## 5 Application of Our Proposed Approach
### Case Study-1: Demand Data for Transformer Tanks
#### 5.1.3 Development of Level Prediction Model
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Variable Name**} & \multicolumn{1}{c}{**Fuzzy Numbers**} & \multicolumn{1}{c}{**Variable Name**} & \multicolumn{1}{c}{**Fuzzy Numbers**} \\ \hline \multirow{5}{*}{Growth Rate, \(b_{T}\)} & E. Low = (0.1, 0.15, 0.2, 1) & & L1 = (13, 13.5, 14; 1) \\  & V. Low = (0.175, 0.25, 0.3; 1) & & L2 = (13.75, 14.5, 15; 1) \\  & Low = (0.275, 0.35, 0.4; 1) & & L3 = (14.75, 15.5, 16; 1) \\  & Avg = (0.375, 0.45, 0.5; 1) & & L4 = (15.75, 16.5, 17; 1) \\  & High = (0.475, 0.55, 0.6;Meanwhile, as adopted by [69], the membership functions and the linguistic fuzzy sets of a particular input \(L_{T}\) are also deduced by using Table 4, Equation (7), Equation (13) and presented through Equations (18)-(24). The membership functions for the remaining input and output variables are also developed in the same manner as shown by Equations (18)-(24). \[\mu_{E.Low}(L_{T})=\left\{ \begin{array}{l}\frac{x-13}{14-13};\;13\leq x\leq 1 4\\ \frac{15-x}{15-14};\;14\leq x\leq 15\\ \\ 0;\;x>15\end{array} \right. \tag{18}\]  
\[\mu_{V.Low}(L_{T})=\left\{ \begin{array}{l}\frac{x-14.75}{16-14.75};\;14.75 \leq x\leq 16\\ \\ \frac{17-x}{17-16};\;16\leq x\leq 17\\ \\ 0;\;x>17\end{array} \right. \tag{19}\]  
\[\mu_{Low}(L_{T})=\left\{ \begin{array}{l}\frac{x-16.75}{18-16.75};\;16.75 \leq x\leq 18\\ \\ \frac{19-x}{19-18};\;18\leq x\leq 19\\ \\ 0;\;x>19\end{array} \right. \tag{20}\]  
\[\mu_{Avg}(L_{T})=\left\{ \begin{array}{l}\frac{x-18.75}{20-18.75};\;18.75 \leq x\leq 20\\ \\ \frac{21-x}{21-20};\;20\leq x\leq 21\\ \\ 0;\;x>21\end{array} \right. \tag{21}\]  
\[\mu_{High}(L_{T})=\left\{ \begin{array}{l}\frac{x-20.75}{22-20.75};\;20.75 \leq x\leq 22\\ \\ \frac{23-x}{23-22};\;22\leq x\leq 23\\ \\ 0;\;x>23\end{array} \right. \tag{22}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Linguistic Variables**} & \multirow{2}{*}{**Type**} & \multicolumn{3}{c}{**Coefficient**} \\ \cline{3-5}  & & & **C1** & **C2** & **C3** \\ \hline \multirow{9}{*}{\(L_{T}\)_pred_} & L1 & Triangular & 13 & 13.5 & 14 \\  & L2 & Triangular & 13.75 & 14.5 & 15 \\  & L3 & Triangular & 14.75 & 15.5 & 16 \\  & L4 & Triangular & 15.75 & 16.5 & 17 \\  & L5 & Triangular & 16.75 & 17.5 & 18 \\  & L6 & Triangular & 17.75 & 18.5 & 19 \\  & L7 & Triangular & 18.75 & 19.5 & 20 \\  & L8 & Triangular & 19.75 & 20.5 & 21 \\  & L9 & Triangular & 20.75 & 21.5 & 22 \\  & L10 & Triangular & 21.75 & 22.5 & 23 \\  & L11 & Triangular & 22.75 & 23.5 & 24 \\  & L12 & Triangular & 23.75 & 24.5 & 25 \\  & L13 & Triangular & 24.75 & 25.5 & 26 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._\[\mu_{V.High}(L_{T})=\left\{ \begin{array}{l}\frac{x-22.75}{24-2275};\ 22.75\leq x\leq 21.87\\ \\ \frac{25-x}{25-24};\ 25\leq x\leq 24\\ \\ 0;\ x\geq 25\end{array} \right. \tag{23}\]  
\[\mu_{E.High}(L_{T})=\left\{ \begin{array}{l}\frac{x-24.75}{26-24.75};\ 24.75\leq x\leq 26\\ \\ \frac{27-x}{27-26};\ 26\leq x\leq 27\\ \\ 0;\ x\geq 27\end{array} \right. \tag{24}\]  
Development of Fuzzy Inference Rules and Process  
For Case-1, 30 decision rules, as presented in Figure 5, are formulated and inserted in the fuzzy logic tool box. Each rule has two distinct parts: premise and conclusion. The input variables are interpreted with the premise part as \"if\" whereas the output variables are taken with conclusion as \"then\". To clarify how a rule works, an example is given for one of the decision rules along with the interpretation:  
Rule 1: If L = E. low and b = V. low and SN = V. low and D = E. low, then L (pred) = L2. This means that if the level is E. low, growth rate is V. low, seasonality is V. low, demand is E. low; then the predicted level is L2. After formulating the decision rules and membership functions, the inference process is carried out. In the first step of the inference, the conclusion of each decision rule is derived for a particular input by a comprehensive comparison among the ranges of input variables.