# 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.

Îŧ_{C}(x*) âĨ Îŧ_{C}(x), âx â 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^{â} = minâĄ{ x | Îŧ_{C}(x) = h(C) }

## Last of Maxima (LoM) method:

Determine the largest value of the domain with maximized membership degree

LoM = Last of Maxima: x^{â} = max{ x | Îŧ_{C}(x) = h(C) }

**Example: First of Maxima and Last of Maxima**

Find the defuzzification value for a given fuzzy set

**First of Maxima:** x^{â} = 1

**Last of Maxima:** x^{â} = 6

## Watch on YouTube: Maxima Methods

## Middle of Maxima (MoM) method:

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

Where, M ={ x_{i} |Â Îŧ_{C}(x_{i}) = 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^{â} = (a + b) / 2

x^{â} = (2 + 5) / 2

x^{â} = 3.5

## Test Your Knowledge:

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

- First of Maxima (FoM)
- Last of Maxima (LoM)
- Middle of Maxima (MoM)

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