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ESE304  Introduction to Optimization (Exam #1)
Fall Semester, 2010
M. Carchidi
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Problem #1 (10 points)
Let
A
,
B
and
C
be three
f
xed
n
×
n
matrices and let
S
be the set of all
n
×
n
matrices
X
such that
AX
+
XB
≤
C
, i.e.,
S
=
{
X

AX
+
XB
≤
C
}
.
Prove that
S
is a convex set.
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Problem #2 (30 points)
Use the
BigM
method to solve the following linear programming problem.
min
3
x
1
+5
x
2
+7
x
3
(Objective Function)
s.t.
2
x
1
+4
x
2
+5
x
3
≥
30
(Constraint #1)
4
x
1
+6
x
2
+7
x
3
≤
44
(Constraint #2)
x
1
,x
2
,x
3
≥
0
(Sign Restrictions)
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Problem #3 (30 points)
Alyssa has a limited budget of
$18
and can only buy two types of deserts.
Desert 1 sells for
$2
per piece and each piece of this desert contains
3
units
of cake and
1
unit of chocolate, while Desert 2 sells for
$1
per piece and
each piece of this desert contains
1
unit of cake and
1
unit of chocolate. Her
goals are to ingest at least
26
units of cake and
16
units of chocolate.
a.)
(6 points)
Can Alyssa meet both of her goals with her present budget?
Explain with graphs.
b.)
(8 points)
By what
minimum
amount must her budget allocation for desert
increase if she wants to meet both of her goals?
c.)
(8 points)
By what
minimum
amount must her cake goal decrease if she
wants to meet her chocolate goal and still stay within her budget?
d.)
(8 points)
By what
minimum
amount must her chocolate goal decrease if
she wants to meet her cake goal and still stay within her budget?
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Problem #4 (30 points)
Highland’s TVRadio store must determine how many TVs and radios to
keep in stock. A TV requires 12 square feet of
f
oor space, whereas a radio
requires 4 square feet, and a total of 260 square feet of
f
oor space is available.
A TV will earn Highland $90 in pro
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This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.
 Winter '11
 MichaelA.Carchidi

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