## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm
Condition 1 (C1): Acceptable advantage

To determine the satisfactory compromise solution, the next correlation is applied:

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

Here, \(A(1)\text{and}A(2)\) are the alternatives, which comprise the first and second positions, correspondingly, while the ranking list is dependent on \(Q_{i}\) values. Meanwhile, \(I\) indicates the count of alternatives. Condition 2 (C2): Acceptable stability

Alternative \(A(1)\) should be ranked as the best by \(S\)- and/or \(R\)-ranked lists in order to guarantee the compromise solution stability regarding the decision making process. The set of compromise solutions or the compromise solution may be addressed according to the previous conditions, which are listed in \[SR_{ij}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{k=1,k\
eq j}^{n}\left|\ln r_{ ik}\right|\biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n \tag{17}\]2

## 5 Experimental Results

The energy consumption profile data of customers in [16] are adopted in this work. These data are collected for five typical households; each household has seven appliances. The power rate and the operational time duration for each appliance of a given household are shown in \[SR_{ij}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{k=1,k\
eq j}^{n}\left|\ln r_{ ik}\right|\biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n \tag{17}\]3 The preferred operation time for each appliance of given users (beginning and ending time (\(B_{h}^{u}\) and \(E_{h}^{u}\), respectively) are shown in \[SR_{ij}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{k=1,k\
eq j}^{n}\left|\ln r_{ ik}\right|\biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n \tag{17}\]4 These preferred time operations for each appliance are indicated by the end-user, which is dependent on the preferred usage. For example, a microwave needs 6 min to complete its work during the preferred time of the first customer (\(H_{1}\)) from 06:00 to 21:00, while it needs about 8 min for finishing its work during 04:00-18:00 regarding the second user (\(H_{2}\)). Two types of pricing rates are considered. A time of use (ToU) of [16] is utilized to realize a fair comparison, as shown in \[SR_{ij}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{k=1,k\
eq j}^{n}\left|\ln r_{ ik}\right|\biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n \tag{17}\]5 The high price rate is in the peak time during two intervals, the first being 07:00-10:00 and the second being 18:00-20:00. Meanwhile, other time slots are considered as off-peak time with a lower price rate. The second price rate is the adaptive consumption level pricing strategy (ACLPS) that was discussed earlier in [27]. Seeking to separate the effect of high demanding users from maximizing the total pricing rate regarding other clients, ACLPS is set to handle different consumption levels for several time intervals. Accordingly, the consumption of any user will impact only their price rate. The pricing model has an acceptable high social fairness between different engaging users. ACLPS has been modified to be well matched with several time operations (peak, and off-peak) and consumption levels, as indicated in \[SR_{ij}=\ln\biggl{(}1+\biggl{(}\frac{1}{n}\sum_{k=1,k\
eq j}^{n}\left|\ln r_{ ik}\right|\biggr{)}\biggr{)},\text{where }i=1,2,\ldots,m\text{ and }j=1,2,\ldots,n \tag{17}\]6

In MAHA, the count of devices represents the number of decision variables. The population size (number of solutions) is set to be 30 for all scenarios. Meanwhile, the maximum number of iterations is represented as 100 based on extensive simulation attempts to obtain the optimal solution.