Table 7 Fuzzy system membership functions and linguistic values

Input variable

V_M

\(\mu \left(x\right)=\left\{\begin{array}{cc}\frac{0.35-x}{0.35}& x\in [0 . 0.35]\\ 0& x >0.35\end{array}\right.\)  

W

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0\\ \frac{x-0.2}{0.3}& x\in [0.2 . 0.5]\\ \frac{0.8-x}{0.3}& x\in [0.5 . 0.8]\\ 0& x>0.8\end{array}\right.\)  

M

\(\mu \left(x\right)=\left\{\begin{array}{cc}\frac{x-0.6}{0.4}& x< 0.6\\ 0& x\in [0.6 . 1]\end{array}\right.\)  

S

ST_M

\(\mu \left(x\right)=\left\{\begin{array}{ll}\frac{0.35-x}{0.35}& x\in [0 . 0.35]\\ 0& x >0.35\end{array}\right.\)  

L

\(\mu \left(x\right)=\left\{\begin{array}{cc}0 x<0\\ \frac{x-0.2}{0.3}& x\in [0.2 . 0.5]\\ \frac{0.8-x}{0.3}& x\in [0.5 . 0.8]\\ 0& x>0.8\end{array}\right.\)  

M

\(\mu \left(x\right)=\left\{\begin{array}{cc}\frac{0.35-x}{0.35}& x\in [0 . 0.35]\\ 0& x >0.35\end{array}\right.\)  

H

SoCH_M

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.65\\ \frac{x-0.65}{0.17}& x\in [0.65 . 0.82]\\ 1& x>0.82\end{array}\right.\)  

B

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.32\\ \frac{x-0.32}{0.18}& x\in [0.32 . 0.5]\\ 1& x\in [0.5 . 0.7]\\ \frac{0.82-x}{0.08}& x\in [0.7 . 0.82]\\ 0& x>0.82\end{array}\right.\)  

M

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.32\\ \frac{x-0.32}{0.18}& x\in [0.32 . 0.5]\\ 1& x\in [0.5 . 0.7]\\ \frac{0.82-x}{0.08}& x\in [0.7 . 0.82]\\ 0& x>0.82\end{array}\right.\)  

G

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.65\\ \frac{x-0.65}{0.17}& x\in [0.65 . 0.82]\\ 1& x>0.82\end{array}\right.\)  

E

Output variable

SoMob_M

\(\mu \left(x\right)=\left\{\begin{array}{cc}1& x<0.1\\ \frac{x-0.1}{0.05}& x\in [0.1 . 0.15]\\ 0& x>0.15\end{array}\right.\)  

VL

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.08\\ \frac{x-0.08}{0.07}& x\in [0.08 . 0.15]\\ 1& x\in [0.15 . 0.25]\\ \frac{0.32-x}{0.07}& x\in [025 . 0.32]\\ 0& x>0.32\end{array}\right.\)  

L

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.2\\ \frac{x-0.2}{0.15}& x\in [0.2 . 0.35]\\ 1& x\in [0.35 . 0.5]\\ \frac{0.65-x}{0.15}& x\in [0.5 . 0.65]\\ 0& x>0.65\end{array}\right.\)  

M

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.45\\ \frac{x-0.45}{0.15}& x\in [0.45 . 0.6]\\ 1& x\in [0.6 . 0.75]\\ \frac{0.9-x}{0.15}& x\in [0.75 . 0.9]\\ 0& x>0.9\end{array}\right.\)  

H

\(\mu \left(x\right)=\left\{\begin{array}{cc}0& x<0.75\\ \frac{x-0.75}{0.1}& x\in [0.75 . 0.85]\\ 1& x>0.85\end{array}\right.\)  

VH