Word Count : 728

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Let's go step by step and perform one iteration of the Binary-Coded Genetic Algorithm (GA) to minimize the function: \[ f(x_1, x_2) = x_1 + x_2 + 2x_1^2 - x_2^2 \] within the range 0 ≤ x₁, x₂ ≤ 5 using Binary-Coded GA. --- ### Step 1: Encoding Variables into Binary Strings We assume 3 bits for each variable. Since each variable takes values between 0 and 5, we map the binary numbers to real values. Using 3-bit binary encoding, the mapping is: \[ x = \frac{\text{Binary Code}}{(2^3 - 1)} \times 5 = \frac{\text{Binary Code}}{7} \times 5 \] | Binary (3-bit) | Decimal Value | Scaled Value \(x = \frac{\text{Decimal}}{7} \times 5\) | |---------------|--------------|----------------------------------| | 000 | 0 | 0 | | 001 | 1 | 0.714 | | 010 | 2 | 1.429 | | 011 | 3 | 2.143 | | 100 | 4 | 2.857 | | 101 | 5 | 3.571 | | 110 | 6 | 4.286 | | 111 | 7 | 5.000 | Each chromosome consists of 6 bits (3 bits for \(x_1\) and 3 ___ ____ _________ _______ ___ ____ __________ ________ ___ _____ ___.
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