## 4 Proposed Hybrid FVIMDE Algorithm
### Mathematical Model of the Hybrid DE-FVIM Optimizer
#### 4.1.3 Mutation and Crossover  
For each individual \(i\), a mutant vector \(\mathbf{V}_{i}\) is generated using the DE/rand/1 strategy, as shown in Equation (5):  
\[\mathbf{V}_{i}=\mathbf{P}_{r_{1}}+F\times(\mathbf{P}_{r_{2}}-\mathbf{P}_{r_{3}}) \tag{5}\]  
where \(r_{1},r_{2},r_{3}\) are distinct indices randomly selected from the population, and \(F\) is the scaling factor.  
Following mutation, a trial vector \(\mathbf{U}_{i}\) is generated through binomial crossover, as defined in Equation (6). The crossover generates a trial vector by combining elements from the mutant vector and the target vector. The goal of this operation is to introduce variability in the population while preserving some of the characteristics of the current solution.  
The crossover operator used is a binomial crossover, where each element of the trial vector \(\mathbf{U}_{i}\) is taken from either the mutant vector \(\mathbf{V}_{i}\) or the target vector \(\mathbf{P}_{i}\), based on a crossover probability \(C_{r}\). The equation used to generate the trial vector is as follows:  
\[\mathbf{U}_{i}(j)=\begin{cases}\mathbf{V}_{i}(j)&\text{if }\text{rand}(0,1) \leq C_{r}\text{ or }j=j_{\text{rand}}\\ \mathbf{P}_{i}(j)&\text{otherwise}\end{cases} \tag{6}\]  
where \(C_{r}\) is the crossover rate and \(j_{\text{rand}}\) is a randomly chosen index to ensure that at least one element is inherited from \(\mathbf{V}_{i}\).  
In this equation, \(\mathbf{U}_{i}(j)\) is the \(j\)-th component of the trial vector for individual \(i\), \(\mathbf{V}_{i}(j)\) is the \(j\)-th component of the mutant vector for individual \(i\), and \(\mathbf{P}_{i}(j)\) is the \(j\)-th component of the target vector for individual \(i\). The crossover rate \(C_{r}\) determines the probability of taking a component from the mutant vector, and \(j_{\text{rand}}\) is a randomly chosen index that ensures at least one element is inherited from the mutant vector. This prevents the trial vector from being identical to the target vector.  
The crossover process operates as follows.  
For each dimension \(j\) of the solution vector, a random number is generated. If this random number is less than or equal to the crossover rate \(C_{r}\), or if the index \(j\) equals \(j_{\text{rand}}\), the corresponding component from the mutant vector \(\mathbf{V}_{i}(j)\) is copied into the trial vector \(\mathbf{U}_{i}(j)\). If neither condition is met, the component from the target vector \(\mathbf{P}_{i}(j)\) is retained in the trial vector.  
To ensure diversity in the population, the inclusion of \(j_{\text{rand}}\) guarantees that at least one component from the mutant vector is always included in the trial vector. This prevents the trial vector from being a direct copy of the target vector, maintaining variability in the population.  
The crossover operation iterates over each individual in the population (using the loop 'for i = 1:pop') and each dimension of the solution vector (using the loop 'for j = 1:dim'). Inside the nested loop, a random index 'jrand' is selected once for each individual, ensuring that at least one component from the mutant vector \(\mathbf{V}_{i}\) is copied into the trial vector \(\mathbf{U}_{i}\). For each dimension, a random number is generated. If this random number is less than or equal to the crossover rate (set to 0.9 in the code), or if the current index \(j\) equals \(j_{\text{rand}}\), the corresponding component from the mutant vector is copied into the trial vector. Otherwise, the component from the target vector is retained.