Homework 2 Solutions (1)
pdf
School
University of Wisconsin, Madison *
*We aren’t endorsed by this school
Course
345
Subject
Mechanical Engineering
Date
Dec 6, 2023
Type
Pages
13
Uploaded by ProfessorRose11914
Homework 2 Solutions
July 8, 2022
1
Homework #2 Solutions
Due July 5 @ 11:59pm
1.1
Submission requirements
Upload a
single PDF or HTML file
of your IJulia notebook for this entire assigment.
Clearly
denote which question each section of your file corresponds to.
1.2
Problem 1 – On your mark, get set, bake!
A bakery mass produces 4 different types of cake: carrot, chocolate, vanilla, and angel-food. The
bakery measures cake characteristics using two different features: sweetness (
S
) and density (
D
).
The following table summarizes the characteristics of each type of cake as well as the (max) num-
ber of each type that are baked each month:
Cake
S
D
Max baked
Carrot
100
15
3810
Chocolate
90
18
2700
Vanilla
86
6
4000
Angel food
110
3
1500
The cakes can either be sold as-is at $14 a cake or combined into a “party platter” for catering.
There are two party platter options –
A
and
B
– that have requirements for average sweetness and
density. No more than 7 cakes can be added to any individual platter. The table below summarizes
these requirements and gives the selling price per platter:
Platter
S
D
Max cakes
Price
A
at least 98
at most 10
7
$20
B
at least 100
at most 8
7
$30
The
S
and
D
values for the party platters are calculated as the weighted averages of the
S
and
D
of the cakes added to the platter.
(a) Formulate a linear program that will yield the largest possible revenue for the bakery. As-
sume it is okay to sell fractional cakes, since we are planning in aggregate.
1
1.2.1
Solution
Let
C
be the set of cake types and
P
be the set of party platter types.
Decision variables
•
x
i
: number of each cake of type
i
∈
C
that are sold as-is
•
y
ij
: number of each cake of type
i
∈
C
that is used in party platter type
j
∈
P
Objective
max 14
∑
i
∈
C
x
i
+
20
∑
i
∈
C
y
iA
+
30
∑
i
∈
C
y
iB
Constraints
• Do not exceed total number of each cake type produced
x
_
{
car
}
+
y
_
{
car
,
A
}
+
y
_
{
car
,
B
} ≤
3810
x
_
{
choc
}
+
y
_
{
choc
,
A
}
+
y
_
{
choc
,
B
} ≤
2700
x
_
{
van
}
+
y
_
{
van
,
A
}
+
y
_
{
van
,
B
} ≤
4000
x
_
{
ang
}
+
y
_
{
ang
,
A
}
+
y
_
{
ang
,
B
} ≤
1500
• Max number of cakes on each party platter
∑
i
∈
Cy
iA
≤
7;
∑
i
∈
Cy
iB
≤
7
• Minimum sweetness on party platters
100
y
_
{
car
,
A
}
+
90
y
_
{
choc
,
A
}
+
86
y
_
{
van
,
A
}
+
110
y
_
{
ang
,
A
} ≥
98
∑
i
∈
Cy
iA
100
y
_
{
car
,
B
}
+
90
y
_
{
choc
,
B
}
+
86
y
_
{
van
,
B
}
+
110
y
_
{
ang
,
B
} ≥
100
∑
i
∈
Cy
iA
• Maximum density on party platters
15
y
_
{
car
,
A
}
+
18
y
_
{
choc
,
A
}
+
6
y
_
{
van
,
A
}
+
3
y
_
{
ang
,
A
} ≤
10
∑
i
∈
Cy
iA
15
y
_
{
car
,
B
}
+
18
y
_
{
choc
,
B
}
+
6
y
_
{
van
,
B
}
+
3
y
_
{
ang
,
B
} ≤
8
∑
i
∈
Cy
iA
(b) Build and solve the problem in Julia. What are the optimalnumbers of each type of cake and
party platter that the bakery should sell?
2
,
→
,
→
,
→
3
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
1.3
Problem 2 – Multi-period Planning
A company who works with car manufacturers produces and sells engines of various types and
sizes. There is one particular style of engine that comes in three different sizes (call them “size 1,”
“size 2,” and “size 3”). The contractor processes raw materials to procude the engines. Processing
a pallet of raw materials costs $700 per pallet and produces 2 size 1 engines, 1 size 2 engine, and 1
size 3 engine. Size 1 engines sell for $160 each, size 2 engines sell for $210 each, and size 3 engines
sell for $300 each. Engines (of any size) can be stored from month to month. Keeping engines in
good working condition costs $5
×
(engine size) per engine per month (i.e.,$5 for size 1, $10 for size
2, $15 for size 3).
The table below shows the maximum amount of each engine the contractor can sell in the next
three months as well as the maximum number of pallets of raw materials it can process in each
month. Demand that is not met in the month it occurs cannot be carried over to a later month (i.e.,
no backlogging is allowed – so the demand amounts in the table represent the maximum that can
be sell – it may not be possible to sell this much). The contractor currently has 10 size 1 engines
and 2 size 3 engines (no size 2 engines) in stock, and wishes to have at least that much – plus at
least one size 2 engine – in stock at the end of month 3.
Month (
t
)
Max raw
materials (RM
t
)
Max size 1 (M1
t
)
Max size 2 (M2
t
)
Max size 3 (M3
t
)
1
15
30
12
15
2
20
15
18
17
3
12
20
13
21
Formulate a linear programming model to help the contractor maximize its profits over the next
three months. Solve the model in Julia/JuMP and display the production plan (how many of each
engine size to produce, store, and sell in each month).
,
→
4
,
→
,
→
,
→
,
→
,
→
5
,
→
,
→
6
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
(b) Now suppose the contractor would like to allow backlogging of demand. In other words,
not all demand must be met in the month it is generated. The contractor pays a fee of $19
per engine per month backlogged, regardless of engine size. Modify your code from part (a)
to include the ability to backlog. Not all demand must be met, but the same ending stock
requirements should be enforced in part (b) and inventory should be nonnegative in the final
month. How, if at all, does this change the optimal objective value and solution? Speculate
as to the reason for the change (or lack of change).
,
→
,
→
7
,
→
,
→
,
→
,
→
,
→
,
→
,
→
,
→
8
9
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
1.4
Problem 3 – Timetabling
The College of Engineering has 5 different levels of classes (e.g., intro level, PhD level, . . . )
it
plans to offer next year in the Summer 2023 session. For simplicity, the class levels are labeled
A
,
B
,
C
,
D
, and
E
. There are 5 different types of professors (based on area of expertise and type of
professorship) who can teach classes next summer:
I
1,
I
2,
I
3,
I
4, and
I
5. Each individual instructor
can teach one course. Professors of type
I
1 can teach any of the 5 levels. Professors of type
I
2 can
teach levels
A
,
C
, and
D
. Professors of type
I
3 can teach levels
B
,
C
, and
E
. Professors of type
I
4
can teach levels
A
and
D
. Finally, Professors of type
I
5 are highly specialized and can only teach
level
E
.
The number of classes at each level that are scheduled to be taught next year are given in the table
below:
Level
A
B
C
D
E
# needed
5
8
12
11
4
The cost of hiring a professor to teach a class and the available number of each type of professor
are listed in the table below (the cost does not depend on the level of class taught - only on the type
of professor). It is not required to use all professors – these number available quantities represent
the maximum of each type that can be used.
Prof. type (
i
)
I
1
I
2
I
3
I
4
I
5
Cost of hiring
$385
$340
$300
$270
$250
# available (
M
i
)
20
7
10
11
3
Formulate a linear program that will help determine how many professors of each type should be
hired to teach the required courses next year at a minimum cost. Solve your model in Julia and
report the solution (make sure you interpret the meaning of your solution). What type of MCNF
problem is this?
1.5
Solution
This is a transportation problem:
,
→
,
→
,
→
,
→
10
,
→
,
→
1.6
Problem 4 – A tricky one
We’re going to model a math puzzle as a shortest path problem. Start with the number 1. At
each step, you can either triple the current number or add one to the current number. What is
the minimal number of operations (steps) needed to get from 1 to 2020? There are a few different
11
ways to solve this problem. For full credit, build an optimization model structured as a shortest
path problem.
,
→
12
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
13
Related Documents
Related Questions
In a hypothetical scenario respond to the following email to reasuure the owner and in an informative and professional way.
Dear Project Manager-
I am so excited to have broken ground on the new mixed use space. As I watch the underground utilities such as water, storm drain, and sewer get installed I find myself worrying about failures. What steps are you taking to ensure those systems will not have to be dug up and repaired later in the construction process? We cannot afford a delay of any kind. Could you please explain how you plan to prevent any delays and ensure that the systems are functional and leak free.
The other concern that has come up that I would like to address now is the comfort of the office staff. Our current building is not comfortable. I am wondering why. We have the temperature appropriately set and it is easily maintained, but people still complain. What can we do to ensure they will be comfortable in the new building? A few months ago I asked one of our…
arrow_forward
13. Here is a list of activity times for a project as well as crashing costs for its activities. Determine
which activities should be crashed and the total cost of crashing if the goal is to shorten the proj-
ect by three weeks as cheaply as possible. First construct a network diagram. You can use either
an AOA or an AON (see Solved Problem 5).
Path
Activity
Duration (weeks)
First Crash
Second Crash
Тор
A
5
$8
$10
9
C
3
14
15
Middle
3
11
7
9
3
14
15
Bottom
5
10
15
G
5
11
13
H
5
12
14
arrow_forward
Please write solution on paper, thank you. No typing, it's hard to follow. I''l give positive feedback.
arrow_forward
Please solve, engineering econ
arrow_forward
You are a biomedical engineer working for a small orthopaedic firm that fabricates rectangular shaped fracture
fixation plates from titanium alloy (model = "Ti Fix-It") materials. A recent clinical report documents some problems with the plates
implanted into fractured limbs. Specifically, some plates have become permanently bent while patients are in rehab and doing partial
weight bearing activities.
Your boss asks you to review the technical report that was generated by the previous test engineer (whose job you now have!) and used to
verify the design. The brief report states the following... "Ti Fix-It plates were manufactured from Ti-6Al-4V (grade 5) and machined into
solid 150 mm long beams with a 4 mm thick and 15 mm wide cross section. Each Ti Fix-It plate was loaded in equilibrium in a 4-point bending
test (set-up configuration is provided in drawing below), with an applied load of 1000N. The maximum stress in this set-up was less than the
yield stress for the Ti-6Al-4V…
arrow_forward
Hartley Electronics, Inc., in Nashville, producesshort runs of custom airwave scanners for the defense industry.The owner, Janet Hartley, has asked you to reduce inventory byintroducing a kanban system. After several hours of analysis, youdevelop the following data for scanner connectors used in onework cell. How many kanbans do you need for this connector?Daily demand 1,000 connectorsLead time 2 daysSafety stock 12 dayKanban size 500 connectors
arrow_forward
A VẬT LÝ ĐẠI CƯƠ X
A TongHopTailieul x
A DaoHam.pdf - Go X
A Chương 1 - Good x
The position vec
My Tutoring | ba x
QUIZ 1: MOTION X
f (2) Facebook
O X
Slide 1
O YouTube
(66) YouTube
+ → C D
A Không bảo mật e-learning.hcmut.edu.vn/pluginfile.php?file=%2F1499051%2Fmod_resource%2Fcontent%2F8%2FChapter%201-%20Kinematics.pdf
A YouTube to MP3 C.
M Hộp thư đến (41) -...
Khóa học: Soft Skill.
O Audio Cambridge 1.
A TẠI LIEU WRITING.
t Link tải Cam 1- 15..
iot IELTS Simulation Te.
A (HV] Chương trình...
A BTC OISP GALA 202..
A GĐ2: OISP YOUTH...
A Tổng hợp đề writin.
A REPORT DE THI IEL.
>>
QUIZ 1: MOTION IN TWO AND THREE DIMENSIONS
1 / 5
195%
in meters and t is in seconds. At t = 3.00 s, find (a) the position of the particle, (b) its velocity, and
(c) its acceleration.
1.2. A motorist drives south at 20.0 m/s for 3.00 min, then turns west and travels at 25.0 m/s for 2.00
min, and finally travels northwest at 30.0 m/s for 1.00 min. For this 6.00-min trip, find (a) the total
vector displacement,…
arrow_forward
.com question/22816263
rch for an answer to any question...
it works
For parents For teachers
Honor code
16 nours agO• Chemistry Hign Scnoo1
You are currently in a stable orbit 9000 km
(1km=1000m) from the center of planet Proxima
Centauri B.
According to the calculations done by NASA, Planet Proxima Centauri B has
a mass of approximately 7.6x10^24kg.
Based on fuel consumption and expected material aboard the ship, your
colony ship lander has a mass of approximately 1.1x10^5 kg.
1. Using Newton's Law of Universal Gravitation, determine what the force of
gravity on your ship at that orbit be using the early measurements
2.Using Newton's second law determine the acceleration of your lander due
to gravity
3.Upon arrival, you measure the actual force of gravity to be 6x10^5 N,which
ic difforent from vourcalclated vauein cuection 1 ldentify 2cnocific
e to search
arrow_forward
No gpt,solve in paper sheet.
Free body diagram is must and should.
arrow_forward
You are an engineer in a company that manufactures and designs several mechanical devices, and your manager asked you to help your customers. In this time, you have two customers, one of them wants to ask about internal combustion engines while the other requires a heat exchanger with particular specifications. Follow the parts in the following tasks to do your job and support your customers.Task 1:Your first customer asked for an internal combustion engine to use it in a designed car. Your role is to describe the operation sequence of different types of available engines, explain their mechanical efficiency, and deliver a detailed technical report which includes the following steps:STEP 1Describe with the aid of diagrams the operational sequence of four stroke spark ignition and four stroke compression ignition engines.STEP 2Explain and compare the mechanical efficiency of two and four-stroke engines.STEP 3Review the efficiency of ideal heat engines operating on the Otto and Diesel…
arrow_forward
XYZ Company Ltd has consulted your firm Zig-Zag Energy over an intention to generate electricity in Ghana. Being the senior project engineer for Zig-Zag Energy, management has asked you to prepare a detailed technical report in respect to the project.
a) Discuss the major factors to take into consideration when consulting on such projects.
A tank containing air is stirred by a paddle wheel.
The work input to the paddle wheels is 2000KJ and the heat transferred to the surrounding from the tank is 6000KJ.
b) Determine the change in the internal energy of the system
There has been a sharp differences between the board of Blue-Blue Energy over which of the energy sources (hydroelectric power plant-HEP and Coal fired plant) is best for power generation. As an energy expert, you were confronted by the debating faction for your opinion. You…
arrow_forward
Compute efficiency and Balance Delay
arrow_forward
Problem 1: You are working in a consulting company that does a lot of hand calculations for designs in
Aerospace Industry for mechanical, thermal, and fluidic systems. You took the Virtual engineering
course, and you want to convince your boss and the team you work to move to modelling and simulation
in computers using a certain software (Ansys, Abaqus, etc). Discuss the benefits and pitfalls of computer
based models used within an industrial environment to solve problems in engineering.
arrow_forward
SEE MORE QUESTIONS
Recommended textbooks for you

Principles of Heat Transfer (Activate Learning wi...
Mechanical Engineering
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Cengage Learning
Related Questions
- In a hypothetical scenario respond to the following email to reasuure the owner and in an informative and professional way. Dear Project Manager- I am so excited to have broken ground on the new mixed use space. As I watch the underground utilities such as water, storm drain, and sewer get installed I find myself worrying about failures. What steps are you taking to ensure those systems will not have to be dug up and repaired later in the construction process? We cannot afford a delay of any kind. Could you please explain how you plan to prevent any delays and ensure that the systems are functional and leak free. The other concern that has come up that I would like to address now is the comfort of the office staff. Our current building is not comfortable. I am wondering why. We have the temperature appropriately set and it is easily maintained, but people still complain. What can we do to ensure they will be comfortable in the new building? A few months ago I asked one of our…arrow_forward13. Here is a list of activity times for a project as well as crashing costs for its activities. Determine which activities should be crashed and the total cost of crashing if the goal is to shorten the proj- ect by three weeks as cheaply as possible. First construct a network diagram. You can use either an AOA or an AON (see Solved Problem 5). Path Activity Duration (weeks) First Crash Second Crash Тор A 5 $8 $10 9 C 3 14 15 Middle 3 11 7 9 3 14 15 Bottom 5 10 15 G 5 11 13 H 5 12 14arrow_forwardPlease write solution on paper, thank you. No typing, it's hard to follow. I''l give positive feedback.arrow_forward
- Please solve, engineering econarrow_forwardYou are a biomedical engineer working for a small orthopaedic firm that fabricates rectangular shaped fracture fixation plates from titanium alloy (model = "Ti Fix-It") materials. A recent clinical report documents some problems with the plates implanted into fractured limbs. Specifically, some plates have become permanently bent while patients are in rehab and doing partial weight bearing activities. Your boss asks you to review the technical report that was generated by the previous test engineer (whose job you now have!) and used to verify the design. The brief report states the following... "Ti Fix-It plates were manufactured from Ti-6Al-4V (grade 5) and machined into solid 150 mm long beams with a 4 mm thick and 15 mm wide cross section. Each Ti Fix-It plate was loaded in equilibrium in a 4-point bending test (set-up configuration is provided in drawing below), with an applied load of 1000N. The maximum stress in this set-up was less than the yield stress for the Ti-6Al-4V…arrow_forwardHartley Electronics, Inc., in Nashville, producesshort runs of custom airwave scanners for the defense industry.The owner, Janet Hartley, has asked you to reduce inventory byintroducing a kanban system. After several hours of analysis, youdevelop the following data for scanner connectors used in onework cell. How many kanbans do you need for this connector?Daily demand 1,000 connectorsLead time 2 daysSafety stock 12 dayKanban size 500 connectorsarrow_forward
- A VẬT LÝ ĐẠI CƯƠ X A TongHopTailieul x A DaoHam.pdf - Go X A Chương 1 - Good x The position vec My Tutoring | ba x QUIZ 1: MOTION X f (2) Facebook O X Slide 1 O YouTube (66) YouTube + → C D A Không bảo mật e-learning.hcmut.edu.vn/pluginfile.php?file=%2F1499051%2Fmod_resource%2Fcontent%2F8%2FChapter%201-%20Kinematics.pdf A YouTube to MP3 C. M Hộp thư đến (41) -... Khóa học: Soft Skill. O Audio Cambridge 1. A TẠI LIEU WRITING. t Link tải Cam 1- 15.. iot IELTS Simulation Te. A (HV] Chương trình... A BTC OISP GALA 202.. A GĐ2: OISP YOUTH... A Tổng hợp đề writin. A REPORT DE THI IEL. >> QUIZ 1: MOTION IN TWO AND THREE DIMENSIONS 1 / 5 195% in meters and t is in seconds. At t = 3.00 s, find (a) the position of the particle, (b) its velocity, and (c) its acceleration. 1.2. A motorist drives south at 20.0 m/s for 3.00 min, then turns west and travels at 25.0 m/s for 2.00 min, and finally travels northwest at 30.0 m/s for 1.00 min. For this 6.00-min trip, find (a) the total vector displacement,…arrow_forward.com question/22816263 rch for an answer to any question... it works For parents For teachers Honor code 16 nours agO• Chemistry Hign Scnoo1 You are currently in a stable orbit 9000 km (1km=1000m) from the center of planet Proxima Centauri B. According to the calculations done by NASA, Planet Proxima Centauri B has a mass of approximately 7.6x10^24kg. Based on fuel consumption and expected material aboard the ship, your colony ship lander has a mass of approximately 1.1x10^5 kg. 1. Using Newton's Law of Universal Gravitation, determine what the force of gravity on your ship at that orbit be using the early measurements 2.Using Newton's second law determine the acceleration of your lander due to gravity 3.Upon arrival, you measure the actual force of gravity to be 6x10^5 N,which ic difforent from vourcalclated vauein cuection 1 ldentify 2cnocific e to searcharrow_forwardNo gpt,solve in paper sheet. Free body diagram is must and should.arrow_forward
- You are an engineer in a company that manufactures and designs several mechanical devices, and your manager asked you to help your customers. In this time, you have two customers, one of them wants to ask about internal combustion engines while the other requires a heat exchanger with particular specifications. Follow the parts in the following tasks to do your job and support your customers.Task 1:Your first customer asked for an internal combustion engine to use it in a designed car. Your role is to describe the operation sequence of different types of available engines, explain their mechanical efficiency, and deliver a detailed technical report which includes the following steps:STEP 1Describe with the aid of diagrams the operational sequence of four stroke spark ignition and four stroke compression ignition engines.STEP 2Explain and compare the mechanical efficiency of two and four-stroke engines.STEP 3Review the efficiency of ideal heat engines operating on the Otto and Diesel…arrow_forwardXYZ Company Ltd has consulted your firm Zig-Zag Energy over an intention to generate electricity in Ghana. Being the senior project engineer for Zig-Zag Energy, management has asked you to prepare a detailed technical report in respect to the project. a) Discuss the major factors to take into consideration when consulting on such projects. A tank containing air is stirred by a paddle wheel. The work input to the paddle wheels is 2000KJ and the heat transferred to the surrounding from the tank is 6000KJ. b) Determine the change in the internal energy of the system There has been a sharp differences between the board of Blue-Blue Energy over which of the energy sources (hydroelectric power plant-HEP and Coal fired plant) is best for power generation. As an energy expert, you were confronted by the debating faction for your opinion. You…arrow_forwardCompute efficiency and Balance Delayarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning

Principles of Heat Transfer (Activate Learning wi...
Mechanical Engineering
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Cengage Learning