Defuzzification methods – Solved examples

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

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

Example: Find crisp value corresponding to the 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


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

Overlapped fuzzy output functions
Labeled fuzzy output function

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

For discussion, the 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 the X axis.

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

The equation of all the lines is 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 the overlapped region.

defuzzification methods - CoG

Center of Sums method:

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

The 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 the area of the largest triangle. From the line equations we have derived,

defuzzification methods - CoA
defuzzification methods - equation

A3 is the largest area. Let us find the center of it.

Watch on YouTube: Example of Defuzzification Methods

Weighted Area Method:

Weighted area method finds the 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 the corresponding membership value of them is 1. Crisp value using this method is computed as,

Maxima Methods:

defuzzification methods - maxima method

Maxima method finds the membership value of peak at a 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 the following two fuzzy output functions using various defuzzification methods.

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

2 Responses

  1. Hafizhah says:

    Thanks for vivid explanation, Sir. But, I’m still confused on the example of CoG, how to get line range if on the graphic, there’s no written about exact number on x absis? In this example case CoG, how can you get the comma number on x absis as range line (1.5 on bc line, 1.8 on cd line, 2.33 on de line)? Is there any formula or rule to determine it? Please, explain to me sir. Thanks a lot

    • codecrucks says:

      Thanks. For that you need to find the intersection of two lines. Probably in defuzzification video, I have explained how to find endpoint coordinates of the line

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