HACETTEPE UNIVERSITY
DEPT. OF ELECTRICAL AND ELECTRONICS ENGINEERING
ELE 704 Optimization
Midterm Examination, 17 April, 2007
Name
: Hardworker
ID #
: # 1
Question
1
2
3
4
5
Total
Mark
20
40
10
20
20
110
Q1. (20pts)
A cardboard box for packing some stu/is to be manufactured as shown in Fig. 1. Find
the dimensions of such a box that maximizes the volume for a given amount of cardboard,
equal to 24
m
2
.
y
x
z
front
(a)
(4pts) Formulate the problem in a standard optimization problem form.
maximize
f
(
x;y;z
) =
xyz
subject to
xy
+
xz
+
yz
= 12
x;y;z
&
0
Since the inequality constraints are rather general and will be shown that they can be embedded to
the problem, you can ignore them.
(b)
r
f
(
x
;y
;z
) +
r
h
(
x
;y
;z
) =
0
2
4
yz
xz
xy
3
5
+
2
4
y
+
z
x
+
z
x
+
y
3
5
=
2
4
0
0
0
3
5
yz
+
(
y
+
z
) = 0
(1)
xz
+
(
x
+
z
) = 0
(2)
xy
+
(
x
+
y
) = 0
(3)
xy
+
xz
+
yz
= 12
1
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(4pts) Find x, y, and z.
(
xy
+
yz
+
xz
) + 2
(
x
+
y
+
z
) = 0
but using the constraint we can write
(
x
+
y
+
z
) =
6
(4)
Obviously,
6
= 0
and
& <
0
since length can not be negative.
If we let
x
= 0
, from Eqn. (2)
y
= 0
and if
y
= 0
from Eqn. (3)
z
= 0
which results in
x
=
y
=
z
= 0
which is not possible. Hence
x
,
y
and
z
are nonzero.
Multiply Eqn. (1) by
x
and the second by
y
and subtract the two to obtain
(
x
y
)
z
= 0
Multiply Eqn. (2) eqn. by
y
and the third by
z
and subtract the two to obtain
(
y
z
)
x
= 0
Since
x
,
y
and
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 Spring '11
 CenkToker
 Electronics Engineering

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