HW3 (1)
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University of Michigan *
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Course
310
Subject
Industrial Engineering
Date
Apr 3, 2024
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3
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IOE 310 – Optimization and Computational Methods (Fall 2023)
Homework
№
3
IOE
: H
№
Due:
September 20th (by 11:59PM ET)
Homework Information:
•
Homework 3 is due
Wednesday September 20, 2023 by 11:59PM ET
.
•
Homework 3 is out of
100 points
.
•
No late assignments will be accepted
.
•
Homework submissions
must
be uploaded to Canvas.
•
Homework submissions
must
comply with the following naming convention. Please create a pdf
file named
LastName_FirstName_IOE310_HW3.pdf
with your solutions.
Note, points may be
deducted is a homework submission does not observe with the naming convention.
•
Students may work together, but should each write up their own solution.
Solutions that are
identical will be treated as an Honor Code violation. See course syllabus for more information.
IOE Department – University of Michigan
Page 1 / 3
IOE 310 – Optimization and Computational Methods (Fall 2023)
Homework
№
3
Problem 1
(5 points)
What are you passionate about?
Problem 2
(5 points) (You will have one question like this in
almost
every homework assignment to ensure that
everyone fully reads the Syllabus.)
What programming language will we be using in this course?
Have you used this language before?
Problem 3
(10 points)
Consider the following optimization problem in standard form:
min
x
3
x
1
-
4
x
2
+ 7
x
3
s.t.
x
1
+ 2
x
2
-
3
x
3
= 7
x
1
+
x
3
= 4
x
1
, x
2
, x
3
≥
0
.
Write out the vector
c
, the vector
b
, the vector
x
vector, and the matrix
A
.
Problem 4
(20 points)
Convert the following linear programs into standard form.
(a)
min
x
x
1
+
x
2
s.t.
x
1
+ 2
x
2
≤ -
7
x
1
-
x
2
≥ -
2
(b)
max
x
4
x
1
+ 2
x
2
-
177
x
3
s.t.
x
1
-
2
x
2
≤ -
4
2
.
5
x
1
-
x
3
= 7
x
1
+
x
2
+
x
3
≥ -
4
x
1
≤
7
x
1
≥
0
, x
3
≤
0
Problem 5
(30 points)
Briarwood Mall in Ann Arbor is considering a major restructure, and management is
considering which stores the mall should include. The total amount of space available for the mall
is 29,000 square feet. Management has decided on the following restrictions:
•
no more than 25% of the space in the mall must be dedicated to stores that serve food and
drink;
•
at least 50% of the space must be dedicated to stores that sell clothes;
•
have at least 1,500 square feet dedicated to stores that sell books;
•
at least 20% of the space dedicated to stores that sell electronics;
•
at least 35% of the space dedicated to stores primarily for kids.
IOE Department – University of Michigan
Page 2 / 3
IOE 310 – Optimization and Computational Methods (Fall 2023)
Homework
№
3
Table 1: Store information
Store
Food/Drink
Clothes
Books
Electronics
For kids
Space
(1000 sq ft)
Bid
($)
Apple
No
No
No
Yes
No
6
$2800
Starbucks
Yes
No
No
No
No
3
$1200
Washtenaw
Dairy
Yes
No
No
No
Yes
2
$1100
The M Den
No
Yes
No
No
Yes
6
$3000
Macy’s
No
Yes
No
Yes
No
15
$3900
Panera Bread
Yes
No
No
No
Yes
4
$1300
H&M
No
Yes
No
No
Yes
10
$3250
Barnes & Noble
No
No
Yes
No
No
5
$2000
The stores listed in Table
1
have applied for space at the mall, each listed with their product offerings,
required space and bid.
The bid amount is the amount the mall will earn if the given store is chosen. For the purposes of this
problem, we may select fractions of stores, such that the bid and space of the store will decrease
proportionally. For example, if half of the Apple store is chosen, the floor space used would be
3,000 square feet and the amount earned would be $1400. More than 100 percent of a store cannot
be chosen. All mall space must be used.
(a) Formulate a linear program that will allow the management to decide what percentage of each
shop should be chosen in order to maximize the total revenue from the stores. Provide the
specific model (data, decision variables, objective function, constraints).
Note: the objective
function and all constraints must be linear.
(b) Formulate the data independent version of the model. Make sure to define all sets, variables,
and parameters clearly.
Problem 6
(30 points)
UM TV is a company that produces televisions (TVs). The company has to decide
on the number of 27 and 20 inch sets to be produced at one of its factories over the next month.
Market research indicates that at most 45 of the 27 inch sets and 35 of the 20 inch sets can be sold
per month. The maximum number of work hours available per month is 800. A 27 inch set requires
15 work hours and a 20 inch set requires 10 work hours. Each 27 inch set produces a profit of
$120
and each 20 inch set produces a profit of
$80
. Assume UM TV can produce fractional numbers of
TV sets.
(a) Formulate a linear program to maximize the profit.
(b) UM TV has the opportunity to increase the demand of TV sets through advertising.
For
every
$20
spent on commercials, 1 more 27 inch TV and 2.5 more 20 inch TVs can be sold.
Formulate a linear program to maximize the profit assuming that UM TV has a
$400
budget
for advertising.
IOE Department – University of Michigan
Page 3 / 3
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