DSCI 3870: Management Science

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DSCI 3870: Management Science
Exam # 3, Version - Green
Date
December 12th,2013
Allotted Time 120 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 37 questions.
Please answer all the questions 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, open notes exam. Laptop computers are also allowed.
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 Be sure to allocate your
time wisely between the multiple choice and T/F questions.
Best of luck!!
1
1. In an all-integer linear program,
A. all variables must be integer*.
B. all objective function coefficients and right-hand side values must be integer.
C. all objective function coefficients must be integer.
D. all right-hand side values must be integer.
2. The objective of the transportation problem is to
A. identify one origin that can satisfy total demand at the destinations and at the
same time minimize total shipping cost.
B. minimize the number of origins used to satisfy total demand at the destinations.
C. minimize the cost of shipping products from several origins to several
destinations*.
D. minimize the number of shipments necessary to satisfy total demand at the
destinations.
3. The assignment problem constraint x31 + x32 + x33 + x34 < 2 means
A. agent 3 can be assigned to 2 tasks.*
B. agent 2 can be assigned to 3 tasks.
C. a mixture of agents 1, 2, 3, and 4 will be assigned to tasks.
D. there is no feasible solution.
4. The measure of risk most often associated with the Markowitz portfolio model is the
A. portfolio average return.
B. portfolio minimum return.
C. portfolio variance.*
D. portfolio standard deviation.
5. We assume in the maximal flow problem that
A. the flow out of a node is equal to the flow into the node.*
B. the source and sink nodes are at opposite ends of the network.
C. the number of arcs entering a node is equal to the number of arcs exiting the
node.
D. None of the alternatives is correct.
6. Dual or shadow prices, similar to those that we get for linear programming models,
are mathematically impossible to derive for integer programming models.
A. True*
B. False
7. Whenever total supply is less than total demand in a transportation problem, the LP
model does not determine how the unsatisfied demand is handled.
A. True*
B. False
2
8. In a model, x1 > 0 and integer, x2 > 0, and x3 = 0, 1. Which solution would not be
feasible?
A. x1 = 5, x2 = 3, x3 = 0
B. x1 = 4, x2 = .389, x3 = 1
C. x1 = 2, x2 = 3, x3 = .578*
D. x1 = 0, x2 = 8, x3 = 0
The next two questions are based on the following case:
Optimal Ovens, Inc. (OOI) makes home toaster ovens at plants in Alabama and
Wisconsin. Completed ovens are shipped by rail to one of OOI’s warehouses in Memphis
and Pittsburgh and then distributed to customer facilities in Fresno, Peoria and Newark.
The two warehouses can also transfer small quantities between themselves using
company trucks. The task is to plan OOI’s distribution over the next month. Each plant
can ship up to 1000 units during this period and none are presently in Fresno, Peoria and
Newark. Fresno, Peoria and Newark customers require 450, 500 and 610 ovens
respectively. Transfers between warehouses are limited to 25 ovens but no cost is
charged. The network flow diagram is given below.
Wisconsin
1
Memphis
5
3
6
2
Fresno
Peoria
4
7
Alabama
Pittsburgh
Newark
Unit costs (cij) of the flow (Fij)on arc (i, j) are detailed in the following tables:
From/To
3. Memphis 4. Pittsburgh
1. Wisconsin
7
8
2. Alabama
4
7
From/To
5. Fresno 6. Peoria 7. Newark
3. Memphis
25
5
17
4. Pittsburgh
29
8
5
3
9. The overall flow constraint at Memphis is given as:
A. - F13 - F23 - F43 + F34 + F35 + F36 + F37 = 0*
B. +F13 + F23 + F43 – F34 – F35 – F36 – F37 <= 0
C. -F13 - F23 - F43 = F34 + F35 + F36 + F37
D. +F13 + F23 + F43 – F34 – F35 – F36 – F37 >= 0
10. Which of the following represents the objective function?
A. Max. 7F31 + 8F41 + 4F32 + 7F42 + 25F53 + 5F63 + 17F73 + 29F54 + 8F64 + 5F74
B. Min. 7F31 + 8F41 + 4F32 + 7F42 + 25F53 + 5F63 + 17F73 + 29F54 + 8F64 + 5F74
C. Min. F13 + F14 + F23 + F24 + F35 + F36 + F37 + F45 + F46 + F47
D. Min. 7F13 + 8F14 + 4F23 + 7F24 + 25F35 + 5F36 + 17F37 + 29F45 + 8F46 + 5F47*
Read the following case and answer the three questions that follow:
Fix-It shop has received four luxury cars Aston Martin (A), Bentley (B), Carrera (C) and
Daimler (D) to repair. Five workers with different talents are available to repair the cars.
The fix-it shop owner has an estimate of the profit if a particular worker does a particular
car. The profits in dollars per hour, which are shown in the table differ because the owner
believes that each worker will differ in speed and skill on these quite varied jobs. Assume
that the variables are Xij where i=A,B,C,D denote cars and j=1,2,3,4,5 denote workers,
and assume that the owner wants to maximize his profits by repairing all the cars and
making sure that each car is assigned to a single worker and each worker works on at
most one car.
Worker #
1
2
3
4
5
A
25
30
28
27
24
B
31
25
24
22
19
C
18
35
31
32
19
D
20
22
25
24
22
Car
11. The constraint for person 5 is given as:
A. 25XA5 + 31XB5 + 18XC5 + 20XD5 =1
B. 25XA5 + 31XB5 + 18XC5 + 20XD5 ≤ 1
C. XA5 + XB5 + XC5 + XD5 ≥ 1
D. XA5 + XB5 + XC5 + XD5 ≤ 1*
4
12. The constraint for Aston Martin is given as:
A. XA1 + XA2 + XA3 + XA4 =1
B. XA1 + XA2 + XA3 + XA4 ≤ 1
C. XA1 + XA2 + XA3 + XA4 + XA5 =1*
D. 25XA1 + 30XA2 + 28XA3 + 27XA4 + 24XA5 ≥ 1
13. Using the Greedy Heuristic discussed in class, what would be the assignment of cars
to workers?
A. Aston Martin5, Bentley1, Carrera2, Daimler4
B. Aston Martin5, Bentley2, Carrera3, Daimler4
C. Aston Martin3, Bentley5, Carrera1, Daimler4
D. Aston Martin3, Bentley1, Carrera2, Daimler4*
14. In class we saw the Max Cover problem for the state of Ohio. Based on the map of
the optimal solution to such a Max Cover problem shown below, how many counties,
excluding the hubs, have been covered to achieve maximum coverage?
A.
B.
C.
D.
25
22
19*
3
5
The next three questions are based on the following case:
Consider the Excel implementation and subsequent Sensitivity Analysis for a typical
portfolio optimization model below. Let x, y and z represent the fraction invested in
stocks AT&T, GM and USS respectively. The “Average Return” and the information in
the “Covariance Matrix” represent basic statistical analysis performed on returns data
provided to us for AT&T, GM and USS. The optimization model embedded in MS Excel
appears below the covariance matrix.
Average Return
Covariance Matrix
AT&T
GM
USS
Decision: Stock %
Requirements
Expected Return
-0.47%
AT&T
0.0014
-0.0016
-0.0006
65.00%
<=65%
-0.30%
7.61%
GM
-0.0016
0.0355
0.0141
5.78%
<=75%
0.44%
33.76%
USS
-0.0006
0.0141
0.0244
29.22%
<=70%
9.86%
Total
100%
=100%
10%
Portfolio
Variance
0.0029
>=10.0%
Sensitivity Analysis
Adjustable Cells
Cell
$C$19
$D$19
$E$19
Name
Decision: Stock % AT&T
Decision: Stock % GM
Decision: Stock % USS
Final
Value
65.00%
5.78%
29.22%
Reduced
Gradient
-0.75%
0.00%
0.00%
Name
Decision: Stock % Total
Expected Return Total
Final
Value
100%
10%
Lagrange
Multiplier
1%
2%
Constraints
Cell
$F$19
$F$21
15. The appropriate objective function is given as:
A. Max. 0.65x2 + 0.0578 y2 + 0.2922 z2
B. Min. 0.0014x2 +0.0355 y2 + 0.0244 z2  0.0032 x y  0.012 x z + 0.0282 y z*
C. Min. 0.65x2 + 0.0578 y2 + 0.2922 z2  0.0016 x y  0.006 x z + 0.0141 y z
D. Max. 0.0014x2 +0.0355 y2 + 0.0244 z2  0.0016 x y  0.006 x z + 0.0141 y z
16. One of the constraints in the problem would be:
A. 0.65x + 0.0578 y + 0.2922 z = 1.0
B. 0.65x + 0.0578 y + 0.2922 z ≤ 0.75
C. 0.003 x + 0.0044 y + 0.0986z = 0.1
D. 0.0047 x + 0.0761 y + 0.3376 z ≥ 0.1*
6
17. If the investor demands one percent more return, then the additional risk that
He/she will have to bear would be approximately:
A. 0.0002*
B. 1.0029%
C. 0.0129
D. 0.02
Consider the shaded feasible regions shown below:
Region A
Region B
18. Which of the following items is correct?
A. both regions are non-convex*
B. both regions are convex
C. Region A is convex and Region B is concave
D. Region A is non-convex and Region B is convex
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.
8
4
2
6
10
12
3
1
5
12
4
3
7
3
7
9
7
6
19. Which of the following represents the constraint for Node 5?
A. X25  X35  X45  X75 + X52 + X53 + X57 = 0*
B. X25  X35  X45  X75 + X52 + X53 + X57 = 1
C. X25  7X35  3X45 + 8X52 + 7X53 + 3X54 + 4X57 = 0
D. X25  X35  X75 + X52 + X53 + X54 + X57 = 0
Read the following case and answer the three questions that follow:
A summer camp recreation director is trying to choose activities for a rainy day.
Information about possible choices is given in the table below. A higher numeric
popularity rating is considered better.
Category
Time
(minutes)
Activity
1 - Painting
30
2 - Drawing
20
3 - Nature craft
30
Music
4 - Rhythm band
20
Sports
5 - Relay races
45
6 - Basketball
60
Computer 7 - Internet
45
8 - Writing
30
9 - Games
40
Let xi = 1 if activity i is chosen, 0 if not, for i = 1, …, 9
Art
Popularity
with
Campers
Popularity
with
Counselors
4
5
3
5
2
1
1
4
1
2
2
1
5
1
3
1
3
2
20. If the objective is to keep the campers happy, what should the objective
function be?
A. Max 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9*
B. Max 30x1 + 20x2 + 30x3 + 20x4 + 45x5 + 60x6 + 45x7 + 30x8 + 40x9
C. Max 2x1 + 2x2 + x3 + 5x4 + 1x5 + 3x6 + 1x7 + 3x8 + 2x9
D. Min 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9
21. Which of the following represents the constraint : “at most one art activity can
be done.”?
A. 4x1 + 5x2 +3 x3  1
B. 2x1 + 2x2 + x3  1
C. x1 + x2 + x3  1*
D. 30x1 + 20x2 + 30x3  1
22. Which of the following represents the constraint : “if basketball is chosen,
then music must be chosen”
A. x6  5x4
B. 60x6  20x4
C. 5x6  x4
D. x6  x4*
8
Max. 54X1 - 9X12 + 78X2 -13 X22
St. X1 < 4
2 X2 < 12
3 X1 + 2 X2 < 18
Xi > 0  i
6
Z = 189
4
2
Z = 162
Z = 117
Feasible Region
2
4
The next two questions are based upon the following case:
Consider the picture above/on the earlier page which shows a “Feasible Region” and
several objective function contours. Answer the next two questions based on this picture.
23. The optimal solution will likely result in:
A. 1 binding constraint
B. 2 binding constraints
C. no binding constraints*
D. 3 binding constraints
24. The optimal solution will likely occur:
A. in the interior of the feasible region*
B. at a corner point
C. at two corner points
D. on the boundary of the feasible region
9
The next three questions are based upon the following case:
The Westfall Company has a contract to produce 10,000 yards of garden hoses for a large
discount chain. Westfall has four different machines that can produce this kind of hose.
Because these machines are from different manufacturers and use differing technologies,
their specifications are not the same.
Fixed Cost to
Variable
Capacity
Set Up
Cost
(yards)
Production Run
Per Hose
1
750
1.25
6000
2
500
1.50
7500
3
1000
1.00
4000
4
300
2.00
5000
Let Pi = the number of hoses produced on machine i and Ui = 1 if machine i is used, = 0
otherwise
Machine
25.
A.
B.
C.
D.
If the company wants to minimize total cost what will be the objective function?
Min 1.25P1 + 1.5P2 + P3 + 2P4
Min750U1 + 500U2 + 1000U3 + 300U4
Min750U1 + 500U2 + 1000U3 + 300U4 + 1.25P1 + 1.5P2 + P3 + 2P4*
Min500U2 + 300U4 + 1.25P1 + 1.5P2 + P3 + 2P4
26.
A.
B.
C.
D.
What does the constraint “P1 + P2 + P3 + P4 ≥ 10000” capture?
Total amount of production
Minimum amount of production
Maximum amount of production
Minimum required contracted amount*
27. If we add the constraint of “U1 + U4 ≤ 1”, what does the constraint capture?
A. Machine 1 and machine 4 should be used.
B. If machine 4 is used, machine 1 cannot be used and vice versa.*
C. Machine 1 and machine 4 should be used together.
D. Neither machine 1 nor machine 4 should be used.
The next four questions are based upon the following case:
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 expected profits for each project are
summarized in the following table:
10
Project
Engineers
Required
Support
Personnel
Required
Profit
($1,000,000s)
1
2
3
4
5
6
20
55
47
38
90
63
15
45
50
40
70
70
1.0
1.8
2.0
1.5
3.6
2.2
28. Which of the following captures the constraint of “If either project 6 or project 4 is
undertaken, then both must be undertaken”?
A. P4 + P6 ≥1
B. P4+P6 = 0
C. P4  P6 ≥ 0
D. P4  P6 = 0*
29. What is the meaning of the constraint “P1 P2 ≥ 0”?
A. Project 2 can be undertaken only if project 1 is undertaken.*
B. If either project 2 or project 1 is undertaken, both must be undertaken
C. If project 2 is undertaken, then project 1 cannot be undertaken.
D. If project 1 is undertaken, then project 2 cannot be undertaken.
30. What is the meaning of the constraint “P3 + P5 ≥ 1”?
A. If either project 3 or project 5 is undertaken, both must be undertaken
B. Either project 3 or 5 must be undertaken.*
C. If project 5 is undertaken, project 3 must not be undertaken and vice versa
D. If project 3 is undertaken, then project 5 cannot be undertaken.
31. Which of the following captures the constraint: “project 2 cannot be undertaken
unless both projects 6 and 4 are undertaken, however, when projects 6 and 4 are
undertaken, then project 2 must be undertaken”
A. P4 + P6 ≥ P2 and P4 ≥ P2, P6 ≥ P2
B. P4 ≥ P2, P6 ≥ P2 and P4 + P6 – 1 ≤ P2*
C. P4 + P6≤ P2 and P4 ≥ P2, P6 ≥ P2
D. P4 ≥ P2, P6 ≥ P2 and P4 + P6 ≤ 2 P2
11
32. Consider a map-based solution of the sales-rep example discussed in class, which
was somewhat different from the Max Cover or Set Cover problems. The map
(where shaded counties are part of the optimal solution), suggests that we are
optimally locating ___ sales reps.
A. 10
B. 5*
C. 3
D. 2
12
The next two questions are based on the following case:
Consider the following integer programming problem and partial graphical solution
which follows. Answer the next two questions based on this information.
Payoff: 3.0 x + 5.0 y = -20.4
y
6
5
4
3
Constraint 1: 2.0 x + 3.0 y = 16.0
Constraint 3: 2.0 x + 0.0 y = 3.0
2
1
Constraint 2: 1.0 x + 4.0 y = 6.0
0
0
1
2
3
4
5
6
7
8
Optimal Decisions(x,y): ( 2.0, 4.0)
Constraint 1: 2.0x + 3.0y <= 16.0
Constraint 2: 1.0x + 4.0y >= 6.0
Constraint 3: 2.0x + 0.0y >= 3.0
Objective function: 3 x + 5 y
Subject to: 2 x > 3
2 x + 3 y < 16
x + 4y > 6
x > 0, y > 0
x - Integer, y - Integer
33. If the objective function were to be minimized, then the linear programming
relaxation of the problem would yield an objective function value:
A. slightly more than 11
B. exactly the same as the integer program
C. slightly less than 11*
D. slightly less than 18
34. If the objective function were to be maximized, then the optimal objective function
value for the integer program would be.
A. 24
B. 25
C. 26*
D. 26.1
13
x
35. Consider below a screen shot of an exercise that we did in class. This represents the
______.
A. Traveling Salesman Game*
B. Transportation Game
C. Shortest Path Game
D. Max Flow Game
36. Consider below a picture that represents an optimal solution to a network model
similar to one we discussed in class and as mentioned, this is one of the few models
that can be solved to optimality using simply a “greedy” approach. We call this a
_________.
A. Shortest Path Model
B. Multicommodity Network Flow Model
C. Traveling Salesman Model
D. Spanning Tree Model*
14
37. Consider below a picture that represents an optimal solution to the “Share of
Choices” model similar to the one that we discussed in class. The optimal pizza
covers __ customers.
A. 5
B. 4
C. 3*
D. 2
Salem Foods
Consumer
1
2
3
4
5
6
7
8
Crust
Cheese
Sauce
Thin
Thick
Mozzarella
Blend
Smooth
Chunky
11
2
6
7
3
17
11
7
15
17
16
26
7
5
8
14
16
7
13
20
20
17
17
14
2
8
6
11
30
20
12
17
11
9
2
30
9
19
12
16
16
25
5
9
4
14
23
16
Design
Variables
L11
0
L21
1
L12
0
L22
1
L13
0
Mild
L23
1
L14
0
Max Share
Constraints
LHS
RHS
1
51
42
80
44
68
4
10
1
1
1
1
>
>
>
>
>
>
>
>
=
=
=
=
1
1
1
1
1
1
1
1
1
1
1
1
26
14
29
25
15
22
30
16
Consumers
Preferring
(Y's)
1
0
0
0
0
0
1
1
15
Sausage
Medium
27
1
16
29
5
12
23
30
L24
1
Hot
8
10
19
10
12
20
19
3
L34
0
Current
Favorite
52
88
66
83
58
70
79
59
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