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.

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

Watch on YouTube: Maxima Methods

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 ={ x_i | μ_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

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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)

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