Defuzzification: What, Why and How?

Fuzzification converts the crisp input into fuzzy value. Defuzzification converts the fuzzy output of fuzzy inference engine into crisp value, so that it can be fed to the controller. The fuzzy results generated can not be used in an application, where decision has to be taken only on crisp values. Controller can only understand the crisp output. So it is necessary to convert the fuzzy output into crisp value.

There is no systematic procedure for choosing a good defuzzification strategy. Selection of defuzzification procedure depends on the properties of the application

Rule base:

Consider the following two rules in the fuzzy rule base.

R₁: If x is A then y is C

R₂: If x is B then y is D

A pictorial representation of the above rule base is shown in the following figures

What is the crisp output for an input say x’ ?

Defuzzification methods:

Lambda Cut Method

Maxima Methods

• Height method
• First of maxima (FoM)
• Last of maxima (LoM)
• Mean of maxima (MoM)

Weighted average method

Centroid methods

• Center of gravity method (CoG)
• Center of sum method (CoS)
• Center of area method (CoA)

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Lambda Cut Method:

This Lambda-cut set A_λ is also called alpha-cut set.

Lambda-cut method is applicable to derive crisp value of a fuzzy set or fuzzy relation.

In this method a fuzzy set A is transformed into a crisp set A_λ for a given value of λ (0 ≤ λ ≤ 1) as,

A_λ = { x |  μ_A(x) ≥ λ }

Example – 1: Lambda-cut for Fuzzy Set

A = { (x₁, 1.0), (x₂, 0.5) , (x₃, 0.3) , (x₄, 0.4) }

For λ = 1: A₁= { x₁ }

For λ = 0.5: A_0.5= { x₁, x₂ }

For λ = 0.4: A₁= { x₁, x₂, x₄ }

Example – 2: Lambda-cut for Fuzzy Relation

Lets define R_λ={ (x, y) | μ_R(x, y) ≥ λ } as a λ cut relation of the fuzzy relation R.

Properties of λ cut sets:

If A and B are two fuzzy sets, defined with the same universe of discourse, then

( A ∪ B )_λ = A_λ ∪ B_λ  

( A ∩ B)_λ = A_λ ∩ B_λ

( A‘)_λ ≠ ( A_λ)’, except for value of λ = 0.5

For any value λ₁ ≥ λ₂ implies A_λ1 ⊆ A_λ2

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Test Your Knowledge:

For data given in the table, apply lambda-cut method and find following:

1. P_0.2, Q_0.3

2. ( P ∪ Q )_0.6

3. ( P ∪ P‘ )_0.8

4. ( P ∩ Q)_0.4