## 3 Proposed Methodology
### Phase 5: Application of the Opposition Nelder-Mead Algorithm
Elitism involves selecting a certain number of the best individuals from the current population and directly transferring them to the next generation without any changes. This ensures that the best solutions found so far are preserved across generations, preventing the loss of fitness during the evolution process. #### 3.6.2 Crossover

The crossover technique used to deal with continuous values in genetic algorithms is known as BLX-\(\alpha\) (Blend Crossover) [76]. BLX-\(\alpha\) is a variation of the traditional crossover operator used in genetic algorithms, which is typically designed for binary or discrete variables. BLX-\(\alpha\) allows for the combination of parent solutions that have continuous values. In the BLX-\(\alpha\) crossover, a new offspring is created by blending the values of corresponding variables from two parent solutions. The process involves selecting a random value within a defined range for each variable and using the blending factor to determine the range of values for the offspring. The blending factor, denoted as alpha (\(\alpha\)), controls the amount of exploration and exploitation during the crossover process. The following steps present a high-level description of the BLX-\(\alpha\) crossover technique used in our methodology. Steps 1, 2, and 3 are iteratively performed until the size of the next population becomes \(N\). It is worth highlighting that two parent individuals will undergo a crossover process based on a specified crossover rate (\(r_{c}\)):

1. Select two parent individuals \(\mathbf{x}_{p_{1}}\) and \(\mathbf{x}_{p_{2}}\) from the mating pool using roulette-wheel selection [73]. 2. Compute the offspring \(\mathbf{x}_{o}\) using Equation (34). \[\mathbf{x}_{o}=[\text{rand}(\lambda_{1}-\alpha\pi_{i},\omega_{1}+\alpha\pi_{i} ),\dots,\text{rand}(\lambda_{D}-\alpha\pi_{D},\omega_{D}+\alpha\pi_{D})]\] (34) \[\lambda_{j}=\min(x_{p_{1}}^{(j)},x_{p_{2}}^{(j)})\] \[\omega_{j}=\max(x_{p_{1}}^{(j)},x_{p_{2}}^{(j)})\] \[\pi_{j}=|\omega_{j}-\lambda_{j}|\] \[\alpha=1-\text{rand}(0,1)\times\left(1-\frac{t_{1}}{\text{IterMax} _{1}}\right)\]
3. Add the new offspring to the next population. The value of \(\alpha\) determines the extent of exploration and exploitation during crossover. A smaller value of \(\alpha\) encourages more exploration, allowing for a wider range of values in the offspring. Conversely, a larger value of \(\alpha\) encourages more exploitation, resulting in offspring closer to the parent solutions. The BLX-\(\alpha\) crossover technique enables the combination of continuous variables in genetic algorithms and provides a way to effectively explore and exploit the search space. #### 3.6.3 Mutation

The mutation technique, commonly used to deal with continuous values in genetic algorithms, is known as Gaussian mutation or normal distribution mutation [77]. This technique introduces random perturbations to the values of the variables in a continuous search space, mimicking the behavior of a Gaussian or normal distribution. In Gaussian mutation, a random value is generated from a Gaussian distribution with a mean of zeroand a predefined standard deviation. This random value is then added to each variable of an individual in the population, causing a small random change in its value. The standard deviation determines the magnitude of the mutation, controlling the exploration and exploitation trade-off during the search process. It is worth emphasizing that an individual will undergo a mutation process based on a designated mutation rate (\(r_{m}\)). The following steps introduce a high-level description of the Gaussian mutation technique used in our methodology:

1. For each variable in an individual, generate a random value from a Gaussian distribution with a mean of zero and a predefined standard deviation. \[\sigma=\left(1-\frac{t_{1}}{\text{IterMax}_{1}}\right)\]
2. Generate a random number drawn from the uniform distribution.