## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm
Due to the preferences among the criteria with it being a mental task, which is achieved by the decision maker, the increasing in criteria leads to inaccuracy preferences among them. In the objective weighting method, the weights of criteria are computed by utilizing a specific computational step and based on the decision matrix. The hybrid weighting methods combine both subjective and objective methods. The adopting weighting method (MEREC) in this proposed research represents one of the objective weighting methods. The MEREC-weighting method exploits the alternatives' performance variation within the whole criteria for computing the weights of each criterion. The steps of the proposed MEREC-weighting method are depicted as follows [24]. _Step 1: Constructing the decision matrix_

The decision matrix (\(DM\)) represents the corner stone in the MCDM process. The \(DM\) combines the performance (\(a_{ij}\)) of \(i^{th}\) alternative in terms of \(i^{th}\) criterion. The dimension of \(DM\) is \(m\times n\), where \(m\) refers to the whole alternative count, and \(n\) is the total number of criteria as presented below. \[DM= \begin{array}{c}A_{1}\\ \vdots\\ A_{m}\end{array}\begin{bmatrix}A_{11}\\ \vdots\\ A_{m1}\end{bmatrix}\begin{bmatrix}A_{11}\\ \vdots\\ A_{m1}\end{bmatrix}\begin{bmatrix}A_{1n}\\ \vdots\\ A_{m1}\end{bmatrix} \\ \end{array} \tag{14}\]

_Step 2: Normalizing the decision matrix_

A simple linear normalization strategy is used for normalizing each element in DM using Equation (15). The normalization process depends on the type of criterion, if it was beneficial or nonbeneficial. However, sets \(BE\) and \(NB\) refer to the beneficial and nonbeneficial criteria, respectively. \[r_{ij}= \begin{cases}\frac{\min\limits_{i=1-\min}a_{ij}}{a_{ij}}&ifj\in BE\\ \frac{a_{ij}^{\prime}}{\max\limits_{i=1-\min}a_{ij}}&ifj\in NB^{\prime}\text{ where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n\\ \end{cases}  \tag{15}\]

_Step 3: Calculating the overall performance_In this step, the comprehensive performance of alternatives with equal criteria weights is computed by using a logarithmic measure as follows. \[S_{i}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{j=1}^{n}\left|\ln r_{ij}\right| \biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m \tag{16}\]

It is worth mentioning that the smaller \(r_{ij}\) values give greater \(S_{i}\) values. _Step 4: Computing the overall performance with removing a criterion_

Similar to Step 3, a logarithmic function is used for computing the overall performance with removing each criterion separately (\(SR_{ij}\)). As a result of the current step, \(SR_{ij}\) is obtained for each \(i^{th}\) alternative considering the removal of \(j^{th}\) criterion by

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

_Step 5: Computing the aggregation of absolute deviations_

The effect of canceling \(j^{th}\) criterion (\(E_{j}\)) is computed as follows:

\[E_{j}=\sum_{i=1}^{m}\left|SR_{ij}-S_{i}\right|,\text{ where }j=1,2,\ldots,n \tag{18}\]

_Step 6: Calculating the weights_

In the last step, the weight of \(j^{th}\) criterion (\(w_{j}\)) is calculated via utilizing the next equation. \[w_{j}=\frac{E_{j}}{\sum_{k=1}^{n}E_{k}},\text{where }j=1,2,\ldots,n \tag{19}\]

### VIKOR Method

Different MCDM models have been developed and used earlier in the literature regarding MCDM problems. Between these models is the VIKOR approach, which is derived from the Serbian name of \"Vlse Kriterijumska Optimizacija Kompromisno Resenje\" that means multi-criteria optimization and compromise solution. The VIKOR method was developed by Serbian citizen [PERSON] and [PERSON] in 2004 [25] as the MCDM method to solve decision problems dominated by non-commensurable and conflicting criteria. VIKOR is considered as one of the pivotal MCDM methods as it delivers a precise solution while maintaining computational simplicity [25].