## 3 Proposed Methodology
### Phase 5: Application of the Opposition Nelder-Mead Algorithm
1 for\(t_{2}\gets 1\)to IterMax\({}_{2}\)do
2 Compute the best vertex \(\mathbf{y}_{1}\) using Equation (30);
3 Compute the worst vertex \(\mathbf{y}_{D+1}\) using Equation (31);
4 Compute the next-worst vertex \(\mathbf{y}_{D}\) using Equation (32);
5 Compute the centroid \(\hat{\mathbf{x}}\) excluding the worst vertex using Equation (33);
6 Compute the reflection vertex \(\mathbf{x}_{t}\) using Equation (5);
7 Compute \(\hat{\mathbf{x}}\) using the selected opposition-based learning strategy (Equations (18)-(22) or (23);
8 if\(\big{(}f(\mathbf{y}_{1})\leq f(\mathbf{x}_{t})<f(\mathbf{y}_{D})\big{)}\)then
9\(\mathbf{x}_{D+1}\leftarrow\text{argmin}\{f(\mathbf{x}_{t}),f(\mathbf{\hat{x}}_ {t})\}\);
10 end if
11 if\(\big{(}f(\mathbf{x}_{t})<f(\mathbf{y}_{1})\big{)}\)then
12 Compute the expansion vertex \(\mathbf{x}_{t}\) using Equation (6);
13 Compute \(\hat{\mathbf{x}}_{t}\) using the selected opposition-based learning strategy (Equations (18)-(22) or (23);
14 if\(\big{(}f(\mathbf{x}_{t})<f(\mathbf{x}_{t})\big{)}\)then
15\(\mathbf{x}_{D+1}\leftarrow\text{argmin}\{f(\mathbf{x}_{t}),f(\mathbf{\hat{x}}_ {t})\}\);
16 end if
17
18 end if
19 if\(\big{(}f(\mathbf{x}_{t})\geq f(\mathbf{y}_{D})\big{)}\)then
20 if\(\big{(}f(\mathbf{y}_{D})\leq f(\mathbf{x}_{t})<f(\mathbf{y}_{D+1})\big{)}\)then
21 Compute the outside contraction vertex \(\mathbf{x}_{oc}\) using Equation (7);
22 Compute \(\hat{\mathbf{x}}_{oc}\) using the selected opposition-based learning strategy (Equations (18)-(22) or (23);
23 if\(\big{(}f(\mathbf{x}_{oc})<f(\mathbf{x}_{t})\big{)}\)then
24\(\mathbf{x}_{D+1}\leftarrow\text{argmin}\{f(\mathbf{x}_{oc}),f(\mathbf{\hat{x} }_{oc})\}\);
25 end if
26 end if
27 if\(\big{(}f(\mathbf{x}_{t})\geq f(\mathbf{y}_{D+1})\big{)}\)then
28 Compute the inside contraction vertex \(\mathbf{x}_{ic}\) using Equation (8);
29 Compute \(\hat{\mathbf{x}}_{ic}\) using the selected opposition-based learning strategy (Equations (18)-(22) or (23);
30 if\(\big{(}f(\mathbf{x}_{ic})<f(\mathbf{y}_{D+1})\big{)}\)then
31\(\mathbf{x}_{D+1}\leftarrow\text{argmin}\{f(\mathbf{x}_{ic}),f(\mathbf{\hat{x} }_{ic})\}\);
32 end if
33
34 end if
35 for\(\mathbf{x}_{i}\in\{\mathbf{x}_{1},\ldots,\mathbf{x}_{D+1}\}\)do
36 Update the vertex \(\mathbf{x}_{i}\) using Equation (9);
37 Compute \(\hat{\mathbf{x}}_{i}\) using the selected opposition-based learning strategy (Equations (18)-(22) or (23);
38\(\mathbf{x}_{i}\leftarrow\text{argmin}\{f(\mathbf{x}_{i}),f(\mathbf{\hat{x} }_{i})\}\);
39 end if
40
41 end for
42
43 end for
```

**Algorithm 3**The pseudocode for the opposition Nelder-Mead algorithm. ### Phase 6: Application of Genetic Operators

The presented study provides a comprehensive understanding of the GA by delineating its key components into distinct sections. Section 3.6.1 elucidates the intricate details of the selection process, where individuals from the population are carefully chosen to pass on to the next generation. Section 3.6.2 focuses on the reproduction process, outlining how the selected parents generate offspring through recombination operations. Lastly, in Section 3.6.3, the mutation process takes center stage, elucidating the mechanisms through which the genetic material of the individuals undergoes random modifications to introduce novel genetic information. It is worth pointing out that if an individual exceeds the boundaries of the search space after undergoing the genetic operators, it is consequently restored to within the valid range using Equation (24). #### 3.6.1 Selection

Genetic algorithms are a type of evolutionary algorithm that mimics the process of natural selection to solve optimization problems. In GAs, the selection mechanism determines which individuals from a given population will be passed to the next generation. The selection process is crucial in driving the search for better solutions over successive generations. Several common selection mechanisms are used in GAs [74]. In our methodology, we use elitism [75].