## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm
\right., \tag{8}\]  
\[D_{j}=1,\text{where}j=1,2,\ldots,D_{P} \tag{9}\]  
where \(r3\) is a random number between \([0,1]\). \(randperm(K)\) is a function to generate a random integer within the interval \([1,K]\), and \(rand_{j}([1,D_{P}])\) is a random integer number belonging to the interval \([1,D_{P}]\). It is worth mentioning that switching among flight modes in a guided foraging or territorial foraging depends on a random number. When the random number (\(r1\)) is less than \(1/3\), the diagonal flight mode is invoked. Meanwhile, the omnidirectional flight mode is invoked when the value of \(r1\) is greater than \(2/3\). Otherwise, the axial flight mode is used. _Guided foraging_  
The hummingbird tends to visit the food source that has the highest visit level (long unvisited time) and the highest nectar-refilling rate. When the food source is specified by the hummingbird, the last will fly toward the food source for feeding purposes. Flying towards the food source can be performed via one of the aforementioned three flight skills. After that, the \(i^{th}\) candidate food source for \(G^{th}\) iteration in the guided foraging case is derived based on the flight patterns' vector and current or target food sources as follows:  
\[V_{i}(G+1)=X_{i,tar}(G)+a.D.(X_{i}(G)-X_{i,tar}(G)) \tag{10}\]  
where \(X_{i}(G)\) refers to the position of \(i^{th}\) food source for \(G^{th}\) iteration, and \(X_{i,tar}(G)\) presents the position target food source, which the \(i^{th}\) hummingbird needs to visit. \(a\) refers to a guided factor with normal distribution within the interval (0, 1). \(V_{i}(G+1)\) is the \((G+1)^{th}\) candidate food source (solution). Thus, the \(i^{th}\) current food source can be updated by using Equation (10). In the AHA model that is based on a single-objective optimization problem, a selection process is achieved between the current and updated food sources by comparing the fitness function (\(f(\cdot)\)) values of them as follows:  
\[X_{i}(G+1)= \begin{cases}X_{i}(G)&if\ f(X_{i}(G))\leq f(V_{i}(G+1)\\ V_{i}(G+1)&else\end{cases}  \tag{11}\]  
In each iteration, when the guided foraging is invoked to update the position of \(i^{th}\) food source, the visit level of other food sources in the visit table is increased by 1, while the visit level regarding the target food source is set to 0. In the multi-objective artificial hummingbird algorithm (MAHA) and to conduct the selection process, the dominance rank concept is used instead of Equation (11). In the dominance rank concept, an accumulative population (\(POP_{a}\)) is created by combining the \(N_{P}\) solutions of the \(POP\) population and the corresponding offspring solutions that were obtained in guided or territorial foraging steps. Thus, the dimension of \(POP_{a}\) will be \(D_{P}\times 2N_{P}\). After initiating \(POP_{a}\), a dominance rank index is calculated for every solution. The dominance rank index of a solution refers to the count of solutions that dominate the required solution besides one. Finally, the \(POP\) with \(D_{P}\times N_{P}\) dimensions for the next generation is updated by selecting the compromise solutions from \(POP_{a}\). The compromise solutions are those that have the lowest dominance rank indices. _Territorial foraging_  
The hummingbird search on a new candidate food source is in the territorial area. The following formula is used for deriving a candidate solution corresponding to \(i^{th}\) solution. \[V_{i}(G+1)=X_{i}(G)+b.D.X_{i}(G) \tag{12}\]where \(b\) refers to a territorial factor with a normal distribution within the interval (0, 1). Although the target food source is not used in Equation (12), the visit table is updated after performing the territorial foraging. _Migration foraging_  
When the hummingbirds frequently visit the same region of food sources, certainly the nectar in the food sources will be little. Therefore, the hummingbirds migrate to other distant food sources looking for nectar.