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School
University of Wisconsin, Madison *
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Course
323
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
12
Uploaded by ChancellorCaterpillarMaster871
NAME:
ISyE 323 – Quiz #1 –
Ted Lasso
October 6, 2021
8:00AM-9:15AM
“
Taking on a challenge is a lot like riding a horse, isn’t it? If
you’re comfortable while you’re doing it, you’re probably doing
it wrong.
” – Ted Lasso
READ THIS!
1.
Write your name at the top of this page, and on your blue book if you use one!
2. This exam has a total of 12 pages.
3. The exam is closed book, and you are allowed to use one side of one sheet of notes.
4.
Write your name on your note sheet and turn it in with your exam.
5. Please silence and put cell phones away.
6. You may choose to write some of your answers to problems 4-6 in a blue book.
If you do, you
must clearly indicate so in the exam, and clearly label each problem (including subpart,
e.g., 4.1) in your blue book.
7. The more clearly you write your answer, the better the chance that I can grade it accurately and
give it full credit. In particular, you may write down any assumptions you make in formulating
your problems.
8. For any true/false or multiple choice question that do not ask for an explanation, you may still
optionally
add an explanation. A correct answer will receive full credit with no explanation. An
incorrect answer
might
receive partial credit, depending on the explanation. (I recommend adding
optional explanations only if you have extra time.)
9. The number of points for each problem is displayed at the end of the problem’s title. The time
required for Zach to complete each question is also listed. Please use your time wisely. We will
collect all the papers
promptly
at 9:15. No extra time.
10.
Good luck!
Don’t panic. We’re rooting for you.
Problem
Points
Zach Time
(min.)
1
16
2
2
9
3
3
15
7
4
18
13
5
22
13
6
20
14
Total:
100
52
Fall 2021, ISyE323 Quiz #1
1
True or false (16 total points)
“
I always thought tea was going to taste like hot brown water. And do you know what? I was right.
” – Ted Lasso
Circle true or false.
1.1 Problem (2 points)
True or False:
The goal of solving a linear programming problem is to determine values of the decision
variables that yield the best objective function value while satisfying all the constraints.
1.2 Problem (2 points)
True or False:
It is always possible to solve a linear program by selecting an extreme point that has the
best objective function value.
1.3 Problem (2 points)
True or False:
It is possible for a linear programming problem to have exactly two di erent optimal
solutions.
1.4 Problem (2 points)
True or False:
If a linear program is unbounded, then it must have an unbounded feasible region.
1.5 Problem (2 points)
True or False:
The
rst step in an operations research project is to formulate a mathematical model of
the problem.
1.6 Problem (2 points)
True or False:
It is possible for a convex set to have in nitely many extreme points.
1.7 Problem (2 points)
True or False:
It is possible for a linear program to be both unbounded and infeasible.
1.8 Problem (2 points)
True or False:
Let
S
be the set of points
(
x
1
, x
2
)
that satisfy the constraints
x
1
and
x
2
are integer-valued,
x
1
≥
0
,
x
2
≥
0
, and
2
x
1
+ 3
x
2
≤
6
. The set
S
is a convex set.
Page 2
Fall 2021, ISyE323 Quiz #1
2
Multiple choice and short answer (9 total points)
2.1 Problem (3 points)
What is the di erence between a parameter and a decision variable?
2.2 Problem (2 points)
If
x
1
and
x
2
are decision variables and
a
and
b
are parameters, which of the assumptions of linear
programming does the constraint constraint
ax
1
+
(
a
+
b
)
x
2
b
≤
b
2
violate? (Circle all that apply).
(a) Proportionality
(b) Additivity
(c) Divisibility
(d) Certainty
(e) None
2.3 Problem (2 points)
x
1
and
x
2
are decision variables representing the pounds of raw material
1
and
2
to buy.
If no raw
materials are purchased, there is no cost. If a positive amount of raw material
1
is ordered, there is
xed
charge of $100 for delivery, plus $2 per pound. If a positive amount of raw material 2 is ordered, there
is a
xed charge of $200 for delivery cost, plus $3 per pound. If both are ordered, the total cost is the
sum of the costs of the individual materials. Which, if any, of the assumptions of linear programming
is violated by the function which gives the cost of purchasing raw materials as a function of
x
1
and
x
2
?
(Circle all that apply.)
(a) Proportionality
(b) Additivity
(c) Divisibility
(d) Certainty
(e) None
2.4 Problem (2 points)
If you attempt to solve a linear program and
nd that it is unbounded, what does this usually mean about
your linear program?
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Fall 2021, ISyE323 Quiz #1
3
Graphical solution of a linear program (15 total points)
"
Does my face look like it’s in the mood for
shape-based jokes?
" - Roy Kent
Consider the two-variable linear program
below.
min 2
x
1
+
x
2
subject to:
x
1
+
x
2
≥
3
-
x
1
+
x
2
≥
0
x
2
≥
2
x
1
, x
2
≥
0
0
1
2
3
4
5
0
1
2
3
4
5
A
B
C
D
E
F
G
x
1
x
2
3.1 Problem (4 points)
Shade the feasible region of this linear program in the plot above.
3.2 Problem (3 points)
Solve this problem using the graphical method.
Circle which (if any) of the points below are optimal
solutions.
A: (0,3)
B: (0,2)
C: (1,2)
D: (1.5,1.5)
E: (3,0)
F: (2,2)
G: (4,4)
None
3.3 Problem (3 points)
Now suppose the objective function is changed to
maximize
2
x
1
+
x
2
(instead of minimize). Circle which
(if any) of the points below are optimal solutions.
A: (0,3)
B: (0,2)
C: (1,2)
D: (1.5,1.5)
E: (3,0)
F: (2,2)
G: (4,4)
None
3.4 Problem (3 points)
Now suppose the objective function is changed to maximize
x
1
-
x
2
. Circle which (if any) of the points
below are optimal solutions.
A: (0,3)
B: (0,2)
C: (1,2)
D: (1.5,1.5)
E: (3,0)
F: (2,2)
G: (4,4)
None
Page 4
Fall 2021, ISyE323 Quiz #1
3.5 Problem (2 points)
Write down a linear inequality which would make this linear program
infeasible
if it were added to the
problem.
Page 5
Fall 2021, ISyE323 Quiz #1
4
The Simple Life for Rebecca? (18 Total points)
“
I lost my way for a minute, but I’m on the road back.
” – Rebecca Welton
Rebecca decided to buy a hobby farm as a distraction from the stress of owning a football club. Of
course she wants her farm to be pro table, so she has asked for your help to manage it. Her farm has
120 acres of land on which she can plant three di erent crops: hay, corn, and oats. Rebecca has at most
$250 thousand available for purchasing seeds now, and over the growing season can purchase at most
110 tons of fertilizer (the $250 thousand limit does
not
apply to fertilizer cost). A ton of fertilizer costs $2
thousand. The table below gives the cost for seeds per acre, the required number of tons of fertilizer per
acre, and the net revenue per acre for each type of crop. It is OK with Rebecca if some of the land does
not have crops planted on it, if that is what is most pro table.
Hay
Corn
Oats
Required fertilizer (tons per acre)
0.5
1.5
1
Seed cost ($ thousand per acre )
2
2.5
3
Revenue ($ thousand per acre)
5
10
9
In the following questions you will formulate a linear program to help Rebecca choose how much of each
crop to plant this year in order to maximize total pro t (revenue less fertilizer and seed costs). Use the
following decision variables in your model:
•
x
h
: acres of hay to plant
•
x
c
: acres of corn to plant
•
x
o
: acres of oats to plant
4.1 Problem (2 points)
What is the objective of the linear program?
4.2 Problem (6 points)
What are the constraints?
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Fall 2021, ISyE323 Quiz #1
4.3 Problem (4 points)
Ted Lasso informs Rebecca that in order to keep her land healthy, at least 40% of the land that is planted
with crops should be planted with hay. Write a constraint (or constraints) to model this.
4.4 Problem (4 points)
Rebecca wishes to have her farm certi ed as “environmentally friendly” (EF). To be EF certi ed, the CO2
emissions per acre, averaged over land that is planted with any of the crops, must be at most 250 pounds
per acre. The CO2 emissions per acre of the three crops are given in the table below.
Hay
Corn
Oats
CO2 Emissions (pounds per acre)
275
300
150
Write a constraint (or constraints) that would ensure Rebecca’s farm is certi ed EF this year.
4.5 Problem (2 points)
Is the certainty assumption reasonable in this problem? Circle yes or no
and
brie y explain your answer.
(a) Yes.
(b) No.
Explanation.
Page 7
Fall 2021, ISyE323 Quiz #1
5
Fan Appreciation Day (22 Total Points)
Ted:
If I were to get
red from my job where I’m putting cleats in the trunk of my car
Coach Beard:
You got the boot from puttin’ boots in the boot
Ted Lasso has decided to host a fan appreciation day for the FC Richmand fans.
The event will
last for 5 hours, from 10AM – 3PM. Ted has decided that each fan’s experience will last 3 consecutive
hours, during which one of the hours will be spent eating a meal, and the other two hours will be spent
doing activities with the team. The hour spent eating a meal must be either the
rst or the last hour
of the experience (i.e., a fan could eat a meal for one hour, then do activities for two hours, or they
can do activities for two hours and then eat a meal for an hour). Unfortunately, due to limited sta
ng,
the number of fans who can be eating a meal or doing activities during each hour cannot exceed the
amounts given in the table below:
Hour index :
1
2
3
4
5
Hour time:
10-11AM
11AM-noon
noon-1PM
1-2PM
2-3PM
Maximum number eating a meal
275
295
320
230
200
Maximum number doing activities
170
160
175
155
140
In this problem, you will help Ted schedule fan experiences for the fan appreciation day in order to
maximize the number of fans who can participate.
5.1 Problem (6 points)
Write the decision variables of your model here.
5.2 Problem (2 points)
Write the objective of your model here.
Page 8
Fall 2021, ISyE323 Quiz #1
5.3 Problem (8 points)
Write the constraints of your model here.
5.4 Problem (2 points)
Is the divisibility assumption reasonable for this model? Answer Yes or No, and explain.
5.5 Problem (4 points)
Nate points out that they should also be careful that they do not run out of chips for the meals. They
have 150 pounds of chips available for the event. The amount a fan eats during their meal depends on
what time they are assigned to eat, as given in the table below:
Hour index :
1
2
3
4
5
Hour time:
10-11AM
11AM-noon
noon-1PM
1-2PM
2-3PM
Pounds of chips eaten/fan
0.5
0.8
0.9
0.9
0.6
Extend the model to make sure they do not use more chips than they have available.
Page 9
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Fall 2021, ISyE323 Quiz #1
6
Marketing Campaign (20 Total Points)
“
I never know how to react when a grown man beatboxes in front of me.
” – Keeley
Rebecca has asked Keeley to conduct a marketing campaign to promote the team. Keeley can spend
money to place ads on a set of media platforms
P
(e.g., television, twitter, etc.) with the goal of showing
ads to people in a set
G
of di erent groups. Each dollar spent on advertising in platform
p
∈
P
leads
to
v
pg
viewers in group
g
∈
G
seeing an ad.
Keeley has
B
total dollars to spend on the marketing
campaign, and can spend at most
u
p
dollars for each individual platform
p
∈
P
. For each group
g
∈
G
Keeley wishes to reach at least the target
b
g
viewers for that group, but unfortunately realizes this may
not be possible given the budget. Thus, Keeley wishes to minimize the total “shortfall” from these targets.
(E.g., if the target for a group
g
is
b
g
= 500
viewers, and the number of viewers for that group due to
the advertising campaign is
350
, then this contributes 500-350 to the shortfall. On the other hand, if
the number of viewers for that group is
500
or any higher number, then this contributes 0 to the total
shortfall.) Formulate a linear program to help Keeley plan the marketing campaign. As a hint to get you
started, your solution should use the following decision variables:
•
y
g
: shortfall of the target viewership for group
g
∈
G
The last page of this quiz gives example data for this problem. If you are stuck writing a general model,
you can write a model for that example data to receive partial credit, but you otherwise do not need to
read that page.
6.1 Problem (4 points)
De ne the additional decision variables needed for this model.
6.2 Problem (2 points)
Write the objective of the model here.
Page 10
Fall 2021, ISyE323 Quiz #1
6.3 Problem (8 points)
Write the constraints of the model here.
6.4 Problem (4 points)
A subset
S
⊆
P
of the media platforms are social media and a subset
N
⊆
P
of the media platforms are
newspapers. In order to keep Trent Crimm (from the newspaper
The Independent
) from writing a nasty
story about the team, Keeley has determined that the total amount spent on newspaper platforms should
be at least 50% more than the total amount spent on social media. Extend the model to consider this
restriction.
6.5 Problem (2 points)
Brie y discuss how the certainty assumption might be violated in this problem.
Page 11
Fall 2021, ISyE323 Quiz #1
Here we give example data for problem 6.
It is not necessary to read this unless you are stuck
with writing the general model as asked for in that problem.
The example data has
P
=
{
Television
,
Twitter
,
Facebook
,
TheIndependent
}
as the set of media
platforms and
G
=
{
Men20
-
29
,
Women20
-
29
,
Men30
-
39
,
Women30
-
39
}
as the set of groups.
The available budget is
B
= 2000
dollars. The table below shows the viewership of each group reached
per dollar spent on each media platform (
v
pg
), the target for each group
g
∈
G
(
b
g
) in thousands of
people, and the maximum that can be spent on each platform.
Viewership (
v
pg
)
Platforms
p
∈
P
TV
Twitter
Facebook
The Indep.
Target (
b
g
)
Men20-29
7.5
6
5
1
5000
Group (
g
∈
G
)
Women20-29
4
7
5.5
2.5
4000
Men30-39
6.5
5
7
4
6000
Women30-39
3
8
6
4.5
6000
Maximum spend (
u
p
)
1400
1500
1200
800
In this example, the social media platforms are
S
=
{
Twitter
,
Facebook
}
, and the newspapre
platforms are the set
N
=
{
TheIndependent
}
.
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