ISE426 Homework 2
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Lehigh University *
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Industrial Engineering
Date
Jan 9, 2024
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8
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ISE 426. Optimization and Applications (Spring 2022)
Prof. Karmel S. Shehadeh
Homework #2–Due via Course Site at 9 AM on Feb 28, 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.
“
You got this! Good luck.
”
ISE 426
Homework #2–Due via Course Site at 9 AM on Feb 28, 2022
Spring 2022
Question 1 (15 points). Large-scale LP Formulation.
Consider the following transportation problem.
A paper manufacturer has
n
mills to produce
newsprint, and
m
printing plants whose demand for the newsprint must be satisfied every week.
Each mill
i
∈ {
1
, . . . , n
}
can produce
s
i
tons of newsprint every week. Each printing plant
j
∈
{
1
, . . . , m
}
must receive
d
j
tons of shipped newsprint. The shipping cost, in dollars per ton, from
mill
i
to printing plant
j
is
c
i,j
, for all
i
∈ {
1
, . . . , n
}
and
j
∈ {
1
, . . . , m
}
. Assume that the total
supply from mills and total demand from printing plants are equal.
Write a general linear programming formulation for the problem, where the aim is to satisfy the
printing plants’ demand while minimizing total transportation costs and satisfying the supply
constraints. In particular, clearly specify the
1.
Decision variables.
2.
Objective.
3.
Constraints.
For simplicity, assume that orders (in ton) from a mill to a printing plant need not be integers.
Lehigh University
Page 1 of 7
ISE 426
Homework #2–Due via Course Site at 9 AM on Feb 28, 2022
Spring 2022
Question 2 (10 points). Graphical Solution Method for LP.
Suppose that the following constraints have been provided for a linear programming model.
3
x
1
+ 2
x
2
≥
6
x
1
-
x
2
≤
3
x
1
≥
0
, x
2
≥
0
1.
(5 points)
Graphically show that the feasible region is unbounded.
2.
(2 points)
If the objective is to maximize
z
=
x
1
-
2
x
2
, does the model have an optimal
solution? If so, find it. If not, explain why not.
3.
(2 points)
If the objective is to maximize
z
=
-
x
1
+ 2
x
2
, does the model have an optimal
solution? If so, find it. If not, explain why not.
4.
(1 point)
For objective functions where this model has no optimal solution, does this mean
that there are no good solutions according to the model?
Explain.
What probably went
wrong when formulating the model?
Lehigh University
Page 2 of 7
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