Use a Genetic Algorithm to find the value of x that maximizes XTC f(x) = sin( in the interval 0 ≤ x ≤ 255 256. where x is restricted to being an integer. Assume that a binary representation for the chromosome is used. Which one is the correct chromosome in random? 01001 0100101 11001101 1010101010
Use a Genetic Algorithm to find the value of x that maximizes XTC f(x) = sin( in the interval 0 ≤ x ≤ 255 256. where x is restricted to being an integer. Assume that a binary representation for the chromosome is used. Which one is the correct chromosome in random? 01001 0100101 11001101 1010101010
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![### Using Genetic Algorithms to Optimize Functions
#### Problem Statement
Use a Genetic Algorithm to find the value of \( x \) that maximizes the function:
\[ f(x) = \sin\left(\frac{x \pi}{256}\right) \]
given that \( x \) is an integer in the interval \( 0 \leq x \leq 255 \).
#### Details
Assume that a binary representation for the chromosome is used. Determine which one of the following binary representations is a valid chromosome at random:
1. \( 01001 \)
2. \( 0100101 \)
3. \( 11001101 \)
4. \( 1010101010 \)
#### Analysis
When using genetic algorithms, chromosomes are typically represented in binary to encode the possible values of \( x \). The binary length of the chromosome must be sufficient to represent all values within the given interval \( 0 \leq x \leq 255 \). Since 255 is represented as \( 11111111 \) in binary, exactly 8 bits are required.
Considering this requirement:
- \( 01001 \) (5 bits, insufficient)
- \( 0100101 \) (7 bits, insufficient)
- \( 11001101 \) (8 bits, valid length)
- \( 1010101010 \) (10 bits, more than sufficient but not necessary)
Thus, the correct chromosome in random among the given options is:
**Option 3: \( 11001101 \)**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f309ce-3c3d-4315-8a6b-c8c8322162e4%2F800529a5-b265-44b4-811a-c0743e71d689%2Flb3ljrk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Genetic Algorithms to Optimize Functions
#### Problem Statement
Use a Genetic Algorithm to find the value of \( x \) that maximizes the function:
\[ f(x) = \sin\left(\frac{x \pi}{256}\right) \]
given that \( x \) is an integer in the interval \( 0 \leq x \leq 255 \).
#### Details
Assume that a binary representation for the chromosome is used. Determine which one of the following binary representations is a valid chromosome at random:
1. \( 01001 \)
2. \( 0100101 \)
3. \( 11001101 \)
4. \( 1010101010 \)
#### Analysis
When using genetic algorithms, chromosomes are typically represented in binary to encode the possible values of \( x \). The binary length of the chromosome must be sufficient to represent all values within the given interval \( 0 \leq x \leq 255 \). Since 255 is represented as \( 11111111 \) in binary, exactly 8 bits are required.
Considering this requirement:
- \( 01001 \) (5 bits, insufficient)
- \( 0100101 \) (7 bits, insufficient)
- \( 11001101 \) (8 bits, valid length)
- \( 1010101010 \) (10 bits, more than sufficient but not necessary)
Thus, the correct chromosome in random among the given options is:
**Option 3: \( 11001101 \)**
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