## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm
Furthermore, the ToU pricing scheme offers R 41.6497 and 5220 W for the cost and peak, respectively. The ToU exhibits a little advancement with the inconvenience compared to the ACLPS pricing scheme. According to Tables A1-A3, MAHA based on the ACLPS pricing scheme converges to the optimal solution faster and candidate the Pareto front, which includes only two different solutions. This indicates \begin{table}
\begin{tabular}{c c c} \hline \hline
**Time** & **Duration** & **Price Rate (R/kWh)** \\ \hline Peak & 07:00–10:00 and 18:00–20:00 & 1.7487 \\ \hline Off-peak & Otherwise & 0.5510 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Incentive rate and pricing [16]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Device Name**} & \multicolumn{6}{c}{**Starting and Ending Time (\(B_{h}^{u}\) and \(E_{h}^{u}\))**} \\ \cline{3-6}  & & \(H_{1}\) & \(H_{2}\) & \(H_{3}\) & \(H_{4}\) & \(H_{5}\) \\ \hline
1 & Microwave & 06:00–21:00 & 04:00–18:00 & 08:00–11:00 & 05:30–09:00 & 01:00–19:00 \\ \hline
2 & Stove & 06:30–15:00 & 06:00–15:00 & 08:00–11:00 & 05:30–09:00 & 01:00–15:00 \\ \hline
3 & Electrical Water Heater & 06:00–15:00 & 09:15–00:00 & 23:00–04:00 & 16:00–23:00 & - \\ \hline
4 & Dishwasher Machine & 10:00–15:00 & 18:00–22:00 & 15:00–17:00 & 16:00–22:00 & 01:00–19:00 \\ \hline
5 & Clothes Dryer & 10:00–15:00 & 18:00–22:00 & 15:00–17:00 & 16:00–22:00 & 01:00–19:00 \\ \hline
6 & Vacuum Cleaner & 10:00–18:00 & 09:00–12:00 & 08:00–15:00 & 08:00–14:00 & 01:00–19:00 \\ \hline
7 & DVD Player & 10:00–23:00 & 08:00–23:00 & 08:00–23:00 & 08:00–22:00 & 01:00–19:00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Favorable operational time for users’ appliances [16]. that the proposed MAHA based on the ACLPS pricing scheme is a reliable, efficient, and powerful combination for solving load scheduling optimization problems. After obtaining the optimal Pareto front solutions from the MAHA, the decision matrix of performance data is ready for deriving the appropriate weights by using the MEREC method. The next step is finding the best compromise solution by applying the VIKOR muti-criteria decision making method. As it has been mentioned earlier, the proposed weighting method derives the weights of cost, inconvenience, and peak criteria based on the decision matrix. Thus, the weights are affected by the Pareto front (decision matrix) of the MAHA. Table 5 tabulates the weights \(w_{1},w_{2},\text{and}w_{3}\) for the cost, inconvenience, and peak of power, respectively, for different pricing schemes. In the ACLPS pricing scheme, the cost has the highest weight corresponding to other criteria as highlighted within the third column of Table 5.

As stated previously, the number of alternatives equals the number of optimal solution sets (\(N_{P}=30\)), which present as a Pareto front in the objective function space. Table 6 shows the acceptable advantage and acceptable stability conditions regarding the VIKOR model for different data sets of users based on the ToU and ACLPS pricing schemes. Referring to Table 6, the condition of the acceptable advantage has not been realized for various users in the ACLPS pricing scheme. Thus, the compromise solution will be a set of solutions. Meanwhile, the first and second conditions of VIKOR are achieved in the first, fourth, fifth, and all-users cases in the ToU pricing scheme. Therefore, there is one compromise solution that realizes equilibrium among the triple criteria. For the second and third users, the acceptable advantage has not been maintained in the ToU pricing scheme. Consequently, multiple sets of compromise solutions may offer comprising among the inconvenience, cost, and peak of power criteria according to the VIKOR method. Table 7 illustrates the criteria values of the compromise solutions for each user data set regarding both pricing schemes, which were obtained by the VIKOR method.