## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm
The migration foraging is invoked when the iteration counter (\(G\)) exceeds the predetermined migration coefficient value. In migration foraging and the worst solutions, which have the worst fitness function values, it will be replaced with a randomly produced solution within the search space. The formula in Equation (5) is used in the migration foraging for replacing the worst solutions. It is worth mentioning that the visit level of the worst solutions in the visit table will be updated at the end of the migration foraging process. Referring to AHA, besides the maximum number of iterations (\(G_{max}\)) and the population size (\(N_{P}\)) parameters, there is another parameter, which is the migration factor (\(M\)) that is used to execute the migration foraging. The migration factor is chosen to be twice the population size (i.e., \(M=2N_{P}\)) [22]. In the guided foraging, the exploration is achieved at the early iterations, while in the later stage of iterations, the exploitation is achieved. The territorial foraging is devoted to exploitation, while the exploration is achieved in the migration foraging. The switching factor between the guided foraging and territorial foraging is a random number (\(r2\)) between (0, 1). The guided foraging strategy is executed when the \(r2\) value is less than 0.5; otherwise, the territorial foraging strategy is invoked. The pseudo-code of MAHA is presented in Algorithm 1. The time computational complexity of the proposed MAHA depends on the initialization, the evaluation of fitness functions (\(FE\)), the update of hummingbirds' position, the population size (\(N_{P}\)), the number of decision variables (\(D_{P}\)), and the maximum number of iterations (\(G_{max}\)). Thus, the overall computational complexity of MAHA can be presented by  
\[ \begin{array}{lll}O(MAHA)&=&O(problem\;definition)+O(initialization)+O(G( function\;evaluation))+\\ &&O(G(guided\; foraging))+O(G(teritorial\; foraging))+\\ &&O(G(migration\; foraging))\\ &=&O(1+N_{P}D_{P}+G_{max}N_{P}FE+\frac{1}{2}N_{P}D_{P}G_{max}+\frac{1}{2}N_{ P}D_{P}G_{max}+\frac{G_{max}}{2N_{P}}D_{P}+2G_{max}N_{P})\end{array}  \tag{13}\]  
```
Input:\(N_{P}\), \(D_{P}\), \(G_{max}\) Output: Global minimum solution Initialization: For \(i^{th}\) hummingbird from 1 to \(N_{P}\) Do \(X_{ji}^{0}=X_{ji}^{Low}+rand\left(X_{ji}^{Up}-X_{ji}^{Low}\right)\) End For For \(i^{th}\) food source from 1 to \(N_{P}\), Do if \(i\
eq j\) Then Vistittable\((i,j)=0\) Else Visit_table\((i,j)=null\) End If End For While \(G\leq G_{max}\) Do For \(i^{th}\) hummingbird from 1 to \(N_{P}\), Do \(POP_{a}(:,1:N_{P})=POP(:,:)\) If \(\mathbf{r1}\leq\frac{1}{3}\) Then perform Equation (7) Else If \(\mathbf{r1}\geq\frac{2}{3}\) Then perform Equation (7)
```  
**Algorithm 1**Pseudo-code of the proposed MAHA algorithm ```
**Algorithm 1**Cont. ```
\begin{tabular}{l} _Eise_ \\ _Perform Equation (9)_ \\ _End If_ \\ _If_ _r2 \textless{} 0.5_ \\ _Guided foraging:_ \\ _Then perform Equation (10)_ \\ \(\mathbf{POP_{a}}(.,N_{P}+i)=\mathbf{V_{i}}(\mathbf{G}+\mathbf{1})\) \\ _Specify_ \(\dot{y}_{i,t}(G)\) \\ _If_ \(f(X_{i



### MEREC-Weighting Method

In general, for evaluating and ranking a set of solutions that are dominated by a set of criteria and based on the MCDM method, four steps should be followed. These steps are (i) defining the criteria and alternatives of the problem, (ii) constructing the decision matrix that includes the performance of each alternative (solution) in terms of each criterion, (iii) computing the weights of each criterion, and (iv) evaluating and ranking the solutions according to the aggregate performance of them for all criteria [23]. Therefore, the weights of criteria play an essential role in evaluating the solutions within the MCDM problem. However, there are three types of weighting methods, namely, subjective, objective, and hybrid, between the first and second types. The subjective weighting methods are represented as inefficient when the number of criteria is increased.