Defuzzification

Defuzzification is the process of conversion of fuzzy values into crisp(Non-fuzzy) values. This is required because a number of engineering application cannot use the fuzzy values for processing.It can be considered rounding off .

Methods for Defuzzification

1.      Lambda-cuts

2.      Max-membership principle

3.      Centroid method

4.      Weighted average method

5.      Mean-max membership

6.      Center of sums

7.      Center of largest area

8.      First of maxima , last of maxima

Lambda-cuts

A_λ is the lambda cut of fuzzy set A it is defined as

Weak lambda cut

A_λ ={x| μ _A (x) ≥ λ}; λ ∈[0,1]

Strong lambda cut

A_λ ={x| μ _A (x) > λ}; λ ∈[0,1]

Max-membership principle

Also known as height method limited to peak output functions.

μ _C (x*) ≥ μ _C (x) for all x ∈ X

 

Centroid method

Most commonly used method. Also known as center of are or center of gravity method.

x* =

 

Weighted average method

This method is valid for symmetrical output membership function only. The function is weighted by its maximum membership value.

x*=

Mean-max membership

Also known as middle of maxima. Related to max-membership method, except that the locations of the maximum membership can be nonunique.

Center of sums

This method employs algebraic sum of individual fuzzy subsets instead of their union. The calculations here are very fast, but the common areas are added twice.

Center of largest area

This method can be used when output consists of at least two convex fuzzy subsets which are not overlapping. The defuzzified value is the center of gravity of the largest area.

C_i  is the largest area.

First of maxima , last of maxima

The first or last value with maximum value is selected

The steps involved are