16.888/ESD 77 Multidisciplinary System Design±
Optimization:±
Assignment 4 Part a) Solution±
Part A1
(a) Increasing mutation rate results in increased population density.
(b) The required number of bits:
n
nbits
=
f
(
n
,
Δ
x d
,
)
=
∑
⎡
⎢
ln
⎡
⎢
⎛
⎜
x
UB
−
x
LB
⎞
⎟
⎤
⎥
ln
d
⎤
⎥
=
17
i
=
1
⎢
⎣⎝
Δ
x
⎠⎦
⎥
(c) Using the same formula as above, we get: nbits = 2.
(d) The binary to decimal conversion can be written as:
d
=
(binary to decimal conversion) +
x
min
Here
x
=1 and therefore to get a decimal number 3, we should look for a binary string
min
that represent decimal number 2. The corresponding binary string is 10 (i.e.,
1.2
2
−
1
+
0.2
0
=
2 ).
(e) The correct answer is (b) – B&C.
±
A is not possible because it’s starting bit is 1, but both parents starting bit is 0. D is not±
possible since it’s 4
th
bit is 1, but that of both parents is 0.
±
F
100
(f)
p
=
1
=
=
0.10 .
F
1
5
∑
F
1000
i
i
=
1±
(g) In this population of size 5, only F
2
has a better fitness than F
1.
Hence, assuming week
dominance, F
2
would be selected in 4 out of 5 instances. The only time it would lose
when it is compared to F
2
. Therefore the probability that F
1
would be selected is 4/5 =
0.80.
(h) A 16 bit variable can have 2
16
possible values and a 4 bit variable can have 2
4
possible±
values. ±
Hence the total number of possible population members = 2
16
. 2
4
=2
20
. ±
1
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View Full DocumentPart A2
This problem has 5 continuous variables (i.e., the c/s dimensions of I beams and
supporting column) and 3 integer variables (i.e., number of beams and material
identifiers). Therefore, we can use heuristic algorithms like Simulated Annealing,
Genetic Algorithm; or Branch and Bound Algorithms, or perform explicit enumeration on
integer variables while optimizing for 5 continuous variables (this would mean 4*4*4 =
64 gradientbased optimization runs for each possible inter parameter settings).
This problem also brings out the role of constraints and bounding characteristics in MDO
problem formulations.
The first step would be to use some “physical” bounds on the crosssectional dimensions.
For example, one cannot produce an Ibeam of thickness 0 or 1 mm or it is impractical
for the height of Ibeam to go over say, 5 m in an overbridge construction.
In addition, for it to be an Ibeam, impose two geometrical constraints:
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 Fall '03
 lDavidMiller
 Optimization, integer variables, explicit enumeration

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