Arrange the following encoding strategies used in Genetic Algorithms(GAs) in…

Correct sequence: Permutation Encoding → Gray coding → Binary Encoding → Real-valued Encoding

1. Permutation Encoding: Start with the problem-specific discrete representation (for example, an ordering for routing or sequencing problems). This captures the combinatorial structure of candidate solutions.

2. Gray coding: Convert integer indices to Gray code when mapping integers to bit patterns. Gray coding reduces Hamming-distance changes between consecutive integer values, smoothing search behavior when using bitwise operators.

3. Binary Encoding: Organize the Gray-coded values into binary strings to form the genotype used by genetic operators (crossover, mutation). This is the encoded representation on which genetic operators operate.

4. Real-valued Encoding (decoding): Finally, decode the binary genotype into real-valued phenotype variables for evaluation in the objective function when the problem domain requires continuous values.

Rationale: Begin with the representation that matches the problem (permutation for ordering tasks), use Gray coding to improve locality when mapping integers to bits, represent the genotype as binary strings for genetic operations, and decode to real values only at the end if the phenotype requires continuous variables.