## 5 Application of Our Proposed Approach
### Case Study-1: Demand Data for Transformer Tanks
#### 5.1.3 Development of Level Prediction Model  
In practice, the Holt-Winters method has been recognized to have relatively higher accuracy and reliability compared to other conventional forecasting approaches. However, the potential lack caused by the consideration of stationary initial values in its recursion process has left scope for incorporating artificial intelligence to improve the forecasting accuracy. In this context, a level prediction model is built via a fuzzy logic-based model. Salient steps in the procedure are described in the following sections. Choice of Inference Mechanism and Variables  
In this research, the developed fuzzy logic-based model is implemented in the Fuzzy Logic Toolbox of MATLAB R2009b where the Mamdani inference system is chosen for predicting the parameter level (\(L_{Tpred}\)). It is also necessary to define the input and output variables since their interrelations will materialize the objective(s) of the fuzzy logic-based model. Input variables i.e., the demand (\(D_{T}\)), the level (\(L_{T}\)), the growth rate (\(b_{T}\)), the seasonal parameter (\(SN_{T}\)), and output variables i.e., the predicted level (\(L_{Tpred}\)), in the MATLAB platform are exhibited by Figure 11 in Appendix A.  
Fuzzification and Formulation of Membership Functions  
In the fuzzification stage, the values of all the linguistic variables, as shown in Table 2, are fuzzified with the aid of membership functions. Usually the membership functions are considered as the core of fuzzy logic-based modelling as they have a direct influence on the strength and weakness of the model [64; 65]. Among the different types of membership functions, the triangular one is selected for the fuzzification purpose due to its ease of construction and versatility for system representations [66]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Linguistic Variables** & **Linguistic Values** & **Ranges** \\ \hline \multirow{8}{*}{Level (\(L_{T}\))} & E. Low & 13\(\sim\)15 \\  & V. Low & 14.75\(\sim\)17 \\  & Low & 16.75\(\sim\)19 \\  & Avg & 18.75\(\sim\)21 \\  & High & 20.75\(\sim\)23 \\  & V. High & 22.75\(\sim\)25 \\  & E. High & 24.75\(\sim\)27 \\ \hline \multirow{8}{*}{Input} & E. Low & 0.1\(\sim\)0.2 \\  & V. Low & 0.175\(\sim\)0.3 \\  & Low & 0.275\(\sim\)0.4 \\ \cline{1-1}  & Growth rate (\(b_{T}\)) & Avg & 0.375\(\sim\)0.5 \\  & High & 0.475\(\sim\)0.6 \\  & V. High & 0.575\(\sim\)0.7 \\  & E. High & 0.675\(\sim\)0.8 \\ \hline \multirow{8}{*}{Seasonal parameter (\(SN_{T}\))} & E. Low & 0.8\(\sim\)0.84 \\  & V. Low & 0.837\(\sim\)0.88 \\ \cline{1-1}  & Low & 0.875\(\sim\)0.92 \\ \cline{1-1}  & Avg & 0.9175\(\sim\)0.96 \\ \cline{1-1}  & High & 0.9575\(\sim\)1 \\ \cline{1-1}  & V. High & 0.9975\(\sim\)1.4 \\ \cline{1-1}  & E. High & 1.375\(\sim\)1.8 \\ \hline \multirow{8}{*}{Demand (\(D_{T}\))} & E. Low & 13\(\sim\)15 \\  & V. Low & 14.75\(\sim\)17 \\ \cline{1-1}  & Low & 16.75\(\sim\)19 \\ \cline{1-1}  & Avg & 18.75\(\sim\)21 \\ \cline{1-1}  & High & 20.75\(\sim\)23 \\ \cline{1-1}  & V. High & 22.75\(\sim\)25 \\ \cline{1-1}  & E. High & 24.75\(\sim\)27 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linguistic variables, their associated values and ranges. However, in this intended development of the fuzzy logic model, the fuzzification of the linguistic parameters is conducted by the following functions:  
\[L_{T}(i_{1})=\left\{ \begin{array}{l}i_{1};\,13\leq i_{1}\leq 27\\ 0;\,\text{Otherwise}\end{array} \right. \tag{13}\]  
\[b_{T}(i_{2})=\left\{ \begin{array}{l}i_{2};\,0.1\leq i_{2}\leq 0.8\\ 0;\,\text{Otherwise}\end{array} \right. \tag{14}\]  
\[SN_{T}(i_{3})=\left\{ \begin{array}{l}i_{3};\,0.8\leq i_{3}\leq 1.8\\ 0;\,\text{Otherwise}\end{array} \right. \tag{15}\]  
\[D_{T}(i_{4})=\left\{ \begin{array}{l}i_{4};\,13\leq i_{4}\leq 27\\ 0;\,\text{Otherwise}\end{array} \right. \tag{16}\]  
\[L_{Tpred}(o_{1})=\left\{ \begin{array}{l}o_{1};\,13\leq o_{1}\leq 26\\ 0;\,\text{Otherwise}\end{array} \right.