# Maxima methods for defuzzification: FoM, LoM and MoM

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.

μ_{C}(x*) ≥ μ_{C}(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 | μ_{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^{∗} = max{ x | μ_{C}(x) = h(C) }

**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

## Middle of Maxima (MoM) method:

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

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

**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:

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

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