# Maxima methods for defuzzification: FoM, LoM and MoM

Maxima methods are quite simple but not as trivial as lambda cut methods. Maxima methods rely on the position of maximum membership of an element at a particular position in a 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 the Max-membership principle and is defined as follows.

μC(x*) ≥ μC(x), ∀x ∈ X

Note: This method is applicable when height is unique.

Example:

## 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) }

## 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) }

Example: First of Maxima and Last of Maxima

Find the defuzzification value for a given fuzzy set

First of Maxima: x = 1

Last of Maxima: x = 6

## Middle of Maxima (MoM) method:

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

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

Note: This method is applicable to symmetric functions only

Example: Middle of maxima

Find the deffuizified value for a given fuzzy set using the middle of the 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 the following methods:

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