## 3 Proposed Model
### Fuzzy Inference System
The estimates (\(x_{i}\) and \(y_{ij}\) coefficients) are used to construct the waste production and determine predictions for further analysis. Thus, waste products can be estimated to enhance waste management and planning. The approach introduces two FIS inputs, namely, fuzzy-rule-based singleton values and FIS. The first input specifies the distance between the global average and an individual, who is represented in Equation (4). \[Dis_{i}=\sqrt{\sum_{d=1}^{N}\left(x_{i}^{d}-GB^{d}\right)^{2}} \tag{5}\]  
where the distance among the global best and \(i^{\text{th}}\) individual is given by \(Dis_{i}\), the dimension \(d^{\text{th}}\) of \(i^{\text{th}}\) individual is specified as \(x_{i}^{d}\), and the \(GB^{d}\) denotes the \(\text{d}^{\text{th}}\) dimension of \(GB\). An error of diversity (\(Err_{div}\)) which is the other input is given below:  
\[Err_{div}=D_{g}-D_{goal,g} \tag{6}\]  
The inputs are updated before applying the \(FIS\) in the proposed method because the magnitude order varies for these inputs while the evolutionary technique is being performed. The equations are represented in (6) and (7) below:  
\[Dis_{std,i}=\left\{ \begin{array}{ll}0,&if\ Dis_{max}-Dis_{min}=0,\\ \frac{Dis_{i}-Dis_{min}}{Dis_{max}-Dis_{min}},&others\end{array} \right. \tag{7}\]  
where the \(\text{Dis}_{\text{std,i}}\) specifies the standardized \(\text{Dis}_{i}\), the \(\text{Dis}_{\text{max}}\) is the highest value of \(\text{Dis}_{i}\), and \(\text{Dis}_{\text{min}}\) is the lowest \(\text{Dis}_{i}\) value. \[Err_{div,std}=\left\{ \begin{array}{ll}1,&\text{if\ }\frac{Err_{div}}{D_{goal,g}} \geq 1,\ D_{goal,g}\
eq 0\\ 0,&\text{if\ }D_{goal,g}=0\\ -1,&\text{if\ }\frac{Err_{div}}{D_{goal,g}}\leq-1,\ D_{goal,g}\
eq 0\\ \frac{Err_{div}}{D_{goal,g}},&others\end{array} \right. \tag{8}\]  
where the standardized error is given by \(\text{Err}_{div,std}\), the range is 0 to 1 for \(\text{Dis}_{\text{std,i}}\), and the error after adjustment is \(-1\) to 1. Algorithm 1 and Figure 5 depict trapezoidal input membership functions. Three fuzzy sets are utilized for the first input, and the second input requires five fuzzy sets. Table 3 represents the singleton fuzzy rule base. \begin{tabular}{c c c c} \hline \hline
**Err\({}_{\text{disstd}}\)** & **Dist\({}_{\text{std},j}\)** & **Close** & **Medium** & **Far** \\ \hline Positive High & 0.7 & 0.8 & 0.9 \\ Positive Low & 0.5 & 0.6 & 0.7 \\ Fit & 0 & 0 & 0 \\ Negative High & 0.3 & 0.4 & 0.5 \\ Negative Low & 0.1 & 0.2 & 0.3 \\ \hline \hline \end{tabular}   
**Algorithm 1. ** Pseudocode of GA-fuzzy inference system. Begin  
// Initialize the parameters  
Initialize the size_population, size_archive, size_elite, size_pool_mating, generation_max,  
Probability_mutation, probability_crossover, pool_offspring, pool_mating, pool_gene;  
Initialize the utilisations_set, slots_total, utilizations_no, slots_total, utilisations_power,  
price_electricity;  
For \(i=1\) to size_population  
For each appliance \(j\) in the appliances_set  
For \(t=1\) to slots_totals;  
Allocate timeslot to the utilisation \(j\) within realistic time slots  
End For  
Initialize the following: Iteration = 1, generation_current = 1;  
For generation_current = 1 to generation_max  
For each chromosome \(i\) in the pool_parent  
For \(t=1\) to slots_total  
Determine the total load at time interval't'  
End For  
Find the best chromosomes  
Copy the best chromosomes into the pool_offspring  
Update the current generation counter as generation_current = generation_current+1;  
Return chromosome fitness value in the pool_parent  
Goto GA Block // Given in Appendix A  
If generation_current equals generation_max then  
Terminate the execution cycle  
Output optimal chromosome  
Else  
Goto Parent Selection Block // Given in Appendix A  
Continue with the execution cycle  
End if  
\(\bullet\) Reinforcement  
Essential genes are selected and inserted into the offspring population. This selection will preserve and update the crucial genes in the successive chromosome population.