1 if the route from node i to node j is taken and 0 otherwise.
Which of the following represents the constraint for Node 5?
a.
b.
c.
d.
e.
X25  X35  X45 + X52 + X53 + X57 = 0
X25  X35  X45 + X52 + X53 + X57 = 1
X25  X35  X45 + X52 + X53 + X54 + X57 = 0*
X25  7X35  3X45 + 8X52 + 7X53 + 3X54 + 4X57 = 0
None of the above.
Answer the next five questions based on the case given below:
Tower Engineering Corporation is considering undertaking several proposed projects for the
next fiscal year. The projects, the number of engineers and the number of support personnel
required for each project, and the net present value (NPV) for each project are summarized in
the following table:
Project
1
2
3
4
5
6
Engineers Required
30
65
53
38
90
63
Support Personnel Required
25
41
30
28
60
70
NPV ($1,000,000s)
2.0
2.8
4.0
3.5
7.6
5.2
Formulate an integer linear program that maximizes Tower's net present value, subject to
constraints, which will be stated in the questions that follow.
Let Pi = 0 or 1, indicate if project i will not be undertaken or will be undertaken respectively.
5. The most appropriate objective function for Tower Engineering Corporation is given as:
a. Max. 2P1 + 2.8P2 + 4P3 + 3.5P4 + 7.6P5 + 5.2P6*
b. Min. 2P1 + 2.8P2 + 4P3 + 3.5P4 + 7.6P5 + 5.2P6
c. Max. P1 + P2 + P3 + P4 + P5 + P6
d. Max. 30P1 + 65P2 + 53P3 + 38P4 + 90P5 + 63P6
e. Max. 25P1 + 41P2 + 30P3 + 28P4 + 60P5 + 70P6
3
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Exam 3
6. No more than four projects can be undertaken. The constraint that captures this is:
a. P1 + P2 + P3 + P4 + P5 + P6 < 4
b. P1 + P2 + P3 + P4 + P5 + P6 > 4
c. P1 + P2 + P3 + P4 + P5 + P6 ≤ 4*
d. P1 + P2 + P3 + P4 + P5 + P6 < 5
e. None of the above.
7. Projects 2 and 3 can only be undertaken if Project 5 is also undertaken. This requirement is
represented by:
a. P2 + P3 = P5
b. P2 < P5 and P3 < P5*
c. P2 + P3 < P5
d. P2 + P3 > P5
e. P2 > P5 and P3 > P5
8. No more than 300 engineers can be used. The constraint that captures this is:
a. 30P1 + 65P2 + 53P3 + 38P4 + 90P5 + 63P6  300*
b. 30P1 + 65P2 + 53P3 + 38P4 + 90P5 + 63P6  300
c. 25P1 + 41P2 + 30P3 + 28P4 + 60P5 + 70P6  300
d. 25P1 + 41P2 + 30P3 + 28P4 + 60P5 + 70P6  300
e. 30P1 + 65P2 + 53P3 + 38P4 + 90P5 + 63P6  300
9. The management of Tower Engineering stipulates that if project 4 is undertaken, then
neither project 1 nor project 6 can be completed. This restriction is capture by:
a. P1 + P4 + P6 < 2
b. P1 > P4 and P1 > P6
c. P1 + P4 + P6 < 1
d. P1 + P4 < 1 and P4 + P6 < 1*
e. P1 < P4 and P1 < P6
10. A minimization Linear Programming (LP) model with two decision variables x1 and x2
has an optimal solution with the objective function value of 22.5. If we restrict x1 and x2 to
be integers, which of the following CANNOT be the new optimal objective function value?
a.
b.
c.
d.
e.
23.5
22.5
45.5
24
Any of the above COULD be the new optimal objective function value*
4
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11.
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Exam 3
The 2-D (i.e. (x,y)) graph of a problem where x can be any non-negative number
while y needs to be a binary variable, has a feasible region _________.
a.
of dots.
b.
of two vertical stripes.
c.
of two horizontal stripes.*
d.
of infinitely many vertical stripes.
e.
of infinitely many horizontal stripes.
12. Which of the following is incorrect?
a. A global optimum is a local optimum in a nonlinear optimization problem.
b. A local maximum may not be the global maximum in a nonlinear optimization problem.
c. A local minimum is the global minimum in a convex nonlinear optimization problem.
d. A local maximum is the global maximum in a convex nonlinear optimization problem.*
e. All of the above is correct.
The next three questions refer to the following case:
Skooters Skateboards produces two models of skateboards, the R and the M. Skateboard
revenue (in $l,000s) for the firm is nonlinear and is stated as:
(6R  0.25R2) + (8M  0.5M2).
Skooters has 100 labor-hours available per week in its paint shop. Each R requires 5 laborhours to paint and each M requires 6 labor-hours. The Management Science team at Skooters
formulates its production planning problem to determine how many R and M skateboards
should be produced per week in order to maximize revenue.
Let R = number of R skateboards to produce per week
M = number of M skateboards to produce per week
13. What would be the revenue if 6 of R and 10 of M were produced per week?
a. $ 116,000
b. $ 65,000
c. $ 45,000
d. $ 60,000
e. $ 57,000*
14. The solution to the unconstrained non-linear programming problem is
R* = 12 and M* = 8. What is the minimum number of additional hours of labor that
Skooters needs per week to make this solution also optimal for the constrained problem?
a.
4
b.
8*
c.
16
d.
32
e.
0
5
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Exam 3
15. Which of the following can NOT be the objective function value of the constrained
maximization problem (assume 100 hours of labor are available)?
a. 69,000. *
b. 65,000.
c. 55,000.
d. 62,000.
e. All of the above could be the objective function value of the constrained problem.
Read the following case and answer the five questions that follow:
A 400-meter medley relay involves four different swimmers, who successively swim 100
meters of the Backstroke, Breaststroke, Butterfly, and Freestyle. A coach has 4 very fast
swimmers whose expected times (in seconds) in the individual events are given in the
following table. The coach wants to assign the swimmers to events so that his chances of
winning are maximized.
Swimmer
Smith
Johnson
Paul
Edwin
Event
Breaststroke Backstroke
72
63
70
67
66
70
76
65
Butterfly
71
64
69
71
Freestyle
55
58
59
57
Let Breaststroke be represented by 1, Backstroke by 2, Butterfly by 3 and Freestyle by 4 and
the swimmers be referred to by the first alphabet of their names i.e. Smith is S etc. Let Xij
represent swimmer “i”(S,J,P or E) assigned to event “j”(1,2,3 or 4).
16.
The objective function is best represented by:
a. Max: 72 XS1 + 63 XS2 + 71 XS3 + 55 XS4 + 70 XJ1 + 67 XJ2 + 64 XJ3 + 58 XJ4 + 66 XP1 +
70 XP2 + 69 XP3 + 59 XP4 + 76 XE1 + 65 XE2 + 71 XE3 + 57 XE4
b. Min: 72 XS1 + 70 XS2 + 66 XS3 + 76 XS4 + 63 XJ1 + 67 XJ2 + 70 XJ3 + 65 XJ4 + 71 XP1 +
64 XP2 + 69 XP3 + 71 XP4 + 55 XE1 + 58 XE2 + 59 XE3 + 57 XE4
c. Min: XS1 + XS2 + XS3 + XS4 + XJ1 + XJ2 + XJ3 + XJ4 + XP1 + XP2 + XP3 + XP4 + XE1 + XE2 +
XE3 + XE4
d. Min: 72 XS1 + 63 XS2 + 71 XS3 + 55 XS4 + 70 XJ1 + 67 XJ2 + 64 XJ3 + 58 XJ4 + 66 XP1 + 70
XP2 + 69 XP3 + 59 XP4 + 76 XE1 + 65 XE2 + 71 XE3 + 57 XE4*
e. None of the above.
17. The constraint for event ‘Freestyle’ is given by:
a. XS4+XJ4+XP4+XE4 = 1*
b. XS4+XJ4+XP4+XE4 < 1
c. XE1+XE2+XE3+XE4 = 1
d. 55XS4+58XJ4+59XP4+57XE4 = 1
e. 55XS4+58XJ4+59XP4+57XE4 = 55
6
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Exam 3
18. The constraint for swimmer ‘Johnson’ is given by:
a. 70XJ1+67XJ2+64XJ3+58XJ4 =58
b. XJ1+XJ2+XJ3+XJ4 =1*
c. XJ1+XJ2+XJ3+XJ4 > 1
d. XJ1-XJ2-XJ3+XJ4 = 0
e. None of the above.
19. Using the Greedy Heuristic discussed in class, what would be the assignment of
swimmers to events?
a. S → 3, J → 4, P → 2, E → 1
b. S → 4, J → 1, P → 2, E → 3
c. S → 4, J → 3, P → 1, E → 2*
d. S → 4, J → 1, P → 3, E → 2
e. S → 2, J → 1, P → 4, E → 3
20. If a swimmer is allowed to compete in more one event without loss of performance, then,
in the optimal assignment, which swimmer(s) would compete in multiple events?
a. Smith and Johnson
b. Smith and Edwin
c. Paul only
d. Johnson only
e. Smith only*
Read the following case and answer the four questions that follow:
Moonbucks coffee company makes two types of Coffee: Arabian Mocha Java and Decaf
Espresso Roast. These two types of coffees are made by blending three varieties of coffee
beans, Plantation X, Plantation Y and Plantation Z. Further, the coffee beans are flavored by
an ingredient called chicory and the coffees must meet restrictions on chicory content. Too
much chicory content spoils the taste of coffee. The three varieties of coffee beans are
shipped, mixed together in two delivery trucks to the retail center. Plantation X and Z can be
shipped in truck 1 and/or truck 2 but Plantation Y is shipped only in truck 2. No more than
1500 pounds of Arabian Mocha Java and 2500 pounds of Decaf Espresso Roast may be sold.
Using the data in the table below, we want to formulate a profit-maximizing non-linear
program.
COFFEE
Arabian Mocha Java
Decaf Espresso Roast
SALES PRICE per POUND ($)
41
30
CHICORY CONTENT (%)
NO MORE THAN 6
NO MORE THAN 3.5
BEAN
COST PER POUND ($)
CHICORY CONTENT (%)
Plantation X
Plantation Y
Plantation Z
25
18
22
4
7
3
7
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Exam 3
Assume the following:
X1, X2 = pounds of Plantation X purchased and shipped in trucks 1 and 2 respectively
Y = pounds of Plantation Y purchased
Z1, Z2 = pounds of Plantation Z purchased and shipped in trucks 1 and 2 respectively
TM1, TE1 = pounds of beans from truck 1 blended into Arabian Mocha Java and
Decaf Espresso Roast respectively
TM2, TE2 = pounds of beans from truck 2 blended into Arabian Mocha Java and
Decaf Espresso Roast respectively
TC1, TC2 = chicory content percentage of beans in trucks 1 and 2, respectively
21. The objective function can be represented as:
a. Min. 41(TM1+TM2)+ 30(TE1+TE2) – 25(X1+ X2) – 18Y – 22(Z1 + Z2)
b.Max. 41(TM1+TE1)+ 30(TM2+TE2) – 25(X1+ X2) – 18Y – 22(Z1 + Z2)
c. Max. 41(TM1+TM2)+ 30(TE1+TE2) – 4(X1+ X2) – 7Y – 3(Z1 + Z2)
d.Min. 25(X1+ X2) + 18Y + 22(Z1 + Z2)
e. Max. 41(TM1+TM2)+ 30(TE1+TE2) – 25(X1+ X2) – 18Y – 22(Z1 + Z2)*
22. The constraint on chicory percentage of beans in Truck 2 is given as:
a. TC2 = (0.04X2 + 0.07Y + 0.03Z2)
b. TC2 = (0.04X2 + 0.03Z2) / (X2 + Z2)
c. TC2 = (0.04X2 + 0.07Y + 0.03Z2) / (X2 + Y + Z2)*
d. TC2 = (0.02X2 + 0.07Y + 0.015Z2) / (X2 + Y + Z2)
e. TC2 = (X2 + Y + Z2) / (X2 + Y + Z2)
23. The coffee shipped in Truck 1 satisfies which of the following constraints:
a. X1 + Z1 > TM1 + TE1
b. X1 + Z1 = TM1 + TE1*
c. X1 + Z1 = TM1 + TM2
d. X1 + Y + Z1 = TM1 + TE1
e. X1 + Z1 = TC1 + TC2*
24. Which of the following CANNOT be the combination of beans used to make Decaf
Espresso Roast?
a. Plantation X and Plantation Y only*
b. Plantation X, Plantation Y and Plantation Z
c. Plantation Z only
d. All of the above can be used.
e. We cannot answer this question without solving the problem first.
8
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Exam 3
Read the following case and answer the four questions that follow:
A University Police Department had decided to install emergency telephones at select
locations on campus. The department wants to install the minimum number of telephones
provided that each of the main campus streets is served by at least one telephone. The figure
below maps the principal streets (A to K) on campus. It is logical to place the telephones at
intersections of streets so that each telephone will serve at least two streets. For example,
Location 2 provides coverage for Streets A, G and H according to the figure below, which
shows the layout of the streets and the telephone locations (encircled numbers).
1 − if a telephone is installed in location j
0 − if a telephone is not installed in location j
6
Street D
Street I
Street C
Street K
7
Street F
5
Street E
4
Street G
2
3
Street B
Street A
1
𝑥𝑗 = {
Street J
𝐿𝑒𝑡,
8
25. The appropriate objective function for the problem is:
a. Max. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11
b. Min. x1+ 2 x2 + 3 x3 +4 x4 + 5 x5 + 6 x6 + 7x7 + 8 x8
c. Min. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11
d. Min. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8 *
e. Max. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8
9
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Exam 3
26. The constraint x2 + x5 ≥ 1 represents the constraint for:
a. Location 2
b. Location 5
c. Street G*
d. Street J
e. None of the above
27. Which location, if it becomes unavailable, would make the problem infeasible?
a. 3
b. 4
c. 5
d. 2
e. None of the above *
28. If the optimal solution to the problem is to place telephones at locations 2, 3, 5 and 7,
then how many streets receive more than the minimum required coverage?
a. 1 street
b. 2 streets*
c. 3 streets
d. 4 streets
e. None
Answer the next three questions based on the case given below:
Hansen Controls has been awarded a contract for a large number of control panels. To meet
this demand, it will use its existing plants in San Diego and Houston, and consider new
plants in Tulsa, St. Louis, and Portland. Finished control panels are to be shipped to Seattle,
Denver, and Kansas City. Pertinent information is given in the table.
Sources
12345-
San Diego
Houston
Tulsa
St. Louis
Portland
Construction
Cost
------450,000
300,000
380,000
Demand
Shipping Cost to Destination:
Kansas
Seattle
Denver
City
1
2
3
4
7
10
9
7
5
8
4
3
10
5
4
3
8
10
13,000
10,000
9,000
Capacity
5,000
5,000
8,000
7,000
9,000
We develop a transportation model that includes provisions for the fixed costs (construction
costs in this case) for the three new plants. The solution of this model would reveal which
plants to build and the optimal shipping schedule.
Let
xij = the number of panels shipped from source i to destination j
yi = 1 if plant i is built, = 0 otherwise (i = 3, 4, 5)
10
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Exam 3
29. The appropriate objective function is given as:
a. Min. 4x11 + 7x12 + 10x13 + 9x21 + 7x22 + 5x23 + 8x31 + 4x32 + 3x33 + 10x41 + 5x42 + 4x43
+ 3x51 + 8x52 + 10x53 +450000 y3 + 300000y4 + 380000y5*
b. Max. 4x11 + 7x12 + 10x13 + 9x21 + 7x22 + 5x23 + 8x31 + 4x32 + 3x33 + 10x41 + 5x42 + 4x43
+ 3x51 + 8x52 + 10x53 +450000 y3 + 300000y4 + 380000y5
c. Max. in. 4x11 + 7x12 + 10x13 + 9x21 + 7x22 + 5x23 + 8x31 + 4x32 + 3x33 + 10x41 + 5x42 +
4x43 + 3x51 + 8x52 + 10x53 350000 y3  200000y4  480000y5
d. Min. 4x11 + 7x12 + 10x13 + 9x21 + 7x22 + 5x23 + 8x31 + 4x32 + 3x33 + 10x41 + 5x42 + 4x43
+ 3x51 + 8x52 + 10x53
e. None of the above.
30.
The constraint for supply from Portland is given as:
a.
x51 + x52 + x53  9000
b.
3x51 + 8x52 + 10x53  9000
c.
3x51 + 8x52 + 10x53  9000y5
d.
x51 + x52 + x53  9000y5 *
e.
x15 + x25 + x35  9000y5
31.
How many new plants would Hansen need to build?
a. Zero
b. One
c. Two
d. Three *
e. We can’t answer this without seeing the optimal solution.
11
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Exam 3
For Questions 32-33, please refer to the Network Diagram below. Let Xij represent the
number of units shipped from i to j, where i= 1, 2, 3 and j=4, 5, 6. The objective function is
to minimize overall transportation cost while making sure all demand is met. Answer each
question independently of the other questions. Unit shipping costs are intentionally hidden
from the diagram.
Supply
32.
Plants
Retailers
Demand
80
1
4
60
30
2
5
25
25
3
6
30
Under which of the following situations (consider them separately) would we NOT
be able to meet all the demand?
I. Route 1-4 is removed.
II. Retailer 6’s demand is increased to 45 units.
a. Only I. *
b. Only II.
c. Both I and II.
d. Neither I nor II.
e. There is insufficient information to answer this question.
33.
Suppose that the original problem has a unique solution. Part of this solution requires
Plant 3 send 15 units to Retailer 4 and nothing to the other retailers. We increase the capacity
of Plant 3 by 10 units. If we solve the new problem, what would happen to the objective
function value (the total transportation cost)?
a.
b.
c.
d.
e.
The objective function value would increase.
The objective function value would not change. *
The objective function value would decrease.
The problem would actually become infeasible.
Both b and c are possible.
12