The Design and Implementation of Adsorptive Removal of Cu(II) from Leachate Using ANFIS

Table 3

The most common MFs and their associated expressions.

Triangular MF, trimf	𝑓(𝑥;𝑎,𝑏,𝑐)=0,𝑥≤𝑎𝑥−𝑎𝑏−𝑎,𝑎≤𝑥≤𝑏𝑐−𝑥𝑐−𝑏,𝑏≤𝑥≤𝑐0,𝑐≤𝑥
Trapezoidal MF, trapmf	𝑓(𝑥;𝑎,𝑏,𝑐,𝑑)=0,𝑥≤𝑎𝑥−𝑎𝑏−𝑎𝑎≤𝑥≤𝑏1,𝑏≤𝑥≤𝑐𝑑−𝑥𝑑−𝑐𝑐≤𝑥≤𝑑0,𝑑≤𝑥
Generalized bell-shaped MF	1𝑓(𝑥;𝑎,𝑏,𝑐)=1+((𝑥−𝑐)/𝑎)^2𝑏
Gaussian curve MF	𝑓(𝑥;𝜎,𝑐)=𝑒^−(𝑥−𝑐)2/2𝜎2
pi-shape MF	2𝑓(𝑥;𝑎,𝑏,𝑐,𝑑)=0,𝑥≤𝑎𝑥−𝑎𝑏−𝑎^2,𝑎≤𝑥≤𝑎+𝑏21−2𝑥−𝑏𝑏−𝑎^2,𝑎+𝑏2≤𝑥≤𝑏1−2𝑥−𝑐𝑑−𝑐^2,𝑐≤𝑥≤𝑐+𝑑22𝑥−𝑑𝑑−𝑐^2𝑐+𝑑2≤𝑥≤𝑑0,𝑥≥𝑑
MF composed of the difference between two sigmoidal MFs, dsigmf	1𝑓(𝑥;𝑎,𝑐)=1+𝑒^−𝑎(𝑥−𝑐)
Product of the two sigmoid MF, psigmf	1𝑓(𝑥;𝑎,𝑐)=1+𝑒^−𝑎(𝑥−𝑐)