Defuzzification methods, as we know, are used to convert defuzzified output put of fuzzy inference engine to crisp value. In previous few articles, we already have discussed maxima methods, Center of Gravity (CoG) method, Center of Sums (CoS) method, Center of Largest Area (CoA) method etc.

In this article, we will see example of various defuzzification techniques.

Example: Find crisp value corresponding to following fuzzy output sets using various defuzzification methods.

Output fuzzy set 1
Output fuzzy set 1
 Output fuzzy set 2
Output fuzzy set 2
 Output fuzzy set 3
Output fuzzy set 3

Solution:

To compute the crisp value, we shall place all the function on same axis.

Overlapped fuzzy output functions
Labeled fuzzy output function

To compute the area of polygon, we need to compute the equation of line enclosing it and integrating it over lines making it.

For discussion, equation of line ab is derived here. Rest can be understand

(y – y1) / (x – x1) = (y2  – y1) / (x2 – x1)

For line ab, (x1, y1) = () and (x2, y2) = ()

(y – 0) / (x – 0) = (1 – 0) / (1.5 – 0)

y / x = 0.67

y = 0.67x   

Line ab ranges from 0 to 1.5 on X axis.

In similar way, we can find equation of all the lines. Click here to see detail.

Equation of all the lines are summarized in the table below:

defuzzification methods - Line equation

Center of Gravity method:

For computing crisp value using center of gravity method, we shall find the area of overlapped region.

defuzzification methods - CoG

Center of Sums method:

For computing crisp value using center of sums method, we shall find the area of individual triangle rather then overlapped region as in center of gravity method.

Area of triangle is computed as, A=1/2×b×h

A1 = 1/2 × 3 × 1 = 1.5

A2 = 1/2 × 2 × 1 = 1

A3 = 1/2 × 4 × 1 = 2

Putting all these values in the equation of center of sums method,

defuzzification methods - CoS

Center of Largest Area method:

For computing crisp value using center of largest area method, we shall find the centroid of area of largest triangle. From the line equations we have derived,

defuzzification methods - CoA
defuzzification methods - equation

A3 is the largest area. Lets find the center of it.

Watch on YouTube: Example of Defuzzification Methods

Weighted Area Method:

Weighted area method finds weighted sum of center of area and its membership value.

Output fuzzy set 1
Output fuzzy set 1
Output fuzzy set 2
Output fuzzy set 2
 Output fuzzy set 3
Output fuzzy set 3

Center of fuzzy sets 1, 2 and 3 is 1.5, 2 and 3 respectively. And corresponding membership value of them is 1. Crisp value using this methods is computed as,

Maxima Methods:

defuzzification methods - maxima method

Maxima method finds the membership value of peak at certain position.

First of Maxima: x* = 1.5

Last of Maxima: x = 3

Middle of Maxima: Not Applicable

Height Method: Not Applicable


Test Your Knowledge:

Example: Find the crisp value for following two fuzzy output functions using various defuzzification methods.

Please post your answer / query / feedback in comment section below !