Maxima methods are quite simple but not as trivial as lambda cut methods. Maxima methods relies on the position of maximum membership of element at particular position in fuzzy set.

The set of methods under maxima methods we will be discussing here are:

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

Height method:

This method is based on Max-membership principle, and defined as follows.

μC(x*) ≥ μC(x), ∀x ∈ X

height method
Height method

Note: This method is applicable when height is unique.

Example:

Example of height method
Example of height method

First of Maxima (FoM) method:

Determine the smallest value of the domain with maximized membership degree

FoM = First of Maxima: x = min⁡{ x | μC(x) = h(C) }

first of maxima
First of maxima

Last of Maxima (LoM) method:

Determine the largest value of the domain with maximized membership degree

LoM = Last of Maxima: x = max{ x | μC(x) = h(C) }

last of maxima
Last of maxima

Example: First of Maxima and Last of Maxima

Find the defuzzification value for given fuzzy set

First of Maxima: x = 1

Last of Maxima: x = 6

Watch on YouTube: Maxima Methods

maxima methods

Middle of Maxima (MoM) method:

In order to find middle of maxima, we have to find the “middle” of elements with maximum membership value

middle of maxima equation

Where, M ={ xi | μC(xi) = h(C) }, Or M is the set of points having highest membership value

middle of maxima
Middle of maxima

Note: This method is applicable to symmetric functions only

Example: Middle of maxima

Find the deffizified value for given fuzzy set using middle of maxima method:

x = (a + b) / 2

x = (2 + 5) / 2

x = 3.5


Test Your Knowledge:

For the given fuzzy set Young, perform defuzzification using following methods:

  1. First of Maxima (FoM)
  2. Last of Maxima (LoM)
  3. Middle of Maxima (MoM)

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