Defuzzification: What, Why and How?

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

There is no systematic procedure for choosing a good defuzzification strategy. The 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 the alpha-cut set.

Lambda-cut method is applicable to derive the 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

Let us 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 the 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 the lambda-cut method and find the following:

1. P_0.2, Q_0.3

2. ( P ∪ Q )_0.6

3. ( P ∪ P‘ )_0.8

4. ( P ∩ Q)_0.4