Each search parameter (ps) in Fuzzy algorithms is specified by:

Pg = the genetic parameter,

1 = the number of bits in the genetic parameter,

R =a specified parameter range, and

O=a specified offset 

Explanation:

Each search parameter (\(p_s\)) in fuzzy algorithms is specified by:

• \(p_g\) = the genetic parameter
• \(l\) = the number of bits in the genetic parameter
• \(R\) = a specified parameter range
• \(O\) = a specified offset

The correct option is:

Option 3: \(p_s = \frac{p_g}{2^l-1}R + O\)

This option accurately describes the relationship between the genetic parameter and the search parameter in fuzzy algorithms. The genetic parameter (\(p_g\)) is normalized by the bit length (\(l\)) and scaled by the parameter range (\(R\)), with an offset (\(O\)) added to the result.

Detailed Solution:

To understand why option 3 is correct, let's break down each component of the formula:

1. **Genetic Parameter (\(p_g\))**: This is the value obtained from the genetic algorithm, representing the encoded solution.

2. **Number of Bits (\(l\))**: This specifies how many bits are used to represent the genetic parameter. The maximum value that can be represented with \(l\) bits is \(2^l - 1\).

3. **Parameter Range (\(R\))**: This is the range within which the search parameter should fall. It defines the span of possible values for the parameter.

4. **Offset (\(O\))**: This is a specified constant value that adjusts the final result, ensuring the parameter falls within the desired range.

The formula \(p_s = \frac{p_g}{2^l-1}R + O\) works as follows:

- The genetic parameter \(p_g\) is divided by \(2^l - 1\) to normalize it to a value between 0 and 1.

- This normalized value is then scaled by the parameter range \(R\), adjusting it to the appropriate magnitude.

- Finally, the offset \(O\) is added to shift the result to the desired starting point.

By using this formula, we can ensure that the search parameter \(p_s\) accurately reflects the encoded genetic parameter within the specified range and offset.

Let's analyze the other options to understand why they are incorrect:

Option 1: \(p_s = \frac{p_g}{2^O-1}R + 1\)

This option incorrectly uses the offset \(O\) in the denominator instead of the number of bits \(l\). The offset should be added at the end, not used in normalization. Additionally, adding 1 at the end does not accurately reflect the scaling and offset adjustment required.

Option 2: \(p_s = \frac{p_g}{2^l-1}O + R\)

This option swaps the roles of the offset \(O\) and the parameter range \(R\). The offset should be added at the end, while the range should be used to scale the normalized genetic parameter.

Option 4: \(p_s = \frac{p_g}{2^R-1}1 + O\)

This option incorrectly uses the parameter range \(R\) in the denominator instead of the number of bits \(l\). The number of bits should be used to normalize the genetic parameter. Additionally, multiplying by 1 has no effect and does not contribute to the correct scaling.

Conclusion:

Understanding the correct relationship between the genetic parameter and the search parameter is crucial for accurately implementing fuzzy algorithms. The correct formula, \(p_s = \frac{p_g}{2^l-1}R + O\), ensures that the genetic parameter is properly normalized, scaled, and adjusted within the specified range and offset. This accurate representation is essential for the effective functioning of fuzzy algorithms in solving optimization problems.