ISE 426 Homework 4
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Jan 9, 2024
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ISE 426: Optimization and Applications (Spring 2022)
Instructor: Dr. Karmel S.Shehadeh
Homework #4-Due via Course Site at 9 AM on April 11, 2022
Instructions
•
Show all of the work
leading to the solution of each problem. Points are allocated to all of
the steps of the solution process, not just the final answer.
•
I strongly encourage you to
type
all your assignment solutions using your favorite typesetting
system (e.g., MS Word, L
A
T
E
X, etc.). For your convenience, a L
A
T
E
X template for typing your
solution is available in the assignment folder on the
Course Site
.
•
If you write an
AMPL
code to solve any problem, you are required to provide a carefully and
detailed commented version of the code in an appendix of the assignment
.
Note that your
code is NOT a substitute
for a detailed written explanation of the approach you take to
solve the problem and your results.
•
Include a title page with assignment number, your name and contact information, and the
names of all students that you discussed your assignment with (if any). Make sure that you
scan/compile your HW work into a single and legible PDF file.
“
Bonne chance!
”
ISE 426
Homework #4-Due via Course Site at 9 AM on April 11, 2022
Spring 2022
Question 1 (25 points). MILP Formulation for an Assignment Problem.
There are two
machines for processing jobs, each with an availability of 6 hours.
Consider the
following seven possible jobs with their respective profits and processing times.
Table 1: Profit and processing time of each job
Job
1
2
3
4
5
6
7
Profit
p
j
10
5
20
12
14
7
8
Processing time
t
j
(in hour)
2.5
1.5
4.5
2.8
3
1.7
1.8
1.1 Formulate a mixed-integer linear program that determines the optimal job assignment on
the two machines that maximize the total profit. In particular, clearly specify the
(a)
(5 points) Decision variables.
(b)
(3 points) Objective.
(c)
(7 points) Constraints.
1.2 Model and solve the problem using
AMPL
. In particular, produce a print-out
of the following
(a)
(5 points) AMPL code (model and data).
(b)
(2.5 points) Optimal solution.
(State clearly, in words, the optimal schedule)
(c)
(2.5 points) Optimal objective value.
Lehigh University
Page 1 of 5
ISE 426
Homework #4-Due via Course Site at 9 AM on April 11, 2022
Spring 2022
Question 2 (25 points). Binary Variables+Logical Constraints+
AMPL
Donki is a rich investor, and he is designing a new portfolio combination. As a smart investor,
Donki selects several potential stocks from the service sector and manufacturing sector.
The
expected return and risk factor for each asset is provided in the following table.
Service
Manufacturing
1
2
3
4
5
6
Estimated return (in %)
5.2
3.1
2.9
4.2
3.7
4.8
Risk factor (per $1M investment)
1.60
1.22
1.18
1.40
1.33
1.42
The total investment amount available is $100 million and the risk factor of the portfolio should
be less than 120.
If an asset is included in the portfolio, the minimum investment amount to
that asset is $20M. Since handling the trading of multiple assets is costly, at most four assets
should be included in the portfolio. However, to achieve diversification, at least one of the assets
in each sector should be included. Moreover, neither assets 5 nor 6 can be included unless asset 4
is included.
2.1 Formulate this problem as a mixed-integer linear program using appropriate binary variables.
In particular, clearly specify the
(a)
(5 points) Decision variables.
(b)
(3 points) Objective.
(c)
(7 points) Constraints.
2.2 Model and solve the problem using
AMPL
to find out which investments should be pursued.
In particular, produce a print-out
of the following
(a)
(5 points) AMPL code (model and data).
(b)
(2.5 points) Optimal solution.
(State clearly, in words, the optimal investment)
(c)
(2.5 points) Optimal objective value.
Lehigh University
Page 2 of 5
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