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

rule base for defuzzification

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 Aλ 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 Aλ for a given value of λ (0 ≤ λ ≤ 1) as,

Aλ = { x |  μA(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: A0.5= { x1, x2 }

For λ = 0.4: A1= { x1, x2, x4 }

Example – 2: Lambda-cut for Fuzzy Relation

Let us define Rλ = { (x, y) | μR(x, y) ≥ λ } as a λ cut relation of the fuzzy relation R.

relation r
Lambda cut of relation r
Lambda cut of relation r

Properties of λ cut sets:

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

( AB )λ = AλBλ  

( AB)λ = AλBλ

( A‘)λ ≠ ( Aλ)’, except for the value of λ = 0.5

For any value λ1 ≥ λ2 implies Aλ1 ⊆ Aλ2

Test Your Knowledge:

defuzzification using lambda cut

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

1. P0.2, Q0.3

2. ( PQ )0.6

3. ( PP‘ )0.8

4. ( PQ)0.4 

1 Response

  1. Aniket says:

    1. {x2, x3, x4, x5}, {x1, x2, x3, x5}
    2. {x1, x2, x3, x5}
    3. {x1, x2}
    4. {x5}

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