## 3 Proposed Methodology
### Phase 3: Augmentation of the Population  
The Nelder-Mead algorithm requires the construction of a simplex with exactly \(D+1\) vertices, where \(D\) represents the dimensionality of the problem. However, in some cases, the number of individuals available in the initial population is smaller than \(D+1\), i.e., \(N<(D+1)\). To overcome this limitation and enable the application of the Nelder-Mead algorithm, we augment the population size by generating additional individuals. By introducing these extra individuals, we ensure that the simplex can be properly formed, allowing the algorithm to proceed as intended. This augmentation step ensures that the optimization process can fully leverage the capabilities of the Nelder-Mead algorithm, even when the initial population size is insufficient to construct the required simplex. In optimization, when the population size is insufficient or does not meet the requirements of certain algorithms, techniques can be employed to augment or expand the population. These techniques aim to increase the diversity, coverage, or exploration capabilities of the population to enhance the optimization process. Some common techniques used to augment a population in optimization include:  
1. **Scaling:** Scaling a vector \(\mathbf{x}\) in an \(n\)-dimensional search space involves adjusting the magnitude of its components uniformly. Mathematically, the scaled vector \(\hat{\mathbf{x}}\) can be obtained by multiplying each component of the original vector \(\mathbf{x}\) by a scaling factor \(s\) (Equation (25)) [72]. \[\hat{\mathbf{x}}=s\times\mathbf{x}\] (2  
its respective axes. In our methodology, the vector \(\mathbf{t}\) is generated randomly from the normal distribution \(\mathcal{N}(0,1-\frac{t_{1}}{\text{IterMax}_{1}})\). 4. **Reflection:** Reflecting a vector \(\mathbf{x}\) in an \(n\)-dimensional search space involves creating its mirror image across a specified line or plane while preserving its magnitude. Mathematically, the reflected vector \(\hat{\mathbf{x}}\) can be obtained by subtracting the original vector \(\mathbf{x}\) from the double of the projection of \(\mathbf{x}\) onto the reflection line or plane (Equation (28)) [72]. \[\hat{\mathbf{x}}=2\times(\mathbf{x}\cdot\mathbf{v})\times\mathbf{v}-\mathbf{x}\] (28) where \(\mathbf{x}\) represents the original vector in the \(n\)-dimensional space, \(\hat{\mathbf{x}}\) represents the reflected vector, \(\mathbf{v}\) represents the normal vector of the reflection line or plane, and - denotes the dot product between two vectors. The reflection operation effectively flips the sign of the component along the reflection axis, resulting in the mirror image of the original vector. In our methodology, the vector \(\mathbf{v}\) is generated randomly from the normal distribution \(\mathcal{N}(0,1-\frac{t_{1}}{\text{IterMax}_{1}})\). 5. **Similarity transformation:** The similarity transformation of a vector \(\mathbf{x}\) in an \(n\)-dimensional search space involves scaling and rotating the vector while preserving its shape. Mathematically, the transformed vector \(\hat{\mathbf{x}}\) can be obtained by first scaling the original vector \(\mathbf{x}\) by a scaling factor \(s\) and then rotating it using a rotation matrix \(R\) (Equation (29)) [72]. \[\hat{\mathbf{x}}=s\times R\times\mathbf{x}\] (29) where \(\mathbf{x}\) represents the original vector in the \(n\)-dimensional space, \(\hat{\mathbf{x}}\) represents the transformed vector, \(s\) is the scaling factor, and \(R\) is the rotation matrix. The similarity transformation allows for modifications in size and orientation while preserving the relative positions of the vector's components. Algorithm 2 serves as a concise representation, capturing the key stages of the augmentation phase. This algorithm effectively distills the fundamental procedures and essential processes involved in augmenting a given population.