DSCI 3870: Management Science Date: 12/15/2009

DSCI 3870: Management Science
Exam # 3
SAMPLE
Date: 12/15/2009
Time: 2 hours (1:30 p.m. – 3:30 p.m.)
Name:
Student ID #:
Please read this carefully
This sample exam consists of 44 (T/F and multiple-choice) questions. (Your final exam will
likely have between 40-44 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 you 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. Best of luck!!
1. In a transshipment problem, shipments
a. cannot occur between two origin nodes.
b. cannot occur between an origin node and a destination node.
c. cannot occur between a transshipment node and a destination node.
d. can occur between any two nodes.*
2. Consider a shortest route problem in which a bank courier must travel between branches and the
main operations center. When represented with a network,
a. the branches are the arcs and the operations center is the node.
b. the branches are the nodes and the operations center is the source.
c. the branches and the operations center are all nodes and the streets are the arcs.*
d. the branches are the network and the operations center is the node.
3. Let x1 , x2 , and x3 be 0-1 binary variables whose values indicate whether the projects are not
done (0) or are done (1). Which constraint below indicates that at least two of the projects
must be done?
a. x1 + x2 + x3 ≥ 2*
b. x1 + x2 + x3 ≤ 2
c. x1 + x2 + x3 = 2
d. x1− x2 = 0
4. Modeling a fixed cost problem as an integer linear program requires
a. adding the fixed costs to the corresponding variable costs in the objective function.
b. using 0-1 variables.*
c. using multiple-choice constraints.
d. using LP relaxation.
5. The maximal flow problem can be formulated as a capacitated transshipment problem.
a. True*
b. False
6. The shortest-route problem is a special case of the transshipment problem.
a. True*
b. False
7. Is it possible that an LP relaxation of the original IP yields a feasible region and an optimal LP
solution, but the IP itself is infeasible and therefore has no solution?
a. Yes*
b. No
c. The question compares “apples and oranges” and can’t be answered.
8. If x1 + x2 ≤ 500 y1 and y1 is binary 0-1, then if y1 is 0, x1 and x2 will be 0.
a. True*
b. False
9 . If the optimal solution to the LP relaxation problem is integer, it is the optimal solution to the
integer linear program.
a. True*
b. False
10. The interpretation of the dual price for nonlinear programming models is different than that for
linear programming models.
a. True
b. False*
11. A function is quadratic if its nonlinear terms have a power of 4.
a. True
b. False*
12. In the LP formulation of a maximal flow problem, a conservation-of-flow constraint ensures
that an arc's flow capacity is not exceeded.
a. True
b. False*
The next two questions refer to the following case:
I recently decided to go on a camping trip and bought a nice tent and backpack from REI in
Dallas. I wanted to carry a number of items in my backpack, but its weight limit was 14 lbs. The
table below lists the items by their weight (in lbs.) and their perceived benefit; with a larger
number representing more benefit than a smaller one.
Item
1
2
3
4
Description
Ultrabrite Flashlight
Energy snack
Map
Camera
Let xi = 1 if I decide to pack item i, 0 if not
where, i = 1,2,3,4
13. The appropriate model formulation is given as:
a. Min. 25 x1 + 20 x2 + 8 x3 + 12 x4
St. 12x1 + 5x2 + 3x3 + 7x4 ≤ 14
b. Max. 25 x1 + 20 x2 + 8 x3 + 12 x4 *
St. 12x1 + 5x2 + 3x3 + 7x4 ≤ 14
c. Min. 12x1 + 5x2 + 3x3 + 7x4
St. 25 x1 + 20 x2 + 8 x3 + 12 x4 ≤ 65
d. Max. 25/12 x1 + 20/5 x2 + 8/3 x3 + 12/7 x4
St. 12x1 + 5x2 + 3x3 + 7x4 ≤ 27
e. Min. 12/25 x1 + 5/20 x2 + 3/8 x3 + 7/12 x4
St. 12x1 + 5x2 + 3x3 + 7x4 ≤ 14
14. The above problem is popularly referred to as:
a. The backpack problem
b. The utility maximization problem
c. The binary knapsack problem*
d. The camping problem
e. The binary weight to benefit problem.
Weight
12
5
3
7
Benefit
25
20
8
12
The next three questions are based on the following case:
Consider the problem faced by a summer camp recreation director who is trying to choose
activities for a rainy day. Information about possible choices is given in the table below.
(Popularity rating numbers indicate that higher is better).
Category
Activity
Art
1 − Painting
2 − Drawing
3 − Nature craft
4 − Rhythm band
5 − Relay races
6 − Basketball
7 − Internet
8 − Creative writing
9 − Games
Music
Sports
Computer
Time
(minutes)
30
20
30
20
45
60
45
30
40
Popularity
with Campers
4
5
3
5
2
1
1
4
1
Popularity
with
Counselors
2
2
1
5
1
3
1
3
2
Let xi = 1 if activity i is chosen, 0 if not, for i = 1, ... , 9
15. The most appropriate objective function is:
a. Max. 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9*
b. Min. 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9
c. Max. 2x1 + 2x2 + 1x3 + 5x4 + 1x5 + 3x6 + 1x7 + 3x8 + 2x9
d. Min. 2x1 + 2x2 + 1x3 + 5x4 + 1x5 + 3x6 + 1x7 + 3x8 + 2x9
16. Which constraint captures the restriction that no more than two computer activities can
be done?
a. x7 + x8 + x9 = 2
b. x7 + x8 + x9 > 2
c. x7 + x8 + x9 ≤ 2*
d. x8 + x7 + x9< 2
17. That the overall counselor popularity rating should be no higher than 10 is captured by
the constraint:
a. 2x1 + 2x2 + 1x3 + 5x4 + 1x5 + 3x6 + 1x7 + 3x8 + 2x9 > 10
b. 2x1 + 2x2 + 1x3 + 5x4 + 1x5 + 3x6 + 1x7 + 3x8 + 2x9 ≤ 10*
c. 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9 > 10
d. 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9 ≤ 10
The next four 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. Please answer the four questions that follow based upon
the information provided.
Portfolio Model
Average Return
Covariance
Matrix
AT&T
GM
USS
Decision: Stock
%
Requirements
Expected Return
AT&T
-30.0%
10.3%
-21.6%
-4.6%
-7.1%
-5.6%
14.9%
26.0%
41.9%
-7.8%
16.9%
-3.5%
2.48%
GM
22.5%
29.0%
21.6%
-27.2%
14.4%
10.7%
3.8%
-8.9%
-9.0%
8.3%
3.5%
-17.6%
4.26%
USS
32.1%
30.5%
19.5%
19.0%
-7.2%
71.5%
13.3%
23.2%
2.1%
13.1%
0.6%
10.8%
19.04%
Year
1
2
3
4
5
6
7
8
9
10
11
12
AT&T
0.0376
-0.0125
-0.0110
GM
-0.0125
0.0269
0.0079
USS
-0.0110
0.0079
0.0377
Total
29.25%
<=75%
0.73%
14.87%
<=75%
0.63%
55.88%
<=75%
10.64%
100%
=100%
12%
Sensitivity Analysis
Adjustable Cells
Cell
$C$19
$D$19
$E$19
Name
Decision: Stock % AT&T
Decision: Stock % GM
Decision: Stock % USS
Final
Value
29.25%
14.87%
55.88%
Reduced
Gradient
0.00%
0.00%
0.00%
Name
Decision: Stock % Total
Expected Return Total
Final
Value
100%
12%
Lagrange
Multiplier
0%
19%
Constraints
Cell
$F$19
$F$21
Portfolio
Variance
0.0122
>=12.0%
18.
The appropriate objective function is given as:
a. Max. 0.0376x2 +0.0269 y2 + 0.0377 z2 - 0.0645 x y + 0.0753 x z - 0.0646 y z
b. Min. 0.0376x2 +0.0269 y2 + 0.0377 z2 - 0.0645 x y + 0.0753 x z - 0.0646 y z
c. Max. 0.0376x2 +0.0269 y2 + 0.0377 z2 - 0.0125 x y + 0.011 x z - 0.0079 y z
d. Max. 0.0376x2 + 0.0269 y2 + 0.0377 z2 + 0.0125 x y + 0.011 x z + 0.0079 y z
e. Min. 0.0376x2 +0.0269 y2 + 0.0377 z2 - 0.025 x y - 0.022 x z + 0.0158 y z*
19. One of the explicit constraints in the problem would be:
a. 0.0376x +0.0269 y + 0.0377 z = 1.0
b. 0.0376x +0.0269 y + 0.0377 z <= 0.75
c. 2.48 x + 4.26 y + 19.04 z >= 12*
d. 0.0248 x + 0.0426 y + 0.1904 z <= 0.12
e. 2.48 x + 4.26 y + 19.04 z = 12
20. It will be lucrative to invest in AT&T stock, if its average return increases to at least:
a. 5%
b. 6%
c. 8%
d. 10%
e. No increase needed. AT&T stock is already invested in.*
21. If the required return is increased to 13% and assuming that its feasible to achieve this
higher return, what would be a good estimate of the new portfolio variance?. (Note:
Formula cells have 12% actually embedded as 0.12 and so on)
a. 0.0141*
b. 0.019
c. 0.19
d. 0.31
e. 0.0153
Read the following case and answer the five 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. Figure 1 shows
layout of the streets and the telephone locations (encircled numbers).
Let, xj = 1-if a telephone is installed at location j, 0-if a telephone is not installed at location j
22. The minimum number of constraints (excluding non-negativity) for the most concise IP
formulation of the above problem would be:
a. 8
b. 9
c. 10
d. 11*
e. 19
23. If the optimal solution to the problem is to place telephones at locations 1, 2, 5 and 7,
then how many streets receive more than the minimum required coverage?
a. 8
b. 6
c. 4
d. 2
e. 1*
24. The appropriate objective function for the problem is:
a. Max.. x1 - x8
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*
d. Min. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11
e. Cannot be stated as the distances between locations have not been provided.
25. The constraint x2 + x4 ≥ 1 represents the constraint for:
a. Location 2
b. Location 4
c. Street H
d. Street I *
e. Streets B, I and C
26. The formulation for this problem may be best classified as a,
a. Max. Covering Model
b. Additional Covering Model
c. Set Covering Model*
d. Traveling Salesman Model
e. Minimum Cost Location Analysis Model
The next two questions are based upon the following case:
Consider the picture below which shows a feasible region and the objective function
contour. Answer the next two questions based on this picture.
Max. 3X1 + 5 X2
St. X1 < 4
9 X12 + 5 X22 < 216
Xi > 0 ∀ i
6
4
Feasible Region
2
2
4
27. The optimal solution will likely result in:
a. 1 binding constraint*
b. 2 binding constraints
c. 3 binding constraints
d. no binding constraints
e. the concept of a binding constraint does not exist in such cases.
28. The optimal solution will likely occur:
a.
at a corner point
b.
at two corner points
c.
on the boundary of the feasible region*
d.
in the interior of the feasible region
e.
outside the feasible region
Read the following case and answer the four questions that follow:
Starbucks 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, is shipped in
truck 1, Plantation Z is shipped in truck 2, and Plantation Y is shipped in truck1 and/or truck
2. No more than 2000 pounds of Arabian Mocha Java and 1000 pounds of Decaf Espresso
Roast may be sold. Using the data in the table below, we want to formulate a profitmaximizing non-linear program.
Table
COFFEE
Arabian Mocha Java
Decaf Espresso Roast
SALES PRICE per POUND ($)
28
32
CHICORY CONTENT (%)
NO MORE THAN 5
NO MORE THAN 7.5
COST PER POUND ($)
Plantation X
Plantation Y
Plantation Z
20
26
15
6
3
9
Assume the following:
X = pounds of Plantation X purchased
Y1, Y2 = pounds of Plantation Y purchased and shipped in trucks 1 and 2 respectively
Z = pounds of Plantation Z purchased
TS1, TT1 = pounds of beans from truck 1 blended into Arabian Mocha Java and
Decaf Espresso Roast respectively
TS2, TT2 = 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
29. The objective function can be represented as:
a. Max. 32 (TS1+TS2) + 28(TT1+TT2)
b.Max. 32(TS1+TS2)+ 28(TT1+TT2)– 7.5 TC1 – 5 TC2
c. Max. 28(TS1+TS2)+ 32(TT1+TT2)– 9X – 3(Y1 + Y2) – 6Z
d.Min. 28(TS1+TS2)+ 32(TT1+TT2)– 20X – 26(Y1 + Y2) – 15Z
e. Max. 28(TS1+TS2)+ 32(TT1+TT2)– 20X – 26(Y1 + Y2) – 15Z*
30. The constraint on chicory percentage of beans in Truck 2 is given as:
a. TC2 = (0.09Z + 0.03Y2) / (Z + Y2)*
b. TC2 = 0.09Z + 0.03Y2
c. TC2 = (0.09Z + 0.03Y) / (Z + 2Y)
d. TC2 = 0.12 (Z + Y2) / (Z + Y2)
e. TC2 = (0.09Z + 0.03Y) / (Z + Y)
31. The coffee shipped in Truck 1 satisfies which of the following constraints:
a. Y2 + Z = TS2 + TT2
b. Y + Z = TS1 + TT1
c. Y1 + X = TS2 + TT2
d. Y1 + X = TS1 + TT1*
e. Z + X = TS1 + TT1+Y1
32.
The restriction on chicory content for Decaf Espresso Roast is captured by the
constraint:
a. TT1 TC1 + TT2 TC2 >= 0.075
b. TT1 TC1 + TT2 TC2 <= 0.075 (TT1+TT2)*
c. (TT1 (0.06X + 0.03Y1) / (X + Y1) + TT2 (0.09Z + 0.03Y2) / (Z + Y2)) <= 0.075
d. (TT1 TC1 + TT2 TC2) <= 0.075 X
e. (TT1 (0.06X + 0.03Y1) / (X + Y1) + TT2 (0.09Z + 0.03Y2) / (Z + Y2)) <= 0.05
33. Consider the shaded feasible regions shown below:
Region B
Region A
a.
b.
c.
d.
e.
both regions are convex
Region A is convex and Region B is concave
Region A is non-convex and Region B is convex
Region A is convex and Region B is non-convex*
both regions are non-convex
The next three questions are based on the following case:
Eastborne Realty has $2 million available for the purchase of a new rental property. After an
initial screening, Eastborne reduced the investment alternatives to townhouses and apartment
buildings. Each townhouse can be purchased for $282,000 and five are available. Each
apartment building can be purchased for $400,000 and the developer will construct as many
buildings as Eastborne wants to purchase. Eastborne’s property manager can devote up to
140 hours per month to these new properties; each townhouse is expected to require 4 hours
per month, and each apartment building is expected to require 40 hours per month. The
annual cash flow, after deducting mortgage payments and operating expenses, is estimated to
be $10,000per townhouse and $15,000 per apartment building. Eastborne’s owner would like
to determine the number of townhouses and the number of apartment building to purchase to
maximize annual cash flow.
Let,
T = number of townhouses purchased
A = number of apartment buildings purchased
As seems practical, the optimal number of townhouses and apartment buildings cannot be
fractional quantities. Eastborne’s problem can be solved graphically. Please find below the
LP relaxation solution of the problem along with the IP feasible points.
A
4
Funds Availab
Manager's
Townhouses
Payoff:
10.00
Tim
A
T
3
2
1
0
0
1
2
3
Optimal Decisions(T,A): (2.48, 3.25)
Funds Available ($000) : 282.00T + 400.00A <= 2000.00
Manager's Time (hours) : 4.00T + 40.00A <= 140.00
Townhouses Available : 1.00T + 0.00A <= 5.00
4
5
T
34.
35.
36.
If Eastborne Realty were to simply round off the LP relaxation solution of their IP
model to the nearest feasible integer point, what is the annual cash flow that they
could expect?
a. $62000
b. $70000
c. $76000
d. $65000*
e. $73574
What is the optimal number of townhouses and apartment buildings that Eastborne
Realty should buy in order to maximize their annual cash flow?
a. 2.48 townhouses and 3.25 apartment buildings
b. 2 townhouses and 3 apartment buildings
c. 4 townhouses and 2 apartment buildings*
d. 5 townhouses and no apartment buildings
e. 3.25 townhouses and 2.48 apartment buildings
In order to solve Eastborne Realty’s problem by complete enumeration, we would
have to calculate the objective function value,
a. 1 time
b. 4 times
c. 5 times
d. 19 times
e. 20 times*
Read the following case and answer the three questions following it.
Sandalo Petroleum Refineries, Inc. (SPR) produces petroleum at plants in Vermont a n d
Edward Island. Refined petroleum products are shipped by rail to SPR’s warehouses in
San Diego and New York and then distributed to regional distributors in Connecticut,
Sacramento, Denver and Dallas. The two warehouses can also transfer small
quantities between themselves using company trucks. The task is to plan SPR’s
distribution over the next month. Each plant can ship up to 3300 barrels during this
period and none are presently in Connecticut, Sacramento, Denver and Dallas.
Connecticut, Sacramento, Denver and Dallas Distributors require 2400, 1200, 1400 and
1600 barrels respectively. Transfers between warehouses are limited to 40 Barrels
but no cost is charged. The network flow diagram is given below.
Vermont
1
San Diego
3
5
Connecticut
6
Sacramento
2
4
7
Edward
New York
Denver
8
Dallas
Unit costs (cij) of the flow (Fij) on arc (i, j) are detailed in the following tables:
From/To
1. Vermont
2. Edward
3. SanDiego
13
8
From/To
5. Connecticut
3. SanDiego
8
4. New York
12
4. New York
21
11
6. Sacramento
41
7
7. Denver
3
29
8.Dallas
7
2
37. Which of the following represent the objective function?
a. Min. 13F31+21F41+8F32+11F42+8F53+41F63+3F73+7F83+12F54+7F64+29F74+2F84
b. Min. F13+F14+F23+F24+F35+F36+F37+F38+F45+F46+F47+F48
c. Min.13F13+21F14 +8F23+11F24+8F35+41F36+3F37+7F38+12F45+7F46+29F47+2F48*
d. Max. F13+F14+F23+F24+F35+F36+F37+F38+F45+F46+F47+F48
e. None of the above.
38. The constraint at New York is given as:
a. +F14+24+F34–F43–F45–F46–F47–F48 < 0
b. –F14–F24–F34=F43+F45+F46+F47+F48
c. +F14+F24+F34–F43–F45–F46–F47–F48 > 0
d. –F14–F24–F34+F43+F45+F46+F47+F48=0*
e. Both b and d are equivalent.
39. The constraint at Denver may be written as:
a. –F37–F47=1400
b. +F37+F47= –1400
c. –F37+F47=1400
d. –F37–F47= –1400*
e. – F37– F47=140
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.
Event
Breaststroke Backstroke Butterfly Freestyle
Swimmer
67
73
61
59
Smith
65
72
65
58
Johnson
68
70
69
55
Paul
71
69
71
57
Edwin
DSCI 3870.001
Exam 3
SAMPLE
12/15/2009
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).
40.
The objective function is best represented by:
a. Max: 67 XS1 + 73 XS2 + 61 XS3 + 59 XS4 + 65 XJ1 + 72 XJ2 + 65 XJ3 + 58 XJ4 + 68 XP1
+ 70 XP2 + 69 XP3 + 55 XP4 + 71 XE1 + 69 XE2 + 71 XE3 + 57 XE4
b. Max: XS1 + XS2 + XS3 + XS4 + XJ1 + XJ2 + XJ3 + XJ4 + XP1 + XP2 + XP3 + XP4 + XE1 +
XE2 + XE3 + XE4
c. Min: XS1 + XS2 + XS3 + XS4 + XJ1 + XJ2 + XJ3 + XJ4 + XP1 + XP2 + XP3 + XP4 + XE1 +
XE2 + XE3 + XE4
d. Min: 67 XS1 + 73 XS2 + 61 XS3 + 59 XS4 + 65 XJ1 + 72 XJ2 + 65 XJ3 + 58 XJ4 + 68 XP1
+ 70 XP2 + 69 XP3 + 55 XP4 + 71 XE1 + 69 XE2 + 71 XE3 + 57 XE4*
e. All of the above.
41. The constraint for event ‘Breaststroke’ is given by:
a. 67XS1+65XJ1+68XP1+71XE1 = 1
b. XS1+XJ1+XP1+XE1 < 1
c. XS1+XJ1+XP1+XE1 =1*
d. X11+X21+X31+X41 = 1
e. X11+X12+X13+X14 = 1
42. The constraint for swimmer ‘Paul’ is given by:
a. XP1+XP2+XP3+XP4 < 1
b. X21+X22+X23+X24 = 1
c. X1P+X2P+X3P+X4P = -1
d. XP1+XP2+XP3+XP4 =1*
e. both a. & c. are valid constraints
43. Using the Greedy Heuristic discussed in class, what would be the assignment of
swimmers to events?
a. S → 1, J → 3, P → 2, E → 4
b. S → 3, J → 1, P → 4, E → 2*
c. S → 3, J → 1, P → 2, E → 4
d. S → 1, J → 2, P → 3, E → 4
e. S → 2, J → 1, P → 4, E → 3
44. If you were to solve this assignment problem by complete enumeration, then you
would have to compare a total of,
a. 6 feasible solutions
b. 8 feasible solutions
c. 16 feasible solutions
d. 24 feasible solutions*
e. 36 feasible solutions
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