## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm
It contains a record of the ranked compromised solutions that have been sorted according to the closeness degree of the ideal solution. For the TOPSIS technique, VIKOR's idea is based on determining negative and positive ideal solutions. Next, the finite solutions can be sorted according to two metrics: the closeness measure to one of the ideal solutions and afar from negative ideal one based on non-commensurable conflicting criteria. The construction of the decision matrix, which is notated by \(DM\), is based on the assessment of ith alternative according to jth criterion function (\(f_{ij}\)). Dimensions of \(DM\) are \(I\times J\), where \(I\) refers to the number of alternatives and \(J\) denotes the total number of attributes (criteria). Depending on the closeness measure to the ideal solution, the compromise ranking could be constructed. In MCDM, the compromise solution is considered as a feasible solution where here it is considered as the closest to the ideal solution. The VIKOR model is based on five stages that are illustrated as follows:

_Stage 1: Finding best and worst evaluation values of all criteria_

The best \(f_{j}^{*}\) and the worst \(f_{j}\) evaluation values regarding every criterion will be calculated, where \(j=1,2,\ldots,J\). \(f_{j}^{*}\) and \(f_{j}\) values depend on the criterion type. Accordingly, if the criterion is useful (maximization), then \(f_{j}^{*}\) and \(f_{j}\) can be addressed as

\[f_{j}^{*}=\max_{i=1\to 1}\bigl{(}f_{ij}\bigr{)},j=1,2,\ldots,J \tag{20}\]

\[f_{j}=\min_{i=1\to 1}\bigl{(}f_{ij}\bigr{)},j=1,2,\ldots,J \tag{21}\]Meanwhile, if the criterion is nonbeneficial (minimization), the worst and best values are described as follows:

\[f_{j}^{*}=\min_{i=1\to I}\bigl{(}f_{ij}\bigr{)},j=1,2,\ldots,J \tag{22}\]

\[\hat{f_{j}}=\max_{i=1\to I}\bigl{(}f_{ij}\bigr{)},j=1,2,\ldots,J \tag{23}\]

_Stage 2: Calculating regret and utility indices_

The utility index \((S_{i})\) is calculated in this stage depending on normalized and weighted Manhattan distances of each alternative in order to find the column vector with \(I\times 1\) dimensions as indicated below:

\[S_{i}=\sum_{j=1}^{I}w_{j}\frac{f_{j}^{*}-f_{ij}}{f_{j}^{*}-f_{j}},i=1,2,\ldots,I \tag{24}\]

where \(w_{j}\) is the weight of jth criterion. On the other hand, by relying on the weighted and normalized Chebyshev distance, the regret index is calculated and is represented as follows:

\[R_{i}=\max_{j=1\to I}\Biggl{(}w_{j}\frac{f_{j}^{*}-f_{ij}}{f_{j}^{*}-f_{j}^{ *}}\Biggr{)},i=1,2,\ldots,I \tag{25}\]

_Stage 3: calculating comprehensive indicator_

In step 3 of the VIKOR model, regarding the group utility function, there will be a comprehensive indicator that is called \(Q_{i}\) and is calculated according to the following formula:

\[SR_{ij}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{k=1,k\
eq j}^{n}\left|\ln r_{ ik}\right|\biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n \tag{17}\]0

where \(S^{*}=\min_{i=1\to I}(S_{i}),R^{*}=\min_{i=1\to I}(R_{i}),S=\max_{i=1\to I}(S_{i}),R=\max_{i=1\to I}(R_{i})\). \(\beta\) is the strategy weight of the majority of attributes or the maximum group utility, where \(\beta\in[0,1]\). The compromise ranking is selected with \(\beta=0.5\) (consensus) or with \(\beta<0.5\) (Veto) or with \(\beta>0.5\) (majority group utility). Nevertheless, the value for \(\beta=0.5\) is used by the majority of researchers in the literature [26]. _Stage 4: Ranking alternatives_

Three alternative-ranking lists are determined in this stage depending on sorting \(S_{i},R_{i},\text{and}Q_{i}\) from minimum to maximum values. _Stage 5: Rank choice acceptance_

According to \(Q,S,\text{and}R\) values, all alternatives are sorted in ascending form. After that, the next conditions have to be evaluated to select the compromise solution (or solutions).